College Physics, Volume 2, Eighth Edition

VOLUME 2 > Chapters 15–30

COLLEGE PHYSICS EIGHTH EDITION

R AYMOND A. SERWAY Emeritus, James Madison University

CHRIS VUILLE Embry-Riddle Aeronautical University

JERRY S. FAUGHN Emeritus, Eastern Kentucky University

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College Physics, Volume 2, Eighth Edition Serway/Vuille Physics Editor: Chris Hall Development Editor: Ed Dodd Assistant Editor: Brandi Kirksey Editorial Assistant: Stefanie Beeck

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CONTENTS OVERVIEW VOLUME 2 PART 4

Electricity and Magnetism

CHAPTER 15 CHAPTER 16 CHAPTER 17 CHAPTER 18 CHAPTER 19 CHAPTER 20 CHAPTER 21

Electric Forces and Electric Fields 497 Electrical Energy and Capacitance 531 Current and Resistance 570 Direct-Current Circuits 594 Magnetism 626 Induced Voltages and Inductance 663 Alternating-Current Circuits and Electromagnetic Waves

PART 5

Light and Optics

CHAPTER 22 CHAPTER 23 CHAPTER 24 CHAPTER 25

Reflection and Refraction of Light Mirrors and Lenses 759 Wave Optics 790 Optical Instruments 823

PART 6

Modern Physics

CHAPTER 26 CHAPTER 27 CHAPTER 28 CHAPTER 29 CHAPTER 30

Relativity 847 Quantum Physics 870 Atomic Physics 891 Nuclear Physics 913 Nuclear Energy and Elementary Particles

APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E

Mathematics Review A.1 An Abbreviated Table of Isotopes A.14 Some Useful Tables A.19 SI Units A.21 MCAT Skill Builder Study Guide A.22

Answers to Quick Quizzes, Example Questions, Odd-Numbered Multiple-Choice Questions, Conceptual Questions, and Problems A.52 Index

iv

I.1

732

937

696

CONTENTS CHAPTER 17

Volume 2 About the Authors Preface

Current and Resistance 570

viii

ix

To the Student

xxvii

MCAT Test Preparation Guide

xxx

Part 4: Electricity and Magnetism

17.1

Electric Current 570

17.2

A Microscopic View: Current and Drift Speed

17.3

Current and Voltage Measurements in Circuits

17.4

Resistance, Resistivity, and Ohm’s Law 575

17.5

Temperature Variation of Resistance

17.6

Electrical Energy and Power

CHAPTER 15

17.7

Superconductors

Electric Forces and Electric Fields 497

17.8

Electrical Activity in the Heart

15.1

CHAPTER 18

Properties of Electric Charges

497

500

15.4 The Electric Field 15.5 Electric Field Lines

585

594

18.3 Resistors in Parallel

15.7 The Millikan Oil-Drop Experiment

515

598

18.4 Kirchhoff’s Rules and Complex DC Circuits 18.6 Household Circuits 18.7 Electrical Safety

523

611

612

18.8 Conduction of Electrical Signals by Neurons Summary

CHAPTER 16

Electrical Energy and Capacitance 531

615

Magnetism 626 531

19.1

Magnets 626

16.2 Electric Potential and Potential Energy Due to Point Charges 538

19.2

Earth’s Magnetic Field 628

19.3

Magnetic Fields

16.3 Potentials and Charged Conductors

19.4

Magnetic Force on a Current-Carrying Conductor 633

542

16.4 Equipotential Surfaces 543 16.5 Applications 544 16.6 Capacitance 546 547

16.8 Combinations of Capacitors

549

19.6

Motion of a Charged Particle in a Magnetic Field 639

19.7

Magnetic Field of a Long, Straight Wire and Ampère’s Law 642

19.8

Magnetic Force Between Two Parallel Conductors 645

19.9

Magnetic Fields of Current Loops and Solenoids 646

16.9 Energy Stored in a Charged Capacitor 555 16.10 Capacitors with Dielectrics 557 562

630

19.5 Torque on a Current Loop and Electric Motors

16.7 The Parallel-Plate Capacitor

613

CHAPTER 19

Potential Difference and Electric Potential

Summary

603

18.5 RC Circuits 607

516

15.9 Electric Flux and Gauss’s Law 517

16.1

584

18.2 Resistors in Series 595

510

15.6 Conductors in Electrostatic Equilibrium 513

Summary

579

580

588

18.1 Sources of emf

505

15.8 The Van de Graaff Generator

574

Direct-Current Circuits 594

15.2 Insulators and Conductors 499 15.3 Coulomb’s Law

Summary

572

636

19.10 Magnetic Domains 650 Summary

652 v

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Contents

CHAPTER 20

22.5 The Rainbow 745

Induced Voltages and Inductance 663

22.6 Huygens’s Principle 746 22.7 Total Internal Reflection 748 Summary

751

20.1 Induced emf and Magnetic Flux 663 20.2 Faraday’s Law of Induction 666

CHAPTER 23

20.3 Motional emf 670

Mirrors and Lenses 759

20.4 Lenz’s Law Revisited (The Minus Sign in Faraday’s Law) 674 20.5 Generators

23.1 Flat Mirrors

759

23.2 Images Formed by Concave Mirrors

676

20.6 Self-Inductance 680

23.3 Convex Mirrors and Sign Conventions

20.7 RL Circuits 683

23.4 Images Formed by Refraction

20.8 Energy Stored in a Magnetic Field 686

23.5 Atmospheric Refraction

Summary

23.6 Thin Lenses

687

Summary

Alternating-Current Circuits and Electromagnetic Waves 696 21.1

Resistors in an AC Circuit

764

769

772

773

23.7 Lens and Mirror Aberrations

CHAPTER 21

762

781

782

CHAPTER 24

Wave Optics

790

24.1 Conditions for Interference

696

790

24.2 Young’s Double-Slit Experiment

791

21.2 Capacitors in an AC Circuit 699

24.3 Change of Phase Due to Reflection

21.3 Inductors in an AC Circuit 701

24.4 Interference in Thin Films 796

21.4 The RLC Series Circuit 702

24.5 Using Interference to Read CDs and DVDs 800

21.5 Power in an AC Circuit 707

24.6 Diffraction

21.6 Resonance in a Series RLC Circuit 708

24.7 Single-Slit Diffraction

21.7

24.8 The Diffraction Grating

The Transformer

710

21.8 Maxwell’s Predictions 712 21.9

713

21.10 Production of Electromagnetic Waves by an Antenna 714

803 805

Summary

815

CHAPTER 25 715

21.12 The Spectrum of Electromagnetic Waves

Optical Instruments 823

720

21.13 The Doppler Effect for Electromagnetic Waves Summary

802

24.9 Polarization of Light Waves 808

Hertz’s Confirmation of Maxwell’s Predictions

21.11 Properties of Electromagnetic Waves

795

722

25.1 The Camera 25.2 The Eye

723

823

824

25.3 The Simple Magnifier

Part 5: Light and Optics

829

25.4 The Compound Microscope 25.5 The Telescope

830

832

25.6 Resolution of Single-Slit and Circular Apertures 835

CHAPTER 22

25.7 The Michelson Interferometer 840

Reflection and Refraction of Light 732 22.1 The Nature of Light

732

22.2 Reflection and Refraction 733 22.3 The Law of Refraction

737

22.4 Dispersion and Prisms

742

Summary

841

Contents

Part 6: Modern Physics

29.5 Natural Radioactivity

CHAPTER 26

29.7 Medical Applications of Radiation

Relativity

29.6 Nuclear Reactions Summary

847

26.1 Galilean Relativity

847

26.2 The Speed of Light

26.3 Einstein’s Principle of Relativity

931

Nuclear Energy and Elementary Particles 937

850

26.4 Consequences of Special Relativity

851

858

30.1 Nuclear Fission 937

26.6 Relativistic Energy and the Equivalence of Mass and Energy 859 26.7 General Relativity

30.2 Nuclear Fusion 941 30.3 Elementary Particles and the Fundamental Forces 943

863

Summary 865

30.4 Positrons and Other Antiparticles 30.5 Classification of Particles

CHAPTER 27

30.6 Conservation Laws 947

Quantum Physics 870

30.7 The Eightfold Way

27.1

929

CHAPTER 30

848

26.5 Relativistic Momentum

926

927

Blackbody Radiation and Planck’s Hypothesis

870

944

945

949

30.8 Quarks and Color 950 30.9 Electroweak Theory and the Standard Model 952

27.2 The Photoelectric Effect and the Particle Theory of Light 872

30.10 The Cosmic Connection 954 30.11 Problems and Perspectives 955

27.3 X-Rays 875 27.4 Diffraction of X-Rays by Crystals 27.5 The Compton Effect

Summary

876

956

Appendix A: Mathematics Review

879

27.6 The Dual Nature of Light and Matter

A.1

Appendix B: An Abbreviated Table of Isotopes A.14

880

27.7 The Wave Function 883

Appendix C: Some Useful Tables

27.8 The Uncertainty Principle 884

Appendix D: SI Units A.21

Summary

Appendix E: MCAT Skill Builder Study Guide

886

Atomic Physics 891 28.2 Atomic Spectra 28.3 The Bohr Model

Index I.1

891

892 894

28.4 Quantum Mechanics and the Hydrogen Atom

899

28.5 The Exclusion Principle and the Periodic Table 902 28.6 Characteristic X-Rays 905 28.7 Atomic Transitions and Lasers Summary

906

908

CHAPTER 29

Nuclear Physics

913

29.1

913

Some Properties of Nuclei

29.2 Binding Energy 916 29.3 Radioactivity

918

29.4 The Decay Processes

921

A.22

Answers to Quick Quizzes, Example Questions, Odd-Numbered Multiple-Choice Questions, Conceptual Questions, and Problems A.52

CHAPTER 28 28.1 Early Models of the Atom

A.19

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ABOUT THE AUTHORS Raymond A. Serway received his doctorate at Illinois Institute of Technology and is Professor Emeritus at James Madison University. In 1990 he received the Madison Scholar Award at James Madison University, where he taught for 17 years. Dr. Serway began his teaching career at Clarkson University, where he conducted research and taught from 1967 to 1980. He was the recipient of the Distinguished Teaching Award at Clarkson University in 1977 and of the Alumni Achievement Award from Utica College in 1985. As Guest Scientist at the IBM Research Laboratory in Zurich, Switzerland, he worked with K. Alex Müller, 1987 Nobel Prize recipient. Dr. Serway also was a visiting scientist at Argonne National Laboratory, where he collaborated with his mentor and friend, Sam Marshall. In addition to earlier editions of this textbook, Dr. Serway is the coauthor of Principles of Physics, fourth edition; Physics for Scientists and Engineers, seventh edition; Essentials of College Physics; and Modern Physics, third edition. He also is the coauthor of the high school textbook Physics, published by Holt, Rinehart and Winston. In addition, Dr. Serway has published more than 40 research papers in the field of condensed matter physics and has given more than 70 presentations at professional meetings. Dr. Serway and his wife, Elizabeth, enjoy traveling, golf, gardening, singing in a church choir, and spending time with their four children and eight grandchildren. Chris Vuille is an associate professor of physics at Embry-Riddle Aeronautical University (ERAU), Daytona Beach, Florida, the world’s premier institution for aviation higher education. He received his doctorate in physics from the University of Florida in 1989 and moved to Daytona after a year at ERAU’s Prescott, Arizona, campus. Although he has taught courses at all levels, including postgraduate, his primary interest has been the delivery of introductory physics. He has received several awards for teaching excellence, including the Senior Class Appreciation Award (three times). He conducts research in general relativity and quantum theory, and was a participant in the JOVE program, a special three-year NASA grant program during which he studied neutron stars. His work has appeared in a number of scientific journals, and he has been a featured science writer in Analog Science Fiction/Science Fact magazine. In addition to this textbook, he is coauthor of Essentials of College Physics. Dr. Vuille enjoys tennis, swimming, and playing classical piano, and he is a former chess champion of St. Petersburg and Atlanta. In his spare time he writes fiction and goes to the beach. His wife, Dianne Kowing, is an optometrist for a local Veterans’ Administration clinic. His daughter, Kira VuilleKowing, is a meteorology/communications double major at ERAU and a graduate of her father’s first-year physics course. He has two sons, Christopher, a cellist and fisherman, and James, avid reader of Disney comics. Jerry S. Faughn earned his doctorate at the University of Mississippi. He is Professor Emeritus and former chair of the Department of Physics and Astronomy at Eastern Kentucky University. Dr. Faughn has also written a microprocessor interfacing text for upper-division physics students. He is coauthor of a nonmathematical physics text and a physical science text for general education students, and (with Dr. Serway) the high-school textbook Physics, published by Holt, Reinhart and Winston. He has taught courses ranging from the lower division to the graduate level, but his primary interest is in students just beginning to learn physics. Dr. Faughn has a wide variety of hobbies, among which are reading, travel, genealogy, and old-time radio. His wife, Mary Ann, is an avid gardener, and he contributes to her efforts by staying out of the way. His daughter, Laura, is in family practice, and his son, David, is an attorney. viii

PREFACE College Physics is written for a one-year course in introductory physics usually taken by students majoring in biology, the health professions, and other disciplines including environmental, earth, and social sciences, and technical fields such as architecture. The mathematical techniques used in this book include algebra, geometry, and trigonometry, but not calculus. This textbook, which covers the standard topics in classical physics and 20thcentury physics, is divided into six parts. Part 1 (Chapters 1–9) deals with Newtonian mechanics and the physics of fluids; Part 2 (Chapters 10–12) is concerned with heat and thermodynamics; Part 3 (Chapters 13 and 14) covers wave motion and sound; Part 4 (Chapters 15–21) develops the concepts of electricity and magnetism; Part 5 (Chapters 22–25) treats the properties of light and the field of geometric and wave optics; and Part 6 (Chapters 26–30) provides an introduction to special relativity, quantum physics, atomic physics, and nuclear physics.

OBJECTIVES The main objectives of this introductory textbook are twofold: to provide the student with a clear and logical presentation of the basic concepts and principles of physics, and to strengthen an understanding of the concepts and principles through a broad range of interesting applications to the real world. To meet those objectives, we have emphasized sound physical arguments and problem-solving methodology. At the same time we have attempted to motivate the student through practical examples that demonstrate the role of physics in other disciplines.

CHANGES TO THE EIGHTH EDITION A number of changes and improvements have been made to this edition. Based on comments from users of the seventh edition and reviewers’ suggestions, a major effort was made to increase the emphasis on conceptual understanding, to add new end-of-chapter questions and problems that are informed by research, and to improve the clarity of the presentation. The new pedagogical features added to this edition are based on current trends in science education. The following represent the major changes in the eighth edition.

Questions and Problems We have substantially revised the end-of-chapter questions and problems for this edition. Three new types of questions and problems have been added: ■

Multiple-Choice Questions have been introduced with several purposes in mind. Some require calculations designed to facilitate students’ familiarity with the equations, the variables used, the concepts the variables represent, and the relationships between the concepts. The rest are conceptual and are designed to encourage conceptual thinking. Finally, many students are required to take multiple-choice tests, so some practice with that form of question is desirable. Here is an example of a multiple-choice question: 12. A truck loaded with sand accelerates along a highway. The driving force on the truck remains constant. What happens to the acceleration of the truck as its trailer leaks sand at a constant rate through a hole in its bottom? (a) It decreases at a steady rate. (b) It increases at a steady rate. (c) It increases and then decreases. (d) It decreases and then increases. (e) It remains constant.

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The instructor may select multiple-choice questions to assign as homework or use them in the classroom, possibly with “peer instruction” methods or in conjunction with “clicker” systems. More than 180 multiple-choice questions are included in Volume 2. Answers to odd-numbered multiple-choice questions are included in the Answers section at the end of the book and answers to all questions are found in the Instructor’s Solutions Manual and on the instructor’s PowerLecture CD-ROM. ■

Enhanced Content problems require symbolic or conceptual responses from the student. A symbolic Enhanced Content problem requires the student to obtain an answer in terms of symbols. In general, some guidance is built into the problem statement. The goal is to better train the student to deal with mathematics at a level appropriate to this course. Most students at this level are uncomfortable with symbolic equations, which is unfortunate because symbolic equations are the most efficient vehicle for presenting relationships between physics concepts. Once students understand the physical concepts, their ability to solve problems is greatly enhanced. As soon as the numbers are substituted into an equation, however, all the concepts and their relationships to one another are lost, melded together in the student’s calculator. The symbolic Enhanced Content problems train students to postpone substitution of values, facilitating their ability to think conceptually using the equations. An example of a symbolic Enhanced Content problem is provided here: 14. ecp An object of mass m is dropped from the roof of a building of height h. While the object is falling, a wind blowing parallel to the face of the building exerts a constant horizontal force F on the object. (a) How long does it take the object to strike the ground? Express the time t in terms of g and h. (b) Find an expression in terms of m and F for the acceleration ax of the object in the horizontal direction (taken as the positive x-direction). (c) How far is the object displaced horizontally before hitting the ground? Answer in terms of m, g, F, and h. (d) Find the magnitude of the object’s acceleration while it is falling, using the variables F, m, and g.

A conceptual Enhanced Content problem encourages the student to think verbally and conceptually about a given physics problem rather than rely solely on computational skills. Research in physics education suggests that standard physics problems requiring calculations may not be entirely adequate in training students to think conceptually. Students learn to substitute numbers for symbols in the equations without fully understanding what they are doing or what the symbols mean. The conceptual Enhanced Content problem combats this tendency by asking for answers that require something other than a number or a calculation. An example of a conceptual Enhanced Concept problem is provided here: 4. ecp A shopper in a supermarket pushes a cart with a force of 35 N directed at an angle of 25 below the horizontal. The force is just sufficient to overcome various frictional forces, so the cart moves at constant speed. (a) Find the work done by the shopper as she moves down a 50.0-m length aisle. (b) What is the net work done on the cart? Why? (c) The shopper goes down the next aisle, pushing horizontally and maintaining the same speed as before. If the work done by frictional forces doesn’t change, would the shopper’s applied force be larger, smaller, or the same? What about the work done on the cart by the shopper?

Preface ■

Guided Problems help students break problems into steps. A physics problem typically asks for one physical quantity in a given context. Often, however, several concepts must be used and a number of calculations are required to get that final answer. Many students are not accustomed to this level of complexity and often don’t know where to start. A Guided Problem breaks a standard problem into smaller steps, enabling students to grasp all the concepts and strategies required to arrive at a correct solution. Unlike standard physics problems, guidance is often built into the problem statement. For example, the problem might say “Find the speed using conservation of energy” rather than only asking for the speed. In any given chapter there are usually two or three problem types that are particularly suited to this problem form. The problem must have a certain level of complexity, with a similar problem-solving strategy involved each time it appears. Guided Problems are reminiscent of how a student might interact with a professor in an office visit. These problems help train students to break down complex problems into a series of simpler problems, an essential problem-solving skill. An example of a Guided Problem is provided here: 32.

GP Two blocks of masses m and m (m m 2) are placed 1 2 1 on a frictionless table in contact with each other. A horizontal force of magnitude F is applied to the block of mass m1 in Figure P4.32. (a) If P is the magnitude of the contact force between the blocks, draw the free-body diagrams for each block. (b) What is the net force on the system consisting of both blocks? (c) What is the net force acting on m1? (d) What is the net force acting on m 2? (e) Write the x-component of Newton’s second law for each block. (f) Solve the resulting system of two equations and two unknowns, expressing the acceleration a and contact force P in terms of the masses and force. (g) How would the answers change if the force had been applied to m 2 instead? (Hint: use symmetry; don’t calculate!) Is the contact force larger, smaller, or the same in this case? Why?

F

m1

m2

FIGURE P4.32

In addition to these three new question and problem types, we carefully reviewed all other questions and problems for this revision to improve their variety, interest, and pedagogical value while maintaining their clarity and quality. Approximately 30% of the questions and problems in this edition are new.

Examples In the last edition all in-text worked examples were reconstituted in a two-column format to better aid student learning and help reinforce physical concepts. For this eighth edition we have reviewed all the worked examples, made improvements, and added a new Question at the end of each worked example. The Questions usually require a conceptual response or determination, or estimates requiring knowledge of the relationships between concepts. The answers for the new Questions can be found at the back of the book. A sample of an in-text worked example follows on the next page, with an explanation of each of the example’s main parts:

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The Problem statement presents the problem itself.

The Goal describes the physical concepts being explored within the worked example.

The Solution section uses a twocolumn format that gives the explanation for each step of the solution in the left-hand column, while giving each accompanying mathematical step in the right-hand column. This layout facilitates matching the idea with its execution and helps students learn how to organize their work. Another benefit: students can easily use this format as a training tool, covering up the solution on the right and solving the problem using the comments on the left as a guide.

Remarks follow each Solution and highlight some of the underlying concepts and methodology used in arriving at a correct solution. In addition, the remarks are often used to put the problem into a larger, real-world context.

EXAMPLE 13.7 Goal

The Strategy section helps students analyze the problem and create a framework for working out the solution.

Measuring the Value of g

Determine g from pendulum motion.

Problem Using a small pendulum of length 0.171 m, a geophysicist counts 72.0 complete swings in a time of 60.0 s. What is the value of g in this location? Strategy First calculate the period of the pendulum by dividing the total time by the number of complete swings. Solve Equation 13.15 for g and substitute values. Solution Calculate the period by dividing the total elapsed time by the number of complete oscillations:

T5

Solve Equation 13.15 for g and substitute values:

T 5 2p g5

Remark

time 60.0 s 5 5 0.833 s # of oscillations 72.0 L Åg

S

T 2 5 4p2

L g

1 39.5 2 1 0.171 m 2 4p2L 5 5 9.73 m/s 2 1 0.833 s 2 2 T2

Measuring such a vibration is a good way of determining the local value of the acceleration of gravity.

QUESTION 13.7 True or False: A simple pendulum of length 0.50 m has a larger frequency of vibration than a simple pendulum of length 1.0 m. EXERCISE 13.7 What would be the period of the 0.171-m pendulum on the Moon, where the acceleration of gravity is 1.62 m/s2 ? Answer

2.04 s

Question New to this edition, each worked example will feature a conceptual question that promotes student understanding of the underlying concepts contained in the example.

Exercise/Answer Every worked example is followed immediately by an exercise with an answer. These exercises allow students to reinforce their understanding by working a similar or related problem, with the answers giving them instant feedback. At the option of the instructor, the exercises can also be assigned as homework. Students who work through these exercises on a regular basis will find the end-of-chapter problems less intimidating.

Many Worked Examples are also available to be assigned as Active Examples in the Enhanced WebAssign homework management system (visit www.serwayphysics.com for more details).

Preface

Online Homework It is now easier to assign online homework with Serway and Vuille using the widely acclaimed program Enhanced WebAssign. All end-of-chapter problems, active figures, quick quizzes, and most questions and worked examples in this book are available in WebAssign. Most problems include hints and feedback to provide instantaneous reinforcement or direction for that problem. We have also added math remediation tools to help students get up to speed in algebra and trigonometry, animated Active Figure simulations to strengthen students’ visualization skills, and video to help students better understand the concepts. Visit www.serwayphysics. com to view an interactive demo of this innovative online homework solution.

Content Changes The text has been carefully edited to improve clarity of presentation and precision of language. We hope that the result is a book both accurate and enjoyable to read. Although the overall content and organization of the textbook are similar to the seventh edition, a few changes were implemented. ■ ■

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Chapter 15, Electric Forces and Electric Fields, has two worked examples that were upgraded with new parts. Chapter 16, Electrical Energy and Capacitance, has a new worked example that illustrates particle dynamics and electric potential. Three other worked examples were upgraded with new parts, and two new quick quizzes were added. Chapter 17, Current and Resistance, was reorganized slightly, putting the subsection on power ahead of superconductivity. It also has two new quick quizzes. Chapter 18, Direct-Current Circuits, has both a new and a reorganized quick quiz. Chapter 19, Magnetism, has a new section on types of magnetic materials as well as a new quick quiz. Chapter 20, Induced Voltages and Inductance, has new material on RL circuits, along with a new example and quick quiz. Chapter 21, Alternating-Current Circuits and Electromagnetic Waves, has a new series of four quick quizzes that were added to drill the fundamentals of AC circuits. The problem-solving strategy for RLC circuits was completely revised, and a new physics application on using alternating electric fields in cancer treatment was added. Chapter 24, Wave Optics, has an improved example and two new quick quizzes. Chapter 26, Relativity, no longer covers relativistic addition of velocities. Three new quick quizzes were added to the chapter. Chapter 27, Quantum Physics, was rewritten and streamlined. Two superfluous worked examples were eliminated (old Examples 27.1 and 27.2) because both are discussed adequately in the text. One of two worked examples on the Heisenberg uncertainty principle was deleted and a new quick quiz was added. The scanning tunneling microscope application was deleted. Chapter 28, Atomic Physics, was rewritten and streamlined, and the subsection on spin was transferred to the section on quantum mechanics. The section on electron clouds was shortened and made into a subsection. The sections on atomic transitions and lasers were combined into a single, shorter section. Chapter 29, Nuclear Physics, was reduced in size by deleting less essential worked examples. Old worked examples 29.1 (Sizing a Neutron Star), 29.4 (Radon Gas), 29.6 (The Beta Decay of Carbon-14), and 29.9 (Synthetic Elements) were eliminated because they were similar to other examples already in the text. The medical application of radiation was shortened, and a new quick quiz was developed. Chapter 30, Nuclear Energy and Elementary Particles, was rewritten and streamlined. The section on nuclear reactors was combined with the one on nuclear fission. The historical section and old Section 30.7 on the meson were eliminated, and the beginning of the section on particle physics was eliminated. The

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section on strange particles and strangeness was combined with the section on conservation laws. The sections on quarks and colored quarks were also combined into Section 30.8, Quarks and Color.

TEXTBOOK FEATURES Most instructors would agree that the textbook assigned in a course should be the student’s primary guide for understanding and learning the subject matter. Further, the textbook should be easily accessible and written in a style that facilitates instruction and learning. With that in mind, we have included many pedagogical features that are intended to enhance the textbook’s usefulness to both students and instructors. The following features are included. QUICK QUIZZES All the Quick Quizzes (see example below) are cast in an objective format, including multiple-choice, true–false, matching, and ranking questions. Quick Quizzes provide students with opportunities to test their understanding of the physical concepts presented. The questions require students to make decisions on the basis of sound reasoning, and some have been written to help students overcome common misconceptions. Answers to all Quick Quiz questions are found at the end of the textbook, and answers with detailed explanations are provided in the Instructor’s Solutions Manual. Many instructors choose to use Quick Quiz questions in a “peer instruction” teaching style.

QUICK QUIZ 4.3 A small sports car collides head-on with a massive truck. The greater impact force (in magnitude) acts on (a) the car, (b) the truck, (c) neither, the force is the same on both. Which vehicle undergoes the greater magnitude acceleration? (d) the car, (e) the truck, (f) the accelerations are the same. A general problem-solving strategy to be followed by the student is outlined at the end of Chapter 1. This strategy provides students with a structured process for solving problems. In most chapters more specific strategies and suggestions (see example below) are included for solving the types of problems featured in both the worked examples and the end-of-chapter problems. This feature helps students identify the essential steps in solving problems and increases their skills as problem solvers. PROBLEM-SOLVING STRATEGIES

PROBLEM -SOLVING STRATEGY NEWTON’S SECOND LAW

Problems involving Newton’s second law can be very complex. The following protocol breaks the solution process down into smaller, intermediate goals: 1. Read the problem carefully at least once. 2. Draw a picture of the system, identify the object of primary interest, and indicate forces with arrows. 3. Label each force in the picture in a way that will bring to mind what physical quantity the label stands for (e.g., T for tension). 4. Draw a free-body diagram of the object of interest, based on the labeled picture. If additional objects are involved, draw separate free-body diagrams for them. Choose convenient coordinates for each object. 5. Apply Newton’s second law. The x- and y-components of Newton’s second law should be taken from the vector equation and written individually. This usually results in two equations and two unknowns. 6. Solve for the desired unknown quantity, and substitute the numbers.

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For biology and pre-med students, icons point the way to various practical and interesting applications of physical principles to biology and medicine. Whenever possible, more problems that are relevant to these disciplines are included.

BIOMEDICAL APPLICATIONS

MCAT SKILL BUILDER STUDY GUIDE The eighth edition of College Physics contains a special skill-building appendix (Appendix E) to help premed students prepare for the MCAT exam. The appendix contains examples written by the text authors that help students build conceptual and quantitative skills. These skillbuilding examples are followed by MCAT-style questions written by test prep experts to make sure students are ready to ace the exam.

Located after the “To the Student” section in the front of the book, this guide outlines 12 concept-based study courses for the physics part of the MCAT exam. Students can use the guide to prepare for the MCAT exam, class tests, or homework assignments.

MCAT TEST PREPARATION GUIDE

APPLYING PHYSICS The Applying Physics features provide students with an additional means of reviewing concepts presented in that section. Some Applying Physics examples demonstrate the connection between the concepts presented in that chapter and other scientific disciplines. These examples also serve as models for students when assigned the task of responding to the Conceptual Questions presented at the end of each chapter. For examples of Applying Physics boxes, see Applying Physics 9.5 (Home Plumbing) on page 299 and Applying Physics 13.1 (Bungee Jumping) on page 435.

Placed in the margins of the text, Tips address common student misconceptions and situations in which students often follow unproductive paths (see example at the right). Thirty-nine Tips are provided in Volume 2 to help students avoid common mistakes and misunderstandings. TIPS

MARGINAL NOTES Comments and notes appearing in the margin (see example at the right) can be used to locate important statements, equations, and concepts in the text.

Although physics is relevant to so much in our modern lives, it may not be obvious to students in an introductory course. Application margin notes (see example at the right) make the relevance of physics to everyday life more obvious by pointing out specific applications in the text. Some of these applications pertain to the life sciences and are marked with a icon.

APPLICATIONS

MULTIPLE-CHOICE QUESTIONS New to this edition are end-of-chapter multiple- choice questions. The instructor may select items to assign as homework or use them in the classroom, possibly with “peer instruction” methods or with “clicker” systems. More than 180 multiple-choice questions are included in Volume 2. Answers to odd-numbered multiple-choice questions are included in the answer section at the end of the book, and answers to all questions are found in the Instructor’s Solutions Manual. CONCEPTUAL QUESTIONS At the end of each chapter there are 10–15 conceptual questions. The Applying Physics examples presented in the text serve as models for students when conceptual questions are assigned and show how the concepts can be applied to understanding the physical world. The conceptual questions provide the student with a means of self-testing the concepts presented in the chapter. Some conceptual questions are appropriate for initiating classroom discussions. Answers to odd-numbered conceptual questions are included in the Answers Section at the end of the book, and answers to all questions are found in the Instructor’s Solutions Manual.

TIP 4.3 Newton’s Second Law Is a Vector Equation In applying Newton’s second law, add all of the forces on the object as vectors and then fi nd the resultant vector acceleration by dividing by m. Don’t find the individual magnitudes of the forces and add them like scalars.

O Newton’s third law APPLICATION Diet Versus Exercise in Weight-loss Programs

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PROBLEMS An extensive set of problems is included at the end of each chapter (in all, 980 problems are provided in Volume 2 of the eighth edition). Answers to odd-numbered problems are given at the end of the book. For the convenience of both the student and instructor, about two-thirds of the problems are keyed to specific sections of the chapter. The remaining problems, labeled “Additional Problems,” are not keyed to specific sections. The three levels of problems are graded according to their difficulty. Straightforward problems are numbered in black, intermediate-level problems are numbered in blue, and the most challengicon identifies problems dealing ing problems are numbered in magenta. The with applications to the life sciences and medicine. Solutions to approximately 12 problems in each chapter are in the Student Solutions Manual/Study Guide. STYLE To facilitate rapid comprehension, we have attempted to write the book in a style that is clear, logical, relaxed, and engaging. The somewhat informal and relaxed writing style is designed to connect better with students and enhance their reading enjoyment. New terms are carefully defined, and we have tried to avoid the use of jargon. INTRODUCTIONS All chapters begin with a brief preview that includes a discussion of the chapter’s objectives and content.

The international system of units (SI) is used throughout the text. The U.S. customary system of units is used only to a limited extent in the chapters on mechanics and thermodynamics. UNITS

PEDAGOGICAL USE OF COLOR Readers should consult the pedagogical color chart (inside the front cover) for a listing of the color-coded symbols used in the text diagrams. This system is followed consistently throughout the text.

Most important statements and definitions are set in boldface type or are highlighted with a background screen for added emphasis and ease of review. Similarly, important equations are highlighted with a tan background screen to facilitate location.

IMPORTANT STATEMENTS AND EQUATIONS

The readability and effectiveness of the text material, worked examples, and end-of-chapter conceptual questions and problems are enhanced by the large number of figures, diagrams, photographs, and tables. Full color adds clarity to the artwork and makes illustrations as realistic as possible. Three-dimensional effects are rendered with the use of shaded and lightened areas where appropriate. Vectors are color coded, and curves in graphs are drawn in color. Color photographs have been carefully selected, and their accompanying captions have been written to serve as an added instructional tool. A complete description of the pedagogical use of color appears on the inside front cover.

ILLUSTRATIONS AND TABLES

SUMMARY The end-of-chapter Summary is organized by individual section headings for ease of reference.

Significant figures in both worked examples and endof-chapter problems have been handled with care. Most numerical examples and problems are worked out to either two or three significant figures, depending on the accuracy of the data provided. Intermediate results presented in the examples are rounded to the proper number of significant figures, and only those digits are carried forward.

SIGNIFICANT FIGURES

Several appendices are provided at the end of the textbook. Most of the appendix material represents a review of mathematical

APPENDICES AND ENDPAPERS

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concepts and techniques used in the text, including scientific notation, algebra, geometry, trigonometry, differential calculus, and integral calculus. Reference to these appendices is made as needed throughout the text. Most of the mathematical review sections include worked examples and exercises with answers. In addition to the mathematical review, some appendices contain useful tables that supplement textual information. For easy reference, the front endpapers contain a chart explaining the use of color throughout the book and a list of frequently used conversion factors. ACTIVE FIGURES Many diagrams from the text have been animated to become Active Figures (identified in the figure legend), part of the Enhanced WebAssign online homework system. By viewing animations of phenomena and processes that cannot be fully represented on a static page, students greatly increase their conceptual understanding. In addition to viewing animations of the figures, students can see the outcome of changing variables to see the effects, conduct suggested explorations of the principles involved in the figure, and take and receive feedback on quizzes related to the figure. All Active Figures are included on the instructor’s PowerLecture CD-ROM for in-class lecture presentation.

TEACHING OPTIONS This book contains more than enough material for a one-year course in introductory physics, which serves two purposes. First, it gives the instructor more flexibility in choosing topics for a specific course. Second, the book becomes more useful as a resource for students. On average, it should be possible to cover about one chapter each week for a class that meets three hours per week. Those sections, examples, and end-of-chapter problems dealing with applications of physics to life sciences are identified with the DNA icon . We offer the following suggestions for shorter courses for those instructors who choose to move at a slower pace through the year. Option A: If you choose to place more emphasis on contemporary topics in physics, you could omit all or parts of Chapter 8 (Rotational Equilibrium and Rotational Dynamics), Chapter 21 (Alternating-Current Circuits and Electromagnetic Waves), and Chapter 25 (Optical Instruments). Option B: If you choose to place more emphasis on classical physics, you could omit all or parts of Part 6 of the textbook, which deals with special relativity and other topics in 20th-century physics. The Instructor’s Solutions Manual offers additional suggestions for specific sections and topics that may be omitted without loss of continuity if time presses.

COURSE SOLUTIONS THAT FIT YOUR TEACHING GOALS AND YOUR STUDENTS’ LEARNING NEEDS Recent advances in educational technology have made homework management systems and audience response systems powerful and affordable tools to enhance the way you teach your course. Whether you offer a more traditional text-based course, are interested in using or are currently using an online homework management system such as WebAssign, or are ready to turn your lecture into an interactive learning environment with an audience response system, you can be confident that the text’s proven content provides the foundation for each and every component of our technology and ancillary package.

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VISUALIZE WHERE YOU WANT TO TAKE YOUR COURSE

WE PROVIDE YOU WITH THE FOUNDATION TO GET THERE Serway/Vuille, College Physics, 8e

Homework Management Systems ENHANCED WEBASSIGN Enhanced WebAssign is the perfect solution to your homework management needs. Designed by physicists for physicists, this system is a reliable and user-friendly teaching companion. Enhanced WebAssign is available for College Physics, giving you the freedom to assign

• every end-of-chapter Problem, Multiple-Choice Question, and Conceptual Question, enhanced with hints and feedback • most worked examples, enhanced with hints and feedback, to help strengthen students’ problem-solving skills • every Quick Quiz, giving your students ample opportunity to test their conceptual understanding • animated Active Figures, enhanced with hints and feedback, to help students develop their visualization skills • a math review to help students brush up on key quantitative concepts Please visit www.serwayphysics.com to view an interactive demonstration of Enhanced WebAssign. The text is also supported by the following Homework Management Systems. Contact your local sales representative for more information. CAPA: A Computer-Assisted Personalized Approach and LON-CAPA, http://www.lon-capa.org/ The University of Texas Homework Service

Audience Response Systems Regardless of the response system you are using, we provide the tested content to support it. Our ready-to-go content includes all the questions from the Quick Quizzes, all the end-of-chapter MultipleChoice Questions, test questions, and a selection of end-of-chapter questions to

AUDIENCE RESPONSE SYSTEM CONTENT

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provide helpful conceptual checkpoints to drop into your lecture. Our Active Figure animations have also been enhanced with multiple-choice questions to help test students’ observational skills. We also feature the Assessing to Learn in the Classroom content from the University of Massachusetts. This collection of 250 advanced conceptual questions has been tested in the classroom for more than ten years and takes peer learning to a new level. Contact your local sales representative to learn more about our audience response software and hardware. Visit www.serwayphysics.com to download samples of our audience response system content.

Lecture Presentation Resources The following resources provide support for your presentations in lecture. An easy-to-use multimedia lecture tool, the PowerLecture CD-ROM allows you to quickly assemble art, animations, digital video, and database files with notes to create fluid lectures. The two-volume set (Volume 1: Chapters 1–14; Volume 2: Chapters 15–30) includes prebuilt PowerPoint® lectures, a database of animations, video clips, and digital art from the text as well as editable electronic files of the Instructor’s Solutions Manual. Also included is the easy-touse test generator ExamView, which features all the questions from the printed Test Bank in an editable format. POWERLECTURE CD-ROM

Each volume contains approximately 100 transparency acetates featuring art from the text. Volume 1 contains Chapters 1 through 14, and Volume 2 contains Chapters 15 through 30.

TRANSPARENCY ACETATES

Assessment and Course Preparation Resources: A number of the resources listed below will help assist with your assessment and preparation processes, and are available to qualified adopters. Please contact your local Thomson • Brooks/Cole sales representative for details. Ancillaries offered in two volumes are split as follows: Volume 1 contains Chapters 1 through 14, and Volume 2 contains Chapters 15 through 30. INSTRUCTOR’S SOLUTIONS MANUAL by Charles Teague and Jerry S. Faughn. Available in two volumes, the Instructor’s Solutions Manual consists of complete solutions to all the problems, multiple-choice questions, and conceptual questions in the text, and full answers with explanations to the Quick Quizzes. An editable version of the complete instructor’s solutions is also available electronically on the PowerLecture CD-ROM.

by Ed Oberhofer. This test bank contains approximately 1 750 multiple-choice problems and questions. Answers are provided in a separate key. The test bank is provided in print form (in two volumes) for the instructor who does not have access to a computer, and instructors may duplicate pages for distribution to students. These questions are also available on the PowerLecture CD-ROM as either editable Word® files (with complete answers and solutions) or via the ExamView test software.

PRINTED TEST BANK

For users of either course management system, we provide our test bank questions in proper WebCT and Blackboard content format for easy upload into your online course.

WEBCT AND BLACKBOARD CONTENT

Consult the instructor’s Web site at www .serwayphysics.com for additional Quick Quiz questions, a problem correlation

INSTRUCTOR’S COMPANION WEB SITE

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guide, images from the text, and sample PowerPoint® lectures. Instructors adopting the eighth edition of College Physics may download these materials after securing the appropriate password from their local Brooks/Cole sales representative.

Student Resources Brooks/Cole offers several items to supplement and enhance the classroom experience. These ancillaries allow instructors to customize the textbook to their students’ needs and to their own style of instruction. One or more of the following ancillaries may be shrink-wrapped with the text at a reduced price: by John R. Gordon, Charles Teague, and Raymond A. Serway. Now offered in two volumes, the Student Solutions Manual/Study Guide features detailed solutions to approximately 12 problems per chapter. Boxed numbers identify those problems in the textbook for which complete solutions are found in the manual. The manual also features a skills section, important notes from key sections of the text, and a list of important equations and concepts. Volume 1 contains Chapters 1 through 14, and Volume 2 contains Chapters 15 through 30.

STUDENT SOLUTIONS MANUAL/STUDY GUIDE

by David Loyd. The Physics Laboratory Manual supplements the learning of basic physical principles while introducing laboratory procedures and equipment. Each chapter of the manual includes a prelaboratory assignment, objectives, an equipment list, the theory behind the experiment, experimental procedures, graphs, and questions. A laboratory report is provided for each experiment so that the student can record data, calculations, and experimental results. To develop their ability to judge the validity of their results, students are encouraged to apply statistical analysis to their data. A complete instructor’s manual is also available to facilitate use of this manual. PHYSICS LABORATORY MANUAL, 3rd edition,

ACKNOWLEDGMENTS In preparing the eighth edition of this textbook, we have been guided by the expertise of many people who have reviewed manuscript or provided prerevision suggestions. We wish to acknowledge the following reviewers and express our sincere appreciation for their helpful suggestions, criticism, and encouragement. Eighth edition reviewers: Gary Blanpied, University of South Carolina Gardner Friedlander, University School of Milwaukee Dolores Gende, Parish Episcopal School Grant W. Hart, Brigham Young University Joey Huston, Michigan State University Mark James, Northern Arizona University Teruki Kamon, Texas A & M University

Mark Lucas, Ohio University Mark E. Mattson, James Madison University J. Patrick Polley, Beloit College Eugene Surdutovich, Wayne State University Marshall Thomsen, Eastern Michigan University David P. Young, Louisiana State University

College Physics, eighth edition, was carefully checked for accuracy by Philip W. Adams, Louisiana State University; Grant W. Hart, Brigham Young University; Thomas K. Hemmick, Stony Brook University; Ed Oberhofer, Lake Sumter Community College; M. Anthony Reynolds, Embry-Riddle Aeronautical University; Eugene Surdutovich, Wayne State University; and David P. Young, Louisiana State University. Although responsibility for any remaining errors rests with us, we thank them for their dedication and vigilance.

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Prior to our work on this revision, we conducted a survey of professors to gauge how they used student assessment in their classroom. We were overwhelmed not only by the number of professors who wanted to take part in the survey, but also by their insightful comments. Their feedback and suggestions helped shape the revision of the end-of-chapter questions and problems in this edition, and so we would like to thank the survey participants: Elise Adamson, Wayland Baptist University; Rhett Allain, Southeastern Louisiana University; Michael Anderson, University of California, San Diego; James Andrews, Youngstown State University; Bradley Antanaitis, Lafayette College; Robert Astalos, Adams State College; Charles Atchley, Sauk Valley Community College; Kandiah Balachandran, Kalamazoo Valley Community College; Colley Baldwin, St. John’s University; Mahmoud Basharat, Houston Community College Northeast; Celso Batalha, Evergreen Valley College; Natalie Batalha, San Jose State University; Charles Benesh, Wesleyan College; Raymond Benge, Tarrant County College Northeast; Lee Benjamin, Marywood University; Edgar Bering, University of Houston; Ron Bingaman, Indiana University East; Jennifer Birriel, Morehead State University; Earl Blodgett, University of Wisconsin– River Falls; Anthony Blose, University of North Alabama; Jeff Bodart, Chipola College; Ken Bolland, The Ohio State University; Roscoe Bowen, Campbellsville University; Shane Brower, Grove City College; Charles Burkhardt, St. Louis Community College; Richard Cardenas, St. Mary’s University; Kelly Casey, Yakima Valley Community College; Cliff Castle, Jefferson College; Marco Cavaglia, University of Mississippi; Eugene Chaffin, Bob Jones University; Chang Chang, Drexel University; Jing Chang, Culver-Stockton College; Hirendra Chatterjee, Camden County College; Soumitra Chattopadhyay, Georgia Highlands College; Anastasia Chopelas, University of Washington; Krishna Chowdary, Bucknell University; Kelvin Chu, University of Vermont; Alice D. Churukian, Concordia College; David Cinabro, Wayne State University; Gary Copeland, Old Dominion University; Sean Cordry, Northwestern College of Iowa; Victor Coronel, SUNY Rockland Community College; Douglas Corteville, Iowa Western Community College; Randy Criss, Saint Leo University; John Crutchfield, Rockingham Community College; Danielle Dalafave, College of New Jersey; Lawrence Day, Utica College; Joe DeLeone, Corning Community College; Tony DeLia, North Florida Community College; Duygu Demirlioglu, Holy Names University; Sandra Desmarais, Daytona Beach Community College; Gregory Dolise, Harrisburg Area Community College; Duane Doyle, Arkansas State University–Newport; James Dull, Albertson College of Idaho; Tim Duman, University of Indianapolis; Arthur Eggers, Community College of Southern Nevada; Robert Egler, North Carolina State University; Steve Ellis, University of Kentucky; Terry Ellis, Jacksonville University; Ted Eltzroth, Elgin Community College; Martin Epstein, California State University, Los Angeles; Florence Etop, Virginia State University; Mike Eydenberg, New Mexico State University at Alamogordo; Davene Eyres, North Seattle Community College; Brett Fadem, Muhlenberg College; Greg Falabella, Wagner College; Michael Faleski, Delta College; Jacqueline Faridani, Shippensburg University; Abu Fasihuddin, University of Connecticut; Scott Fedorchak, Campbell University; Frank Ferrone, Drexel University; Harland Fish, Kalamazoo Valley Community College; Kent Fisher, Columbus State Community College; Allen Flora, Hood College; James Friedrichsen, Austin Community College; Cynthia Galovich, University of Northern Colorado; Ticu Gamalie, Arkansas State University–LRAFB; Andy Gavrin, Indiana University Purdue University Indianapolis; Michael Giangrande, Oakland Community College; Wells Gordon, Ohio Valley University; Charles Grabowski, Carroll Community College; Robert Gramer, Lake City Community College; Janusz Grebowicz, University of Houston–Downtown; Morris Greenwood, San Jacinto College Central; David Groh, Gannon University; Fred Grosse, Susquehanna University; Harvey Haag, Penn State DuBois; Piotr Habdas, Saint Joseph’s University; Robert Hagood, Washtenaw Community College; Heath Hatch, University of Massachusetts Amherst; Dennis Hawk, Navarro College; George Hazelton, Chowan University; Qifang He, Arkansas State University at Beebe; Randall Headrick, University of Vermont; Todd Holden, Brooklyn College; Susanne Holmes-Koetter; Doug Ingram, Texas Christian University; Dwain Ingram, Texas State Technical College; Rex Isham, Sam Houston State University; Herbert Jaeger, Miami University; Mohsen Janatpour, College of San Mateo; Peter Jeschofnig, Colorado Mountain College; Lana Jordan, Merced College; Teruki Kamon, Texas A & M University; Charles Kao, Columbus State University; David Kardelis, College of Eastern Utah; Edward Kearns, Boston University; Robert Keefer, Lake Sumter Community College; Mamadou Keita, Sheridan College, Gillette Campus; Luke Keller, Ithaca College; Andrew Kerr, University of Findlay; Kinney Kim, North Carolina Central University; Kevin Kimberlin, Bradley University; George Knott, Cosumnes River College; Corinne Krauss, Dickinson State University; Christopher Kulp, Eastern Kentucky University; A. Anil Kumar, Prairie View A & M University; Josephine Lamela, Middlesex County College; Eric Lane, University of Tennessee; Gregory Lapicki, East Carolina University; Byron Leles, Snead State

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Community College; David Lieberman, Queensborough Community College; Marilyn Listvan, Normandale Community College; Rafael Lopez-Mobilia, University of Texas at San Antonio; Jose Lozano, Bradley University; Mark Lucas, Ohio University; Ntungwa Maasha, Coastal Georgia Community College; Keith MacAdam, University of Kentucky; Kevin Mackay, Grove City College; Steve Maier, Northwestern Oklahoma State University; Helen Major, Lincoln University; Igor Makasyuk, San Francisco State University; Gary Malek, Johnson County Community College; Frank Mann, Emmanuel College; Ronald Marks, North Greenville University; Perry Mason, Lubbock Christian University; Mark Mattson, James Madison University; John McClain, Panola College; James McDonald, University of Hartford; Linda McDonald, North Park University; Ralph V. McGrew, Broome Community College; Janet McLarty-Schroeder, Cerritos College; Rahul Mehta, University of Central Arkansas; Mike Mikhaiel, Passaic County Community College; Laney Mills, College of Charleston; John Milton, DePaul University; Stephen Minnick, Kent State University, Tuscarawas Campus; Dominick Misciascio, Mercer County Community College; Arthur Mittler, University of Massachusetts Lowell; Glenn Modrak, Broome Community College; Toby Moleski, Muskegon Community College; G. David Moore, Reinhardt College; Hassan Moore, Johnson C. Smith University; David Moran, Breyer State University; Laurie Morgus, Drew University; David Murdock, Tennessee Technological University; Dennis Nemeschansky, University of Southern California; Bob Nerbun, University of South Carolina Sumter; Lorin Neufeld, Fresno Pacific University; K. W. Nicholson, Central Alabama Community College; Charles Nickles, University of Massachusetts Dartmouth; Paul Nienaber, Saint Mary’s University of Minnesota; Ralph Oberly, Marshall University; Terry F. O’Dwyer, Nassau Community College; Don Olive, Gardner-Webb University; Jacqueline Omland, Northern State University; Paige Ouzts, Lander University; Vaheribhai Patel, Tomball College; Bijoy Patnaik, Halifax Community College; Philip Patterson, Southern Polytechnic State University; James Pazun, Pfeiffer University; Chuck Pearson, Shorter College; Todd Pedlar, Luther College; Anthony Peer, Delaware Technical & Community College; Frederick Phelps, Central Michigan University; Robert Philbin, Trinidad State Junior College; Joshua Phiri, FlorenceDarlington Technical College; Cu Phung, Methodist College; Alberto Pinkas, New Jersey City University; Ali Piran, Stephen F. Austin State University; Marie Plumb, Jamestown Community College; Dwight Portman, Miami University Middletown; Rose Rakers, Trinity Christian College; Periasamy Ramalingam, Albany State University; Marilyn Rands, Lawrence Technological University; Tom Richardson, Marian College; Herbert Ringel, Borough of Manhattan Community College; Salvatore Rodano, Harford Community College; John Rollino, Rutgers University– Newark; Fernando Romero-Borja, Houston Community College–Central; Michael Rulison, Oglethorpe University; Marylyn Russ, Marygrove College; Craig Rutan, Santiago Canyon College; Jyotsna Sau, Delaware Technical & Community College; Charles Sawicki, North Dakota State University; Daniel Schoun, Kettering College of Medical Arts; Andria Schwortz, Quinsigamond Community College; David Seely, Albion College; Ross Setze, Pearl River Community College; Bart Sheinberg; Peter Sheldon, Randolph-Macon Woman’s College; Wen Shen, Community College of Southern Nevada; Anwar Shiekh, Dine College; Marllin Simon, Auburn University; Don Sparks, Pierce College; Philip Spickler, Bridgewater College; Fletcher Srygley, Lipscomb University; Scott Steckenrider, Illinois College; Donna Stokes, University of Houston; Laurence Stone, Dakota County Technical College; Yang Sun, University of Notre Dame; Gregory Suran, Raritan Valley Community College; Vahe Tatoian, Mt. San Antonio College; Alem Teklu, College of Charleston; Paul Testa, Tompkins Cortland Community College; Michael Thackston, Southern Polytechnic State University; Melody Thomas, Northwest Arkansas Community College; Cheng Ting, Houston Community College–Southeast; Donn Townsend, Penn State Shenango; Herman Trivilino; Gajendra Tulsian, Daytona Beach Community College; Rein Uritam, Boston College; Daniel Van Wingerden, Eastern Michigan University; Ashok Vaseashta, Marshall University; Robert Vaughn, Graceland University; Robert Warasila, Suffolk County Community College; Robert Webb, Texas A & M University; Zodiac Webster, Columbus State University; Brian Weiner, Penn State DuBois; Jack Wells, Thomas More College; Ronnie Whitener, Tri-County Community College; Tom Wilbur, Anne Arundel Community College; Sam Wiley, California State University, Dominguez Hills; Judith Williams, William Penn University; Mark Williams; Don Williamson, Chadron State College; Neal Wilsey, College of Southern Maryland; Lowell Wood, University of Houston; Jainshi Wu; Pei Xiong-Skiba, Austin Peay State University; Ming Yin, Benedict College; David Young, Louisiana State University; Douglas Young, Mercer University; T. Waldek Zerda, Texas Christian University; Peizhen Zhao, Edison Community College; Steven Zides, Wofford College; and Ulrich Zurcher, Cleveland State University.

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Finally, we would like to thank the following people for their suggestions and assistance during the preparation of earlier editions of this textbook: Gary B. Adams, Arizona State University; Marilyn Akins, Broome Community College; Ricardo Alarcon, Arizona State University; Albert Altman, University of Lowell; John Anderson, University of Pittsburgh; Lawrence Anderson-Huang, University of Toledo; Subhash Antani, Edgewood College; Neil W. Ashcroft, Cornell University; Charles R. Bacon, Ferris State University; Dilip Balamore, Nassau Community College; Ralph Barnett, Florissant Valley Community College; Lois Barrett, Western Washington University; Natalie Batalha, San Jose State University; Paul D. Beale, University of Colorado at Boulder; Paul Bender, Washington State University; David H. Bennum, University of Nevada at Reno; Ken Bolland, The Ohio State University; Jeffery Braun, University of Evansville; John Brennan, University of Central Florida; Michael Bretz, University of Michigan, Ann Arbor; Michael E. Browne, University of Idaho; Joseph Cantazarite, Cypress College; Ronald W. Canterna, University of Wyoming; Clinton M. Case, Western Nevada Community College; Neal M. Cason, University of Notre Dame; Kapila Clara Castoldi, Oakland University; Roger W. Clapp, University of South Florida; Giuseppe Colaccico, University of South Florida; Lattie F. Collins, East Tennessee State University; Lawrence B. Colman, University of California, Davis; Andrew Cornelius, University of Nevada, Las Vegas; Jorge Cossio, Miami Dade Community College; Terry T. Crow, Mississippi State College; Yesim Darici, Florida International University; Stephen D. Davis, University of Arkansas at Little Rock; John DeFord, University of Utah; Chris J. DeMarco, Jackson Community College; Michael Dennin, University of California, Irvine; N. John DiNardo, Drexel University; Steve Ellis, University of Kentucky; Robert J. Endorf, University of Cincinnati; Steve Ellis, University of Kentucky; Hasan Fakhruddin, Ball State University/Indiana Academy; Paul Feldker, Florissant Valley Community College; Leonard X. Finegold, Drexel University; Emily Flynn; Lewis Ford, Texas A & M University; Tom French, Montgomery County Community College; Albert Thomas Frommhold, Jr., Auburn University; Lothar Frommhold, University of Texas at Austin; Eric Ganz, University of Minnesota; Teymoor Gedayloo, California Polytechnic State University; Simon George, California State University, Long Beach; James R. Goff, Pima Community College; Yadin Y. Goldschmidt, University of Pittsburgh; John R. Gordon, James Madison University; George W. Greenlees, University of Minnesota; Wlodzimierz Guryn, Brookhaven National Laboratory; Steve Hagen, University of Florida; Raymond Hall, California State University, Fresno; Patrick Hamill, San Jose State University; Joel Handley; James Harmon, Oklahoma State University; Grant W. Hart, Brigham Young University; James E. Heath, Austin Community College; Grady Hendricks, Blinn College; Christopher Herbert, New Jersey City University; Rhett Herman, Radford University; John Ho, State University of New York at Buffalo; Aleksey Holloway, University of Nebraska at Omaha; Murshed Hossain, Rowan University; Robert C. Hudson, Roanoke College; Joey Huston, Michigan State University; Fred Inman, Mankato State University; Mark James, Northern Arizona University; Ronald E. Jodoin, Rochester Institute of Technology; Randall Jones, Loyola College in Maryland; Drasko Jovanovic, Fermilab; George W. Kattawar, Texas A & M University; Joseph Keane, St. Thomas Aquinas College; Frank Kolp, Trenton State University; Dorina Kosztin, University of Missouri–Columbia; Joan P. S. Kowalski, George Mason University; Ivan Kramer, University of Maryland, Baltimore County; Sol Krasner, University of Chicago; Karl F. Kuhn, Eastern Kentucky University; David Lamp, Texas Tech University; Harvey S. Leff, California State Polytechnic University; Joel Levine, Orange Coast College; Michael Lieber, University of Arkansas; Martha Lietz, Niles West High School; James Linbald, Saddleback Community College; Edwin Lo; Bill Lochslet, Pennsylvania State University; Rafael Lopez-Mobilia, University of Texas at San Antonio; Michael LoPresto, Henry Ford Community College; Bo Lou, Ferris State University; Jeffrey V. Mallow, Loyola University of Chicago; David Markowitz, University of Connecticut; Joe McCauley, Jr., University of Houston; Steven McCauley, California State Polytechnic University, Pomona; Ralph V. McGrew, Broome Community College; Bill F. Melton, University of North Carolina at Charlotte; John A. Milsom, University of Arizona; Monty Mola, Humboldt State University; H. Kent Moore, James Madison University; John Morack, University of Alaska, Fairbanks; Steven Morris, Los Angeles Harbor College; Charles W. Myles, Texas Tech University; Carl R. Nave, Georgia State University; Martin Nikolo, Saint Louis University; Blaine Norum, University of Virginia; M. E. Oakes, University of Texas at Austin; Lewis J. Oakland, University of Minnesota; Ed Oberhofer, Lake Sumter Community College; Lewis O’Kelly, Memphis State University; David G. Onn, University of Delaware; J. Scott Payson, Wayne State University; Chris Pearson, University of Michigan–Flint; Alexey A. Petrov, Wayne State University; T. A. K. Pillai, University of Wisconsin, La Crosse; Lawrence S. Pinsky, University of Houston; William D. Ploughe, The Ohio State University; Patrick Polley, Beloit College; Brooke M. Pridmore, Clayton State University; Joseph Priest, Miami University; James Purcell, Georgia State University; W. Steve Quon, Ventura College; Michael Ram, State University of

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New York at Buffalo; Kurt Reibel, The Ohio State University; M. Anthony Reynolds, Embry-Riddle Aeronautical University; Barry Robertson, Queen’s University; Virginia Roundy, California State University, Fullerton; Larry Rowan, University of North Carolina, Chapel Hill; Dubravka Rupnik, Louisiana State University; William R. Savage, University of Iowa; Reinhard A. Schumacher, Carnegie Mellon University; Surajit Sen, State University of New York at Buffalo; John Simon, University of Toledo; Marllin L. Simon, Auburn University; Matthew Sirocky; Donald D. Snyder, Indiana University at Southbend; George Strobel, University of Georgia; Carey E. Stronach, Virginia State University; Thomas W. Taylor, Cleveland State University; Perry A. Tompkins, Samford University; L. L. Van Zandt, Purdue University; Howard G. Voss, Arizona State University; James Wanliss, Embry-Riddle Aeronautical University; Larry Weaver, Kansas State University; Donald H. White, Western Oregon State College; Bernard Whiting, University of Florida; George A. Williams, University of Utah; Jerry H. Wilson, Metropolitan State College; Robert M. Wood, University of Georgia; and Clyde A. Zaidins, University of Colorado at Denver.

Gerd Kortemeyer and Randall Jones contributed several end-of-chapter problems, especially those of interest to the life sciences. Edward F. Redish of the University of Maryland graciously allowed us to list some of his problems from the Activity Based Physics Project. We are extremely grateful to the publishing team at the Brooks/Cole Publishing Company for their expertise and outstanding work in all aspects of this project. In particular, we thank Ed Dodd, who tirelessly coordinated and directed our efforts in preparing the manuscript in its various stages, and Sylvia Krick, who transmitted all the print ancillaries. Jane Sanders Miller, the photo researcher, did a great job finding photos of physical phenomena, Sam Subity coordinated the media program for the text, and Rob Hugel helped translate our rough sketches into accurate, compelling art. Katherine Wilson of Lachina Publishing Services managed the difficult task of keeping production moving and on schedule. Mark Santee, Teri Hyde, and Chris Hall also made numerous valuable contributions. Mark, the book’s marketing manager, was a tireless advocate for the text. Teri coordinated the entire production and manufacturing of the text, in all its various incarnations, from start to finish. Chris provided just the right amount of guidance and vision throughout the project. We also thank David Harris, a great team builder and motivator with loads of enthusiasm and an infectious sense of humor. Finally, we are deeply indebted to our wives and children for their love, support, and longterm sacrifices. Raymond A. Serway St. Petersburg, Florida Chris Vuille Daytona Beach, Florida

ENGAGING APPLICATIONS Although physics is relevant to so much in our modern lives, it may not be obvious to students in an introductory course. In this eighth edition of College Physics, we continue a design feature begun in the seventh edition. This feature makes the relevance of physics to everyday life more obvious by pointing out specific applications in the form of a marginal note. Some of these applications pertain to the life sciences and are marked with the DNA icon . The list below is not intended to be a complete listing of all the applications of the principles of physics found in this textbook. Many other applications are to be found within the text and especially in the worked examples, conceptual questions, and end-of-chapter problems.

Chapter 15 Measuring atmospheric electric fields, p. 512 Lightning rods, p. 514 Driver safety during electrical storms, p. 515

Chapter 16 Automobile batteries, p. 537 The electrostatic precipitator, p. 544 The electrostatic air cleaner, p. 545 Xerographic copiers, p. 545 Laser printers, p. 546 Camera flash attachments, p. 547 Computer keyboards, p. 547 Electrostatic confinement, p. 547 Defibrillators, p. 556 Stud finders, p. 559

Chapter 17 Dimming of aging lightbulbs, p. 578 Lightbulb failures, p. 582 Electrical activity in the heart, pp. 585–587 Electrocardiograms, p. 585 Cardiac pacemakers, p. 586 Implanted cardioverter defibrillators, p. 586

Chapter 18 Christmas lights in series, p. 596 Circuit breakers, p. 600 Three-way lightbulbs, p. 601 Timed windshield wipers, p. 608 Bacterial growth, p. 608 Roadway flashers, p. 608 Fuses and circuit breakers, p. 612 Third wire on consumer appliances, p. 612 Conduction of electrical signals by neurons, pp. 613–615

Chapter 19 Dusting for fingerprints, p. 628 Magnetic bacteria, p. 629 Labeling airport runways, p. 629 Compasses down under, p. 630 Loudspeaker operation, p. 634 Electromagnetic pumps for artificial hearts and kidneys, p. 635 Lightning strikes, p. 635 Electric motors, p. 638 Mass spectrometers, p. 641

Chapter 20 Ground fault interrupters, p. 668 Electric guitar pickups, p. 669 Apnea monitors, p. 669 Space catapult, p. 671

Magnetic tape recorders, p. 675 Alternating-current generators, p. 676 Direct-current generators, p. 677 Motors, p. 679

Finding the concentrations of solutions by means of their optical activity, p. 813 Liquid crystal displays (LCDs), p. 813

Chapter 21

The camera, pp. 823–824 The eye, pp. 824–829 Using optical lenses to correct for defects, p. 826 Prescribing a corrective lens for a farsighted patient, pp. 827–828 A corrective lens for nearsightedness, p. 828 Vision of the invisible man, p. 828 Cat’s eyes, p. 836

Electric fields and cancer treatment, p. 699 Shifting phase to deliver more power, p. 707 Tuning your radio, p. 708 Metal detectors at the courthouse, p. 709 Long-distance electric power transmission, p. 711 Radio-wave transmission, p. 714 Solar system dust, p. 717 A hot tin roof (solar-powered homes), p. 718 The sun and the evolution of the eye, p. 722

Chapter 22 Seeing the road on a rainy night, p. 734 Red eyes in flash photographs, p. 735 The colors of water ripples at sunset, p. 735 Double images, p. 735 Refraction of laser light in a digital video disk (DVD), p. 741 Identifying gases with a spectrometer, p. 743 Submarine periscopes, p. 749 Fiber optics in medical diagnosis and surgery, p. 750 Fiber optics in telecommunications, p. 750 Design of an optical fiber, p. 751

Chapter 23 Day and night settings for rearview mirrors, p. 761 Illusionist’s trick, p. 762 Concave vs. convex, p. 766 Reversible waves, p. 766 Underwater vision, p. 770 Vision and diving masks, p. 776

Chapter 24 A smoky Young’s experiment, p. 794 Television signal interference, p. 794 Checking for imperfections in optical lenses, p. 798 The physics of CDs and DVDs, p. 800 Diffraction of sound waves, p. 804 Prism vs. grating, p. 806 Rainbows from a CD, p. 807 Tracking information on a CD, p. 807 Polarizing microwaves, p. 810 Polaroid sunglasses, p. 812

Chapter 25

Chapter 26 Faster clocks in a “mile-high city,” p. 865

Chapter 27 Star colors, p. 871 Photocells, p. 875 Using x-rays to study the work of master painters, p. 876 Electron microscopes, p. 882 X-ray microscopes, p. 883

Chapter 28 Discovery of helium, p. 893 Thermal or spectral, p. 893 Auroras, p. 894 Laser technology, p. 908

Chapter 29 Binding nucleons and electrons, p. 917 Energy and half-life, p. 921 Carbon dating, p. 924 Smoke detectors, p. 925 Radon pollution, p. 925 Medical applications of radiation, pp. 929–931 Occupational radiation exposure limits, p. 930 Irradiation of food and medical equipment, p. 930 Radioactive tracers in medicine, p. 930 Magnetic resonance imaging (MRI), p. 931

Chapter 30 Unstable products, p. 938 Nuclear reactor design, p. 940 Fusion reactors, p. 941 Positron emission tomography (PET scanning), p. 945 Breaking conservation laws, p. 949 Conservation of meson number, p. 951

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LIST OF ACTIVE FIGURES Chapter 15 Active Figures 15.6, 15.11, 15.16, 15.21, and 15.28 Chapter 16 Active Figures 16.7, 16.18, and 16.20 Chapter 17 Active Figures 17.4 and 17.9 Chapter 18 Active Figures 18.1, 18.2, 18.6, 18.16, and 18.17 Chapter 19 Active Figures 19.2, 19.17, 19.19, 19.20, and 19.23 Chapter 20 Active Figures 20.4, 20.13, 20.20, 20.22, 20.27, and 20.28 Chapter 21 Active Figures 21.1, 21.2, 21.6, 21.7, 21.8, 21.9, and 21.20 Chapter 22 Active Figures 22.4, 22.6, 22.7, 22.20, and 22.25 Chapter 23 Active Figures 23.2, 23.13, 23.16, and 23.25 Chapter 24 Active Figures 24.1, 24.16, 24.20, 24.21, and 24.26 Chapter 25 Active Figures 25.7, 25.8, and 25.15 Chapter 26 Active Figures 26.4, 26.6, and 26.9 Chapter 27 Active Figures 27.2, 27.3, and 27.4 Chapter 28 Active Figures 28.7, 28.8, and 28.17 Chapter 29 Active Figures 29.1, 29.6, and 29.7 Chapter 30 Active Figures 30.2 and 30.8

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TO THE STUDENT As a student, it’s important that you understand how to use this book most effectively and how best to go about learning physics. Scanning through the preface will acquaint you with the various features available, both in the book and online. Awareness of your educational resources and how to use them is essential. Although physics is challenging, it can be mastered with the correct approach.

HOW TO STUDY Students often ask how best to study physics and prepare for examinations. There is no simple answer to this question, but we’d like to offer some suggestions based on our own experiences in learning and teaching over the years. First and foremost, maintain a positive attitude toward the subject matter. Like learning a language, physics takes time. Those who keep applying themselves on a daily basis can expect to reach understanding and succeed in the course. Keep in mind that physics is the most fundamental of all natural sciences. Other science courses that follow will use the same physical principles, so it is important that you understand and are able to apply the various concepts and theories discussed in the text. They’re relevant!

CONCEPTS AND PRINCIPLES Students often try to do their homework without first studying the basic concepts. It is essential that you understand the basic concepts and principles before attempting to solve assigned problems. You can best accomplish this goal by carefully reading the textbook before you attend your lecture on the covered material. When reading the text, you should jot down those points that are not clear to you. Also be sure to make a diligent attempt at answering the questions in the Quick Quizzes as you come to them in your reading. We have worked hard to prepare questions that help you judge for yourself how well you understand the material. Pay careful attention to the many Tips throughout the text. They will help you avoid misconceptions, mistakes, and misunderstandings as well as maximize the efficiency of your time by minimizing adventures along fruitless paths. During class, take careful notes and ask questions about those ideas that are unclear to you. Keep in mind that few people are able to absorb the full meaning of scientific material after only one reading. Your lectures and laboratory work supplement your textbook and should clarify some of the more difficult material. You should minimize rote memorization of material. Successful memorization of passages from the text, equations, and derivations does not necessarily indicate that you understand the fundamental principles. Your understanding will be enhanced through a combination of efficient study habits, discussions with other students and with instructors, and your ability to solve the problems presented in the textbook. Ask questions whenever you think clarification of a concept is necessary.

STUDY SCHEDULE It is important for you to set up a regular study schedule, preferably a daily one. Make sure you read the syllabus for the course and adhere to the schedule set by your instructor. As a general rule, you should devote about two hours of study time for every one hour you are in class. If you are having trouble with the course, seek the advice of the instructor or other students who have taken the course. You

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may find it necessary to seek further instruction from experienced students. Very often, instructors offer review sessions in addition to regular class periods. It is important that you avoid the practice of delaying study until a day or two before an exam. One hour of study a day for 14 days is far more effective than 14 hours the day before the exam. “Cramming” usually produces disastrous results, especially in science. Rather than undertake an all-night study session immediately before an exam, briefly review the basic concepts and equations and get a good night’s rest. If you think you need additional help in understanding the concepts, in preparing for exams, or in problem solving, we suggest you acquire a copy of the Student Solutions Manual/Study Guide that accompanies this textbook; this manual should be available at your college bookstore.

USE THE FEATURES You should make full use of the various features of the text discussed in the preface. For example, marginal notes are useful for locating and describing important equations and concepts, and boldfaced type indicates important statements and definitions. Many useful tables are contained in the appendices, but most tables are incorporated in the text where they are most often referenced. Appendix A is a convenient review of mathematical techniques. Answers to all Quick Quizzes and Example Questions, as well as odd-numbered multiple-choice questions, conceptual questions, and problems, are given at the end of the textbook. Answers to selected end-of-chapter problems are provided in the Student Solutions Manual/Study Guide. Problem-Solving Strategies included in selected chapters throughout the text give you additional information about how you should solve problems. The contents provides an overview of the entire text, and the index enables you to locate specific material quickly. Footnotes sometimes are used to supplement the text or to cite other references on the subject discussed. After reading a chapter, you should be able to define any new quantities introduced in that chapter and to discuss the principles and assumptions used to arrive at certain key relations. The chapter summaries and the review sections of the Student Solutions Manual/Study Guide should help you in this regard. In some cases, it may be necessary for you to refer to the index of the text to locate certain topics. You should be able to correctly associate with each physical quantity the symbol used to represent that quantity and the unit in which the quantity is specified. Further, you should be able to express each important relation in a concise and accurate prose statement.

PROBLEM SOLVING R. P. Feynman, Nobel laureate in physics, once said, “You do not know anything until you have practiced.” In keeping with this statement, we strongly advise that you develop the skills necessary to solve a wide range of problems. Your ability to solve problems will be one of the main tests of your knowledge of physics, so you should try to solve as many problems as possible. It is essential that you understand basic concepts and principles before attempting to solve problems. It is good practice to try to find alternate solutions to the same problem. For example, you can solve problems in mechanics using Newton’s laws, but very often an alternate method that draws on energy considerations is more direct. You should not deceive yourself into thinking you understand a problem merely because you have seen it solved in class. You must be able to solve the problem and similar problems on your own. We have cast the examples in this book in a special, two-column format to help you in this regard. After studying an example, see if you can cover up the right-hand side and do it yourself, using only the written descriptions on the left as hints. Once you succeed at that, try solving the example completely on your own. Finally, answer the question and solve the exercise. Once you have accomplished

Preface

all these steps, you will have a good mastery of the problem, its concepts, and mathematical technique. After studying all the Example Problems in this way, you are ready to tackle the problems at the end of the chapter. Of these, the Guided Problems provide another aid to learning how to solve some of the more complex problems. The approach to solving problems should be carefully planned. A systematic plan is especially important when a problem involves several concepts. First, read the problem several times until you are confident you understand what is being asked. Look for any key words that will help you interpret the problem and perhaps allow you to make certain assumptions. Your ability to interpret a question properly is an integral part of problem solving. Second, you should acquire the habit of writing down the information given in a problem and those quantities that need to be found; for example, you might construct a table listing both the quantities given and the quantities to be found. This procedure is sometimes used in the worked examples of the textbook. After you have decided on the method you think is appropriate for a given problem, proceed with your solution. Finally, check your results to see if they are reasonable and consistent with your initial understanding of the problem. General problem-solving strategies of this type are included in the text and are highlighted with a surrounding box. If you follow the steps of this procedure, you will find it easier to come up with a solution and will also gain more from your efforts. Often, students fail to recognize the limitations of certain equations or physical laws in a particular situation. It is very important that you understand and remember the assumptions underlying a particular theory or formalism. For example, certain equations in kinematics apply only to a particle moving with constant acceleration. These equations are not valid for describing motion whose acceleration is not constant, such as the motion of an object connected to a spring or the motion of an object through a fluid.

EXPERIMENTS Because physics is a science based on experimental observations, we recommend that you supplement the text by performing various types of “hands-on” experiments, either at home or in the laboratory. For example, the common Slinky™ toy is excellent for studying traveling waves, a ball swinging on the end of a long string can be used to investigate pendulum motion, various masses attached to the end of a vertical spring or rubber band can be used to determine their elastic nature, an old pair of Polaroid sunglasses and some discarded lenses and a magnifying glass are the components of various experiments in optics, and the approximate measure of the free-fall acceleration can be determined simply by measuring with a stopwatch the time it takes for a ball to drop from a known height. The list of such experiments is endless. When physical models are not available, be imaginative and try to develop models of your own.

An Invitation to Physics It is our hope that you too will find physics an exciting and enjoyable experience and that you will profit from this experience, regardless of your chosen profession. Welcome to the exciting world of physics! To see the World in a Grain of Sand And a Heaven in a Wild Flower, Hold infinity in the palm of your hand And Eternity in an hour. —William Blake, “Auguries of Innocence”

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Welcome to your MCAT Test Preparation Guide The MCAT Test Preparation Guide makes your copy of College Physics, eighth edition, the most comprehensive MCAT study tool and classroom resource in introductory physics. The grid, which begins below and continues on the next two pages, outlines 12 concept-based study courses for the physics part of your MCAT exam. Use it to prepare for the MCAT, class tests, and your homework assignments.

Vectors

Force

Skill Objectives: To calculate distance, angles between vectors, and magnitudes.

Skill Objectives: To know and understand Newton’s laws and to calculate resultant forces and weight.

Review Plan: Distance and Angles:
Chapter 1, Sections 1.7, 1.8
Active Figure 1.6
Chapter Problems 35, 41, 44 Using Vectors:
Chapter 3, Sections 3.1, 3.2
Quick Quizzes 3.1, 3.2
Examples 3.1–3.3
Active Figure 3.3
Chapter Problems 8, 13

MCAT Test Preparation Guide

Newton’s Laws:
Chapter 4, Sections 4.1–4.4
Quick Quizzes 4.1, 4.3
Examples 4.1–4.4
Active Figure 4.6
Chapter Problems 5, 7, 11 Resultant Forces:
Chapter 4, Section 4.5
Quick Quizzes 4.4, 4.5
Examples 4.7, 4.9, 4.10
Chapter Problems 19, 27, 37

Motion

Equilibrium

Skill Objectives: To understand motion in two dimensions and to calculate speed and velocity, centripetal acceleration, and acceleration in free-fall problems.

Skill Objectives: To calculate momentum and impulse, center of gravity, and torque.

Review Plan:

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Review Plan:

Motion in One Dimension: Chapter 2, Sections 2.1–2.6
Quick Quizzes 2.1–2.8
Examples 2.1–2.10
Active Figure 2.15
Chapter Problems 3, 10, 23, 31, 50, 59


Motion in Two Dimensions:
Chapter 3, Sections 3.3, 3.4
Quick Quizzes 3.4–3.7
Examples 3.3–3.7
Active Figures 3.14, 3.15
Chapter Problems 27, 33 Centripetal Acceleration:
Chapter 7, Section 7.4
Quick Quizzes 7.6, 7.7
Example 7.6

Review Plan: Momentum:
Chapter 6, Sections 6.1–6.3
Quick Quizzes 6.2–6.6
Examples 6.1–6.4, 6.6
Active Figures 6.7, 6.10, 6.13
Chapter Problems 20, 23 Torque:
Chapter 8, Sections 8.1–8.4
Examples 8.1–8.7
Chapter Problems 5, 9

Work

Matter

Skill Objectives: To calculate friction, work, kinetic energy, potential energy, and power.

Skill Objectives: To calculate pressure, density, specific gravity, and flow rates.

Review Plan:

Review Plan:

Friction:
Chapter 4, Section 4.6
Quick Quizzes 4.6–4.8
Active Figure 4.19 Work: Chapter 5, Section 5.1
Quick Quiz 5.1
Example 5.1
Active Figure 5.5
Chapter Problem 17


Energy: Chapter 5, Sections 5.2, 5.3
Examples 5.4, 5.5
Quick Quizzes 5.2, 5.3


Power: Chapter 5, Section 5.6
Examples 5.12, 5.13


Properties:
Chapter 9, Sections 9.1–9.3
Quick Quiz 9.1
Examples 9.1, 9.3, 9.4
Active Figure 9.3
Chapter Problem 7 Pressure:
Chapter 9, Sections 9.3–9.6
Quick Quizzes 9.2–9.6
Examples 9.4–9.9
Active Figures 9.19, 9.20
Chapter Problems 25, 43 Flow Rates:
Chapter 9, Sections 9.7, 9.8
Quick Quiz 9.7
Examples 9.11–9.14
Chapter Problem 46

Sound

Skill Objectives: To understand interference of waves and to calculate basic properties of waves, properties of springs, and properties of pendulums.

Skill Objectives: To understand interference of waves and to calculate properties of waves, the speed of sound, Doppler shifts, and intensity.

Review Plan:

Review Plan:

Wave Properties:
Chapters 13, Sections 13.1–13.4, 13.7–13.11
Quick Quizzes 13.1–13.6
Examples 13.1, 13.6, 13.8–13.10
Active Figures 13.1, 13.8, 13.12, 13.13, 13.24, 13.26, 13.32, 13.33, 13.34, 13.35
Chapter Problems 11, 17, 25, 33, 45, 55, 61 Pendulum:
Chapter 13, Section 13.5
Quick Quizzes 13.7–13.9
Example 13.7
Active Figures 13.15, 13.16
Chapter Problem 39

Sound Properties:
Chapter 14, Sections 14.1–14.4, 14.6
Quick Quizzes 14.1, 14.2
Examples 14.1, 14.2, 14.4, 14.5
Active Figures 14.6, 14.11
Chapter Problems 7, 27 Interference/Beats:
Chapter 14, Sections 14.7, 14.8, 14.11
Quick Quiz 14.7
Examples 14.6, 14.11
Active Figures 14.18, 14.25
Chapter Problems 37, 41, 57

MCAT Test Preparation Guide

Waves

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Light

Circuits

Skill Objectives: To understand mirrors and lenses, to calculate the angles of reflection, to use the index of refraction, and to find focal lengths.

Skill Objectives: To understand and calculate current, resistance, voltage, power, and energy and to use circuit analysis. Review Plan:

Review Plan: Reflection and Refraction:
Chapter 22, Sections 22.1–22.4
Quick Quizzes 22.2–22.4
Examples 22.1–22.4
Active Figures 22.4, 22.6, 22.7
Chapter Problems 11, 17, 19, 25 Mirrors and Lenses:
Chapter 23, Sections 23.1–23.6
Quick Quizzes 23.1, 23.2, 23.4–23.6
Examples 23.7, 23.8, 23.9
Active Figures 23.2, 23.16, 23.25
Chapter Problems 25, 31, 35, 39

Electrostatics Skill Objectives: To understand and calculate the electric field, the electrostatic force, and the electric potential.

MCAT Test Preparation Guide

Review Plan:

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Coulomb’s Law:
Chapter 15, Sections 15.1–15.3
Quick Quiz 15.2
Examples 15.1–15.3
Active Figure 15.6
Chapter Problems 11 Electric Field:
Chapter 15, Sections 15.4, 15.5
Quick Quizzes 15.3–15.6
Examples 15.4, 15.5
Active Figures 15.11, 15.16
Chapter Problems 19, 23, 27 Potential:
Chapter 16, Sections 16.1–16.3
Quick Quizzes 16.1, 16.3–16.7
Examples 16.1, 16.4
Active Figure 16.7
Chapter Problems 7, 15

Ohm’s Law:
Chapter 17, Sections 17.1–17.4
Quick Quizzes 17.1, 17.3, 17.5
Example 17.1
Chapter Problem 15 Power and Energy:
Chapter 17, Section 17.6
Quick Quizzes 17.7–17.9
Example 17.5
Active Figure 17.9
Chapter Problem 38 Circuits:
Chapter 18, Sections 18.2, 18.3
Quick Quizzes 18.3, 18.5, 18.6
Examples 18.1–18.3
Active Figures 18.2, 18.6

Atoms Skill Objectives: To calculate half-life and to understand decay processes and nuclear reactions. Review Plan: Atoms:
Chapter 29, Sections 29.1, 29.2 Radioactive Decay:
Chapter 29, Sections 29.3–29.5
Examples 29.2, 29.5
Active Figures 29.6, 29.7
Chapter Problems 15, 19, 25, 31 Nuclear Reactions:
Chapter 29, Section 29.6
Quick Quiz 29.4
Example 29.6
Chapter Problems 35, 39

15 © Keith Kent/Science Photo Library/Photo Researchers, Inc.

This nighttime view of multiple bolts of lightning was photographed in Tucson, Arizona. During a thunderstorm, a high concentration of electrical charge in a thundercloud creates a higher-than-normal electric field between the thundercloud and the negatively charged Earth’s surface. This strong electric field creates an electric discharge—an enormous spark—between the charged cloud and the ground. Other discharges observed in the sky include cloud-to-cloud discharges and the more frequent intracloud discharges.

ELECTRIC FORCES AND ELECTRIC FIELDS Electricity is the lifeblood of technological civilization and modern society. Without it, we revert to the mid-nineteenth century: no telephones, no television, none of the household appliances that we take for granted. Modern medicine would be a fantasy, and due to the lack of sophisticated experimental equipment and fast computers—and especially the slow dissemination of information—science and technology would grow at a glacial pace. Instead, with the discovery and harnessing of electric forces and fields, we can view arrangements of atoms, probe the inner workings of the cell, and send spacecraft beyond the limits of the solar system. All this has become possible in just the last few generations of human life, a blink of the eye compared to the million years our kind spent foraging the savannahs of Africa. Around 700 B.C. the ancient Greeks conducted the earliest known study of electricity. It all began when someone noticed that a fossil material called amber would attract small objects after being rubbed with wool. Since then we have learned that this phenomenon is not restricted to amber and wool, but occurs (to some degree) when almost any two nonconducting substances are rubbed together. In this chapter we use the effect of charging by friction to begin an investigation of electric forces. We then discuss Coulomb’s law, which is the fundamental law of force between any two stationary charged particles. The concept of an electric field associated with charges is introduced and its effects on other charged particles described. We end with discussions of the Van de Graaff generator and Gauss’s law.

15.1

Properties of Electric Charges

15.2

Insulators and Conductors

15.3

Coulomb’s Law

15.4 The Electric Field 15.5

Electric Field Lines

15.6

Conductors in Electrostatic Equilibrium

15.7 The Millikan Oil-Drop Experiment 15.8

The Van de Graaff Generator

15.9

Electric Flux and Gauss’s Law

15.1 PROPERTIES OF ELECTRIC CHARGES After running a plastic comb through your hair, you will find that the comb attracts bits of paper. The attractive force is often strong enough to suspend the paper

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Chapter 15

Electric Forces and Electric Fields FIGURE 15.1 (a) A negatively charged rubber rod, suspended by a thread, is attracted to a positively charged glass rod. (b) A negatively charged rubber rod is repelled by another negatively charged rubber rod.

– –– – F F + + Glass + ++ + (a)

Image not available due to copyright restrictions

Like charges repel; unlike charges attract. R

Charge is conserved R

Rubber

Rubber F – –– – – –– –

– – Rubber F (b)

from the comb, defying the gravitational pull of the entire Earth. The same effect occurs with other rubbed materials, such as glass and hard rubber. Another simple experiment is to rub an inflated balloon against wool (or across your hair). On a dry day, the rubbed balloon will then stick to the wall of a room, often for hours. These materials have become electrically charged. You can give your body an electric charge by vigorously rubbing your shoes on a wool rug or by sliding across a car seat. You can then surprise and annoy a friend or coworker with a light touch on the arm, delivering a slight shock to both yourself and your victim. (If the coworker is your boss, don’t expect a promotion!) These experiments work best on a dry day because excessive moisture can facilitate a leaking away of the charge. Experiments also demonstrate that there are two kinds of electric charge, which Benjamin Franklin (1706–1790) named positive and negative. Figure 15.1 illustrates the interaction of the two charges. A hard rubber (or plastic) rod that has been rubbed with fur is suspended by a piece of string. When a glass rod that has been rubbed with silk is brought near the rubber rod, the rubber rod is attracted toward the glass rod (Fig. 15.1a). If two charged rubber rods (or two charged glass rods) are brought near each other, as in Figure 15.1b, the force between them is repulsive. These observations may be explained by assuming the rubber and glass rods have acquired different kinds of excess charge. We use the convention suggested by Franklin, where the excess electric charge on the glass rod is called positive and that on the rubber rod is called negative. On the basis of such observations, we conclude that like charges repel one another and unlike charges attract one another. Objects usually contain equal amounts of positive and negative charge; electrical forces between objects arise when those objects have net negative or positive charges. Nature’s basic carriers of positive charge are protons, which, along with neutrons, are located in the nuclei of atoms. The nucleus, about 10
15 m in radius, is surrounded by a cloud of negatively charged electrons about ten thousand times larger in extent. An electron has the same magnitude charge as a proton, but the opposite sign. In a gram of matter there are approximately 1023 positively charged protons and just as many negatively charged electrons, so the net charge is zero. Because the nucleus of an atom is held firmly in place inside a solid, protons never move from one material to another. Electrons are far lighter than protons and hence more easily accelerated by forces. Further, they occupy the outer regions of the atom. Consequently, objects become charged by gaining or losing electrons. Charge transfers readily from one type of material to another. Rubbing the two materials together serves to increase the area of contact, facilitating the transfer process. An important characteristic of charge is that electric charge is always conserved. Charge isn’t created when two neutral objects are rubbed together; rather, the objects become charged because negative charge is transferred from one object to the other. One object gains a negative charge while the other loses an equal amount of negative charge and hence is left with a net positive charge. When

15.2

a glass rod is rubbed with silk, as in Figure 15.2, electrons are transferred from the rod to the silk. As a result, the glass rod carries a net positive charge, the silk a net negative charge. Likewise, when rubber is rubbed with fur, electrons are transferred from the fur to the rubber. In 1909 Robert Millikan (1886–1953) discovered that if an object is charged, its charge is always a multiple of a fundamental unit of charge, designated by the symbol e. In modern terms, the charge is said to be quantized, meaning that charge occurs in discrete chunks that can’t be further subdivided. An object may have a charge of e, 2e, 3e, and so on, but never1 a fractional charge of 0.5e or 0.22e. Other experiments in Millikan’s time showed that the electron has a charge of
e and the proton has an equal and opposite charge of e. Some particles, such as a neutron, have no net charge. A neutral atom (an atom with no net charge) contains as many protons as electrons. The value of e is now known to be 1.602 19  10
19 C. (The SI unit of electric charge is the coulomb, or C.)

15.2 INSULATORS AND CONDUCTORS

Insulators and Conductors

– –

499

–

+ ++ ++ – + – –

FIGURE 15.2 When a glass rod is rubbed with silk, electrons are transferred from the glass to the silk. Because of conservation of charge, each electron adds negative charge to the silk, and an equal positive charge is left behind on the rod. Also, because the charges are transferred in discrete bundles, the charges on the two objects are e, 2e, 3e, and so on.

Substances can be classified in terms of their ability to conduct electric charge. In conductors, electric charges move freely in response to an electric force. All other materials are called insulators. Glass and rubber are insulators. When such materials are charged by rubbing, only the rubbed area becomes charged, and there is no tendency for the charge to move into other regions of the material. In contrast, materials such as copper, aluminum, and silver are good conductors. When such materials are charged in some small region, the charge readily distributes itself over the entire surface of the material. If you hold a copper rod in your hand and rub the rod with wool or fur, it will not attract a piece of paper. This might suggest that a metal can’t be charged. However, if you hold the copper rod with an insulator and then rub it with wool or fur, the rod remains charged and attracts the paper. In the first case, the electric charges produced by rubbing readily move from the copper through your body and finally to ground. In the second case, the insulating handle prevents the flow of charge to ground. Semiconductors are a third class of materials, and their electrical properties are somewhere between those of insulators and those of conductors. Silicon and germanium are well-known semiconductors that are widely used in the fabrication of a variety of electronic devices.

Rubber – – – – –– –– –– ––

+ + + + + + + +

Charging by Induction An object connected to a conducting wire or copper pipe buried in the Earth is said to be grounded. The Earth can be considered an infinite reservoir for electrons; is strong evidence for the existence of fundamental particles called quarks that have charges of e/3 or 2e/3. The charge is still quantized, but in units of e/3 rather than e. A more complete discussion of quarks and their properties is presented in Chapter 30.

1There

– – – – – – –

(a) Before

– –

–

–

– –

Charging by Conduction Consider a negatively charged rubber rod brought into contact with an insulated neutral conducting sphere. The excess electrons on the rod repel electrons on the sphere, creating local positive charges on the neutral sphere. On contact, some electrons on the rod are now able to move onto the sphere, as in Figure 15.3, neutralizing the positive charges. When the rod is removed, the sphere is left with a net negative charge. This process is referred to as charging by conduction. The object being charged in such a process (the sphere) is always left with a charge having the same sign as the object doing the charging (the rubber rod).

–

– – – – – –

(b) Contact

– – –

–

–

–

–

–

– –

–

–

(c) After breaking contact FIGURE 15.3 Charging a metallic object by conduction. (a) Just before contact, the negative rod repels the sphere’s electrons, inducing a localized positive charge. (b) After contact, electrons from the rod flow onto the sphere, neutralizing the local positive charges. (c) When the rod is removed, the sphere is left with a negative charge.

500

Chapter 15

Electric Forces and Electric Fields

+ – + + – – – – + + – – + – + + (a) + – + –

+ –

–

–

+ +

– – + – – – + – +

(b) + +

+ –

–

–

+

+ +

+

– – – –

+ (c) +

–

+ –

–

–

+ –

+

– +

+

+ – +

(d) + – +

+ – + – +

+

– + +

(e) FIGURE 15.4 Charging a metallic object by induction. (a) A neutral metallic sphere, with equal numbers of positive and negative charges. (b) The electrons on a neutral sphere are redistributed when a charged rubber rod is placed near the sphere. (c) When the sphere is grounded, some of its electrons leave it through the ground wire. (d) When the ground connection is removed, the sphere has excess positive charge that is nonuniformly distributed. (e) When the rod is removed, the remaining electrons redistribute uniformly and there is a net uniform distribution of positive charge on the sphere.

in effect, it can accept or supply an unlimited number of electrons. With this idea in mind, we can understand the charging of a conductor by a process known as induction. Consider a negatively charged rubber rod brought near a neutral (uncharged) conducting sphere that is insulated, so there is no conducting path to ground (Fig. 15.4). Initially the sphere is electrically neutral (Fig. 15.4a). When the negatively charged rod is brought close to the sphere, the repulsive force between the electrons in the rod and those in the sphere causes some electrons to move to the side of the sphere farthest away from the rod (Fig. 15.4b). The region of the sphere nearest the negatively charged rod has an excess of positive charge because of the migration of electrons away from that location. If a grounded conducting wire is then connected to the sphere, as in Figure 15.4c, some of the electrons leave the sphere and travel to ground. If the wire to ground is then removed (Fig. 15.4d), the conducting sphere is left with an excess of induced positive charge. Finally, when the rubber rod is removed from the vicinity of the sphere (Fig. 15.4e), the induced positive charge remains on the ungrounded sphere. Even though the positively charged atomic nuclei remain fixed, this excess positive charge becomes uniformly distributed over the surface of the ungrounded sphere because of the repulsive forces among the like charges and the high mobility of electrons in a metal. In the process of inducing a charge on the sphere, the charged rubber rod doesn’t lose any of its negative charge because it never comes in contact with the sphere. Furthermore, the sphere is left with a charge opposite that of the rubber rod. Charging an object by induction requires no contact with the object inducing the charge. A process similar to charging by induction in conductors also takes place in insulators. In most neutral atoms or molecules, the center of positive charge coincides with the center of negative charge. In the presence of a charged object, however, these centers may separate slightly, resulting in more positive charge on one side of the molecule than on the other. This effect is known as polarization. The realignment of charge within individual molecules produces an induced charge on the surface of the insulator, as shown in Figure 15.5a. This property explains why a balloon charged through rubbing will stick to an electrically neutral wall or why the comb you just used on your hair attracts tiny bits of neutral paper. QUICK QUIZ 15.1 A suspended object A is attracted to a neutral wall. It’s also attracted to a positively charged object B. Which of the following is true about object A? (a) It is uncharged. (b) It has a negative charge. (c) It has a positive charge. (d) It may be either charged or uncharged.

15.3 COULOMB’S LAW In 1785 Charles Coulomb (1736–1806) experimentally established the fundamental law of electric force between two stationary charged particles. An electric force has the following properties: 1. It is directed along a line joining the two particles and is inversely proportional to the square of the separation distance r, between them. 2. It is proportional to the product of the magnitudes of the charges,
q 1
and
q 2
, of the two particles. 3. It is attractive if the charges are of opposite sign and repulsive if the charges have the same sign. From these observations, Coulomb proposed the following mathematical form for the electric force between two charges:

15.3

+

– +

+

– + – +

+ Charged balloon

Induced charges

© 1968 Fundamental Photographs

+ +

501

FIGURE 15.5 (a) The charged object on the left induces charges on the surface of an insulator. (b) A charged comb attracts bits of paper because charges are displaced in the paper.

Wall – + – + – +

+

Coulomb’s Law

(b)

(a)

O Coulomb’s law

The magnitude of the electric force F between charges q 1 and q 2 separated by a distance r is given by F 5 ke

0 q1 0 0 q2 0

r2 where ke is a constant called the Coulomb constant.

[15.1]

Equation 15.1, known as Coulomb’s law, applies exactly only to point charges and to spherical distributions of charges, in which case r is the distance between the two centers of charge. Electric forces between unmoving charges are called electrostatic forces. Moving charges, in addition, create magnetic forces, studied in Chapter 19. The value of the Coulomb constant in Equation 15.1 depends on the choice of units. The SI unit of charge is the coulomb (C). From experiment, we know that the Coulomb constant in SI units has the value [15.2]

This number can be rounded, depending on the accuracy of other quantities in a given problem. We’ll use either two or three significant digits, as usual. The charge on the proton has a magnitude of e  1.6  10
19 C. Therefore, it would take 1/e  6.3  1018 protons to create a total charge of 1.0 C. Likewise, 6.3  1018 electrons would have a total charge of
1.0 C. Compare this charge with the number of free electrons in 1 cm3 of copper, which is on the order of 1023. Even so, 1.0 C is a very large amount of charge. In typical electrostatic experiments in which a rubber or glass rod is charged by friction, there is a net charge on the order of 10
6 C (  1 mC). Only a very small fraction of the total available charge is transferred between the rod and the rubbing material. Table 15.1 lists the charges and masses of the electron, proton, and neutron. When using Coulomb’s force law, remember that force is a vector quantity and must be treated accordingly. Active Figure 15.6a (page 502) shows the electric

CHARLES COULOMB (1736–1806)

TABLE 15.1

Charge and Mass of the Electron, Proton, and Neutron Particle Electron Proton Neutron

Photo courtesy of AIP Niels Bohr Library, E. Scott Barr Collection

ke  8.987 5  109 N  m2/C2

Charge (C) 10
19


1.60  1.60  10
19 0

Mass (kg) 9.11  10
31 1.67  10
27 1.67  10
27

Coulomb’s major contribution to science was in the field of electrostatics and magnetism. During his lifetime, he also investigated the strengths of materials and identified the forces that affect objects on beams, thereby contributing to the field of structural mechanics.

502

Chapter 15

r

Electric Forces and Electric Fields

+

F12

q2 + F21

q1

(a)

– q2 F12 + q1

F21 (b)

ACTIVE FIGURE 15.6 Two point charges separated by a distance r exert a force on each other given by Coulomb’s law. The force on q 1 is equal in magnitude and opposite in direction to the force on q 2. (a) When the charges are of the same sign, the force is repulsive. (b) When the charges are of opposite sign, the force is attractive.

force of repulsion between two positively charged particles. Like other forces, elecS S tric forces obey Newton’s third law; hence, the forces F and F are equal in mag12 21 S nitude but opposite in direction. S(The notation F 12 denotes the force exerted by particle 1 on particle 2; likewise, F 21 is the force exerted by particle 2 on particle 1.) From Newton’s third law, F 12 and F 21 are always equal regardless of whether q 1 and q 2 have the same magnitude. QUICK QUIZ 15.2 Object A has a charge of 2 mC, and object B has a charge of 6 mC. Which statement is true? S

S

S

S

S

S

(a) F AB 5 23F BA (b) F AB 5 2F BA (c) 3F AB 5 2F BA The Coulomb force is similar to the gravitational force. Both act at a distance without direct contact. Both are inversely proportional to the distance squared, with the force directed along a line connecting the two bodies. The mathematical form is the same, with the masses m1 and m 2 in Newton’s law replaced by q 1 and q 2 in Coulomb’s law and with Newton’s constant G replaced by Coulomb’s constant ke . There are two important differences: (1) electric forces can be either attractive or repulsive, but gravitational forces are always attractive, and (2) the electric force between charged elementary particles is far stronger than the gravitational force between the same particles, as the next example shows.

EXAMPLE 15.1 Forces in a Hydrogen Atom Goal

Contrast the magnitudes of an electric force and a gravitational force.

Problem The electron and proton of a hydrogen atom are separated (on the average) by a distance of about 5.3  10
11 m. (a) Find the magnitudes of the electric force and the gravitational force that each particle exerts on the other, and the ratio of the electric force Fe to the gravitational force Fg . (b) Compute the acceleration caused by the electric force of the proton on the electron. Repeat for the gravitational acceleration. Strategy Solving this problem is just a matter of substituting known quantities into the two force laws and then finding the ratio.

Solution (a) Compute the magnitudes of the electric and gravitational forces, and find the ration Fe /Fg . Substitute
q 1

q 2
 e and the distance into Coulomb’s law to find the electric force:

Fe 5 k e

0e02 r2

5 a8.99 3 109

N # m2 1 1.6 3 10219 C 2 2 b 1 5.3 3 10211 m 2 2 C2

 8.2 3 1028 N Substitute the masses and distance into Newton’s law of gravity to find the gravitational force:

Fg 5 G

m em p r2

5 a6.67 3 10211  3.6 3 10247 N

Find the ratio of the two forces:

Fe 5 2.3 3 1039 Fg

231 227 N # m2 1 9.11 3 10 kg 2 1 1.67 3 10 kg 2 b 1 5.3 3 10211 m 2 2 kg2

15.3

Coulomb’s Law

503

(b) Compute the acceleration of the electron caused by the electric force. Repeat for the gravitational acceleration. Use Newton’s second law and the electric force found in part (a):

m e a e 5 Fe

S

ae 5

Use Newton’s second law and the gravitational force found in part (a):

me a g 5 Fg

S

ag 5

Fe 8.2 3 10 28 N 5 5 9.0 3 1022 m/s 2 me 9.11 3 10 231 kg Fg me

5

3.6 3 10 247 N 5 4.0 3 10 217 m/s 2 9.11 3 10 231 kg

Remarks The gravitational force between the charged constituents of the atom is negligible compared with the electric force between them. The electric force is so strong, however, that any net charge on an object quickly attracts nearby opposite charges, neutralizing the object. As a result, gravity plays a greater role in the mechanics of moving objects in everyday life. QUESTION 15.1 If the distance between two charges is doubled, by what factor is the magnitude of the electric force changed? EXERCISE 15.1 Find the magnitude of the electric force between two protons separated by 1 femtometer (10
15 m), approximately the distance between two protons in the nucleus of a helium atom. The answer may not appear large, but if not for the strong nuclear force, the two protons would accelerate in opposite directions at over 1  1029 m/s2! Answer 2  102 N

The Superposition Principle When a number of separate charges act on the charge of interest, each exerts an electric force. These electric forces can all be computed separately, one at a time, then added as vectors. This is another example of the superposition principle. The following example illustrates this procedure in one dimension.

EXAMPLE 15.2 Finding Electrostatic Equilibrium Goal

Apply Coulomb’s law in one dimension. 2.0 m

Problem Three charges lie along the x-axis as in Figure 15.7. The positive charge q 1  15 mC is at x  2.0 m, and the positive charge q 2  6.0 mC is at the origin. Where must a negative charge q 3 be placed on the x-axis so that the resultant electric force on it is zero?

+ q2

Strategy If q 3 is to the right orSleft of the other two charges, the net force on S q 3 can’t be zero because then F 13 and F 23 act in the same direction. ConseS S quently, q 3 must lie between the two other charges. Write F 13 and F 23 in terms of the unknown coordinate position x, then sum them and set them equal to zero, solving for the unknown. The solution can be obtained with the quadratic formula. Solution S Write the x-component of F 13: S

F13x 5 1k e

Write the x-component of F 23:

F23x 5 2k e

Set the sum equal to zero:

ke

x

2.0 m – x

– F23 q 3

F13

+ q1

FIGURE 15.7 (Example 15.2) Three point charges are placed along the x-axis. The charge q 3 is negative, whereas q 1 and q 2 are positive. If S the resultant force on q 3 is zero, the force F 13 exerted by q 1 on q 3 must be equalSin magnitude and opposite the force F 23 exerted by q 2 on q 3.

1 15 3 1026 C 2 0 q3 0 1 2.0 m 2 x 2 2

1 6.0 3 1026 C 2 0 q3 0 x2

1 15 3 1026 C 2 0 q3 0 1 2.0 m 2 x 2 2

x

2 ke

1 6.0 3 1026 C 2 0 q3 0 x2

50

504

Chapter 15

Electric Forces and Electric Fields

Cancel ke , 10
6, and q 3 from the equation and rearrange terms (explicit significant figures and units are temporarily suspended for clarity):

(1) 6(2
x)2  15x 2

Put this equation into standard quadratic form, ax 2  bx  c  0:

6(4
4x  x 2)  15x 2

Apply the quadratic formula:

x5

Only the positive root makes sense:

x  0.77 m

:

2(4
4x  x 2)  5x 2

3x 2  8x
8  0 28 6 !64 2 1 4 2 1 3 2 1 28 2 24 6 2!10 5 # 3 2 3

Remarks Notice that physical reasoning was required to choose between the two possible answers for x, which is nearly always the case when quadratic equations are involved. Use of the quadratic formula could have been avoided by taking the square root of both sides of Equation (1), however this short cut is often unavailable. QUESTION 15.2 If q 1 has the same magnitude as before but is negative, in what region along the x-axis would it be possible for the net electric force on q 3 to be zero? (a) x < 0 (b) 0 < x < 2 m (c) 2 m < x EXERCISE 15.2 Three charges lie along the x-axis. A positive charge q1  10.0 mC is at x  1.00 m, and a negative charge q2 
2.00 mC is at the origin. Where must a positive charge q 3 be placed on the x-axis so that the resultant force on it is zero? Answer x 
0.809 m

EXAMPLE 15.3 A Charge Triangle Goal

Apply Coulomb’s law in two dimensions.

y F13

Problem Consider three point charges at the corners of a triangle, as shown in Figure 15.8, where q 1  6.00  10
9 C, q 2 
2.00  10
9 C, andS q 3  5.00  10
9 C. (a) Find the components of the force F 23 exerted by q2 on q3. S (b) Find the components of the force F 13 exerted by q 1 on q 3. (c) Find the resultant force on q 3, in terms of components and also in terms of magnitude and direction. Strategy Coulomb’s law gives the magnitude of each force, which can be split with right-triangle trigonometry into x- and y-components. Sum the vectors componentwise and then find the magnitude and direction of the resultant vector.

q2 –

S

F23

+ q3

3.00 m

q1 +

37°

F 13 sin 37°

F 13 cos 37°

5.00 m

x S

FIGURE 15.8 (Example 15.3) The force exerted by q 1 on q 3 is F 13 . S S The force exerted by q 2 on q 3 is F 23. The resultant force F 3 exerted S S on q 3 is the vector sum F 13  F 23.

Solution (a) Find the components of the force exerted by q 2 on q 3. Find the magnitude of F 23 with Coulomb’s law:

4.00 m

F23 5 k e

0 q2 0 0 q3 0 r2

5 1 8.99 3 109 N # m2 /C2 2 F23 5 5.62 3 1029 N

1 2.00 3 1029 C 2 1 5.00 3 1029 C 2 1 4.00 m 2 2

15.4 S

Because F 23 is horizontal and points in the negative x-direction, the negative of the magnitude gives the x-component, and the y-component is zero:

S

505

F23x 5 25.62 3 1029 N F23y 5 0

(b) Find the components of the force exerted by q 1 on q 3. Find the magnitude of F 13:

The Electric Field

F13 5 k e

0 q1 0 0 q3 0 r2

5 1 8.99 3 109 N # m2 /C2 2

1 6.00 3 1029 C 2 1 5.00 3 1029 C 2 1 5.00 m 2 2

F13 5 1.08 3 1028 N Use the given triangle to find the comS ponents of F 13:

F13x 5 F13 cos u 5 1 1.08 3 1028 N 2 cos 1 37° 2  8.63 3 1029 N

F13y 5 F13 sin u 5 1 1.08 3 1028 N 2 sin 1 37° 2  6.50 3 1029 N (c) Find the components of the resultant vector. Sum the x-components to find the resultant Fx:

Fx 
5.62  10
9 N  8.63  10
9 N

Sum the y-components to find the resultant Fy:

Fy  0  6.50  10
9 N  6.50 3 1029 N

Find the magnitude of the resultant force on the charge q 3, using the Pythagorean theorem:

 3.01 3 1029 N

0 F 0 5 "Fx2 1 Fy 2 S

5 " 1 3.01 3 1029 N 2 2 1 1 6.50 3 1029 N 2 2  7.16 3 1029 N

Find the angle the resultant force makes with respect to the positive x-axis:

Fy 6.50 3 1029 N u 5 tan21 a b 5 tan21 a b 65.2° Fx 3.01 3 1029 N

Remark The methods used here are just like those used with Newton’s law of gravity in two dimensions. QUESTION 15.3 Without actually calculating the electric force on q2, determine the quadrant into which the electric force vector points. EXERCISE 15.3 Using the same triangle, find the vector components of the electric force on q 1 and the vector’s magnitude and direction. Answers Fx 
8.63  10
9 N, Fy  5.50  10
9 N, F  1.02  10
8 N, u  147°

15.4 THE ELECTRIC FIELD The gravitational force and the electrostatic force are both capable of acting through space, producing an effect even when there isn’t any physical contact between the objects involved. Field forces can be discussed in a variety of ways, but an approach developed by Michael Faraday (1791–1867) is the most practical. In this approach an electric field is said to exist in the region of space around a charged object. The electric field exerts an electric force on any other charged object within the field. This differs from the Coulomb’s law concept of a force

506

Chapter 15

Electric Forces and Electric Fields

Q + + + + +

+ + + +

+ +

q0 + E

+

+ + Test charge

FIGURE 15.9 A small object with a positive charge q 0 placed near an object with a larger positive charge S Q is subject to an electric field E directed as shown. The magnitude of the electric field at the location of q 0 is defined as the electric force on q 0 divided by the charge q 0.

exerted at a distance in that the force is now exerted by something—the field— that is in the same location as the charged object. Figure 15.9 shows an object with a small positive charge q 0 placed near a second object with a much larger positive charge Q. S

Q at the location of a small “test” The electric field E produced by a charge S charge q 0 is defined as the electric force F exerted by Q on q 0 divided by the test charge q 0: S

S

F q0

E;

[15.3]

SI unit: newton per coulomb (N/C) Conceptually and experimentally, the test charge q 0 is required to be very small (arbitrarily small, in fact), so it doesn’t cause any significant rearrangement of the S charge creating the electric field E. Mathematically, however, the size of the test charge makes no difference: the calculation comes out the same, regardless. In view of this, using q 0  1 C in Equation 15.3 can be convenient if not rigorous. When a positive test charge is used, the electric field always has the same direction as the electric force on the test charge, which follows from Equation 15.3. Hence, in Figure 15.9, the direction of the electric field is horizontal and to the right. The electric field at point A in Figure 15.10a is vertical and downward because at that point a positive test charge would be attracted toward the negatively charged sphere. Once the electric field due to a given arrangement of charges is known at some point, the force on any particle with charge q placed at that point can be calculated from a rearrangement of Equation 15.3: S

S

[15.4]

F 5 qE

Here q 0 has been replaced by q, which need not be a S mere test charge. As shown in Active Figure 15.11, the direction of E is the direction of the force that acts on a positive test charge q 0 placed in the field. We say that an electric field exists at a point if a test charge at that point is subject to an electric force there. Consider a point charge q located a distance r from a test charge q 0. According to Coulomb’s law, the magnitude of the electric force of the charge q on the test charge is F 5 ke

0 q 0 0 q0 0

[15.5]

r2

Because the magnitude of the electric field at the position of the test charge is defined as E  F/q 0, we see that the magnitude of the electric field due to the charge q at the position of q 0 is E 5 ke

0q0

[15.6]

r2

Equation 15.6 points out an important property of electric fields that makes them useful quantities for describing electrical phenomena. As the equation indicates, FIGURE 15.10 (a) The electric field at A due to the negatively charged sphere is downward, toward the negative charge. (b) The electric field at P due to the positively charged conducting sphere is upward, away from the positive charge. (c) A test charge q 0 placed at P will cause a rearrangement of charge on the sphere unless q 0 is very small compared with the charge on the sphere.

E

A

q0

P

+

P

E – – –

– – –

– (a)

+ – –

–

+ +

+ + +

+

+ + +

(b)

+ +

+ + + + (c)

+ +

15.4

an electric field at a given point depends only on the charge q on the object setting up the field and the distance r from that object to a specific point in space. As a result, we can say that an electric field exists at point P in Active Figure 15.11 whether or not there is a test charge at P. The principle of superposition holds when the electric field due to a group of point charges is calculated. We first use Equation 15.6 to calculate the electric field produced by each charge individually at a point and then add the electric fields together as vectors. It’s also important to exploit any symmetry of the charge distribution. For example, if equal charges are placed at x  a and at x 
a, the electric field is zero at the origin, by symmetry. Similarly, if the x -axis has a uniform distribution of positive charge, it can be guessed by symmetry that the electric field points away from the x -axis and is zero parallel to that axis. QUICK QUIZ 15.3 A test charge of 3 mC is at a point P where the electric field due to other charges is directed to the right and has a magnitude of 4  106 N/C. If the test charge is replaced with a charge of
3 mC, the electric field at P (a) has the same magnitude as before, but changes direction, (b) increases in magnitude and changes direction, (c) remains the same, or (d) decreases in magnitude and changes direction.

The Electric Field

q0

507

E

P q +

r (a)

q0 P q –

E (b)

ACTIVE FIGURE 15.11 A test charge q 0 at P is a distance r from a point charge q. (a) If q is positive, the electric field at P points radially outwards from q. (b) If q is negative, the electric field at P points radially inwards toward q.

QUICK QUIZ 15.4 A circular ring of charge of radius b has a total charge q uniformly distributed around it. Find the magnitude to the electric field in the center of the ring. (a) 0 (b) keq/b 2 (c) keq 2/b 2 (d) keq 2/b (e) None of these answers is correct. QUICK QUIZ 15.5 A “free” electron and a “free” proton are placed in an identical electric field. Which of the following statements are true? (a) Each particle is acted upon by the same electric force and has the same acceleration. (b) The electric force on the proton is greater in magnitude than the electric force on the electron, but in the opposite direction. (c) The electric force on the proton is equal in magnitude to the electric force on the electron, but in the opposite direction. (d) The magnitude of the acceleration of the electron is greater than that of the proton. (e) Both particles have the same acceleration.

EXAMPLE 15.4 Goal

Electrified Oil

Use electric forces and fields together with Newton’s second law in a one-dimensional problem.

Problem Tiny droplets of oil acquire a small negative charge while dropping through a vacuum (pressure  0) in an experiment. An electric field of magnitude 5.92  104 N/C points straight down. (a) One particular droplet is observed to remain suspended against gravity. If the mass of the droplet is 2.93  10
15 kg, find the charge carried by the droplet. (b) Another droplet of the same mass falls 10.3 cm from rest in 0.250 s, again moving through a vacuum. Find the charge carried by the droplet. S

Strategy We use Newton’s second law with both gravitational and electric forces. In both parts the electric field E is pointing down, taken as the negative direction, as usual. In part (a) the acceleration is equal to zero. In part (b) the acceleration is uniform, so the kinematic equations yield the acceleration. Newton’s law can then be solved for q. Solution (a) Find the charge on the suspended droplet. Apply Newton’s second law to the droplet in the vertical direction:

(1) ma 

 F 
mg  Eq

508

Chapter 15

Electric Forces and Electric Fields

E points downward, hence is negative. Set a  0 in Equation (1) and solve for q:

q5

mg E

5

1 2.93 3 10215 kg 2 1 9.80 m/s 2 2 25.92 3 104 N/C

 24.85 3 10219 C (b) Find the charge on the falling droplet. Use the kinematic displacement equation to find the acceleration:

Dy 5 12at 2 1 v 0t

Substitute y 
0.103 m, t  0.250 s, and v 0  0:

20.103 m 5 12 a 1 0.250 s 2 2

Solve Equation (1) for q and substitute:

q5 5

S

a 5 23.30 m/s 2

m1a 1 g2 E

1 2.93 3 10215 kg 2 1 23.30 m/s 2 1 9.80 m/s 2 2 25.92 3 104 N/C

 23.22 3 10219 C

Remarks This example exhibits features similar to the Millikan Oil-Drop experiment discussed in Section 15.7, which determined the value of the fundamental electric charge e. Notice that in both parts of the example, the charge is very nearly a multiple of e. QUESTION 15.4 What would be the acceleration of the oil droplet in part (a) if the electric field suddenly reversed direction without changing in magnitude? EXERCISE 15.4 Suppose a droplet of unknown mass remains suspended against gravity when E 
2.70  105 N/C. What is the minimum mass of the droplet? Answer 4.41  10
15 kg

PROBLEM -SOLVING STRATEGY CALCULATING ELECTRIC FORCES AND FIELDS

The following procedure is used to calculate electric forces. The same procedure can be used to calculate an electric field, a simple matter of replacing the charge of interest, q, with a convenient test charge and dividing by the test charge at the end: 1. Draw a diagram of the charges in the problem. 2. Identify the charge of interest, q, and circle it. 3. Convert all units to SI, with charges in coulombs and distances in meters, so as to be consistent with the SI value of the Coulomb constant ke . 4. Apply Coulomb’s law. For each charge Q, find the electric force on the charge of interest, q. The magnitude of the force can be found using Coulomb’s law. The vector direction of the electric force is along the line of the two charges, directed away from Q if the charges have the same sign, toward Q if the charges have the opposite sign. Find the angle u this vector makes with the positive x-axis. The x-component of the electric force exerted by Q on q will be F cos u, and the y-component will be F sin u. 5. Sum all the x-components, getting the x-component of the resultant electric force.

15.4

The Electric Field

509

6. Sum all the y-components, getting the y-component of the resultant electric force. 7. Use the Pythagorean theorem and trigonometry to find the magnitude and direction of the resultant force if desired.

EXAMPLE 15.5 Electric Field Due to Two Point Charges Goal Use the superposition principle to calculate the electric field due to two point charges. Problem Charge q 1  7.00 mC is at the origin, and charge q 2 
5.00 mC is on the x-axis, 0.300 m from the origin (Fig. 15.12). (a) Find the magnitude and direction of the electric field at point P, which has coordinates (0, 0.400) m. (b) Find the force on a charge of 2.00  10
8 C placed at P. Strategy Follow the problem-solving strategy, finding the electric field at point P due to each individual charge in terms of x- and y-components, then adding the components of each type to get the x- and y-components of the resultant electric field at P. The magnitude of the force in part (b) can be found by simply multiplying the magnitude of the electric field by the charge.

y E1 E

φ P

θ E2

0.400 m

0.500 m

θ + q1

Solution (a) Calculate the electric field at P. S

Find the magnitude of E1 with Equation 15.6:

–

0.300 m

E1 5 k e

0 q1 0 r

2 1

x

q2

FIGURE 15.12 (Example S 15.5) The resultant electric fi eld ES at P S equals the vector sum E 1  E 2, where S E 1 is the field due to the positive S charge q 1 and E 2 is the field due to the negative charge q 2.

5 1 8.99 3 109 N # m2 /C2 2

1 7.00 3 1026 C 2 1 0.400 m 2 2

5 3.93 3 105 N/C S

The vector E1 is vertical, making an angle of 90° with respect to the positive x -axis. Use this fact to find its components: S

Next, find the magnitude of E2, again with Equation 15.6: S

Obtain the x-component of E2, using the triangle in Figure 15.12 to find cos u:

E1x  E1 cos (90°)  0 E1y  E1 sin (90°)  3.93  105 N/C

E2 5 k e

0 q2 0 r 22

5 1 8.99 3 109 N # m2 /C2 2

1 5.00 3 1026 C 2 1 0.500 m 2 2

5 1.80 3 105 N/C cos u 5

adj hyp

5

0.300 5 0.600 0.500

E 2x  E 2 cos u  (1.80  105 N/C)(0.600)  1.08  105 N/C

Obtain the y-component in the same way, but a minus sign has to be provided for sin u because this component is directed downwards:

sin u 5

opp hyp

5

0.400 5 0.800 0.500

E 2y  E 2 sin u  (1.80  105 N/C)(
0.800) 
1.44  105 N/C

Sum the x -components to get the x -component of the resultant vector:

Ex  E1x  E 2x  0  1.08  105 N/C  1.08  105 N/C

510

Chapter 15

Electric Forces and Electric Fields

Sum the y -components to get the y-component of the resultant vector: Use the Pythagorean theorem to find the magnitude of the resultant vector: The inverse tangent function yields the direction of the resultant vector:

Ey  E1y  E 2y  0  3.93  105 N/C
1.44  105 N/C Ey  2.49  105 N/C E 5 "Ex2 1 Ey 2 5 2.71 3 105 N/C f 5 tan21 a

Ey Ex

b 5 tan21 a

2.49 3 105 N/C b  66.6° 1.08 3 105 N/C

(b) Find the force on a charge of 2.00  10
8 C placed at P. Calculate the magnitude of the force (the direcS tion is the same as that of E because the charge is positive):

F  Eq  (2.71  105 N/C)(2.00  10
8 C)  5.42 3 1023 N

Remarks There were numerous steps to this problem, but each was very short. When attacking such problems, it’s important to focus on one small step at a time. The solution comes not from a leap of genius, but from the assembly of a number of relatively easy parts. QUESTION 15.5 Suppose q 2 were moved slowly to the right. What would happen to the angle f? EXERCISE 15.5 (a) Place a charge of
7.00 mC at point P and find the magnitude and direction of the electric field at the location of q 2 due to q 1 and the charge at P. (b) Find the magnitude and direction of the force on q 2. Answer (a) 5.84  105 N/C, f  20.2 (b) F  2.92 N, f  200.

15.5

ELECTRIC FIELD LINES

A convenient aid for visualizing electric field patterns is to draw lines pointing in the direction of the electric field vector at any point. These lines, introduced by Michael Faraday and called electric field lines, are related to the electric field in any region of space in the following way: S

1. The electric field vector E is tangent to the electric field lines at each point. 2. The number of lines per unit area through a surface perpendicular to the lines is proportional to the strength of the electric field in a given region. S

Note that E is large when the field lines are close together and small when the lines are far apart. Figure 15.13a shows some representative electric field lines for a single positive point charge. This two-dimensional drawing contains only the field lines that lie in the plane containing the point charge. The lines are actually directed radially outward from the charge in all directions, somewhat like the quills of an angry porcupine. Because a positive test charge placed in this field would be repelled by the charge q, the lines are directed radially away from the positive charge. The electric field lines for a single negative point charge are directed toward the charge (Fig. 15.13b) because a positive test charge is attracted by a negative charge. In either case the lines are radial and extend all the way to infinity. Note that the lines are closer together as they get near the charge, indicating that the strength of the field is increasing. Equation 15.6 verifies that this is indeed the case. The rules for drawing electric field lines for any charge distribution follow directly from the relationship between electric field lines and electric field vectors:

15.5

q

+

–

(a)

Electric Field Lines

511

–q Image not available due to copyright restrictions

(b)

FIGURE 15.13 The electric field lines for a point charge. (a) For a positive point charge, the lines radiate outward. (b) For a negative point charge, the lines converge inward. Note that the figures show only those field lines which lie in the plane containing the charge.

1. The lines for a group of point charges must begin on positive charges and end on negative charges. In the case of an excess of charge, some lines will begin or end infinitely far away. 2. The number of lines drawn leaving a positive charge or ending on a negative charge is proportional to the magnitude of the charge. 3. No two field lines can cross each other. Figure 15.14 shows the beautifully symmetric electric field lines for two point charges of equal magnitude but opposite sign. This charge configuration is called an electric dipole. Note that the number of lines that begin at the positive charge must equal the number that terminate at the negative charge. At points very near either charge, the lines are nearly radial. The high density of lines between the charges indicates a strong electric field in this region. Figure 15.15 (page 512) shows the electric field lines in the vicinity of two equal positive point charges. Again, close to either charge the lines are nearly radial. The same number of lines emerges from each charge because the charges are equal in magnitude. At great distances from the charges, the field is approximately equal to that of a single point charge of magnitude 2q. The bulging out of the electric field lines between the charges reflects the repulsive nature of the electric force between like charges. Also, the low density of field lines between the charges indicates a weak field in this region, unlike the dipole. Finally, Active Figure 15.16 (page 512) is a sketch of the electric field lines associated with the positive charge 2q and the negative charge
q. In this case the number of lines leaving charge 2q is twice the number terminating on charge
q. Hence, only half of the lines that leave the positive charge end at the negative charge. The remaining half terminate on negative charges that we assume to be located at infinity. At great distances from the charges (great compared with the charge separation), the electric field lines are equivalent to those of a single charge q.

TIP 15.1 Electric Field Lines Aren’t Paths of Particles Electric field lines are not material objects. They are used only as a pictorial representation of the electric field at various locations. Except in special cases, they do not represent the path of a charged particle released in an electric field.

FIGURE 15.14 (a) The electric field lines for two equal and opposite point charges (an electric dipole). Note that the number of lines leaving the positive charge equals the number terminating at the negative charge.

+

–

(a)

Image not available due to copyright restrictions

512

Chapter 15

Electric Forces and Electric Fields

B

A

+

C

+

Image not available due to copyright restrictions

+2q

+

–

–q

(a) ACTIVE FIGURE 15.16 The electric field lines for a point charge of 2q and a second point charge of
q. Note that two lines leave the charge 2q for every line that terminates on
q.

FIGURE 15.15 (a) The electric field lines for two positive point charges. The points A, B, and C are discussed in Quick Quiz 15.6.

QUICK QUIZ 15.6 Rank the magnitudes of the electric field at points A, B, and C in Figure 15.15, with the largest magnitude first. (a) A, B, C (b) A, C, B (c) C, A, B (d) The answer can’t be determined by visual inspection.

APPLYING PHYSICS 15.1

MEASURING ATMOSPHERIC ELECTRIC FIELDS

The electric field near the surface of the Earth in fair weather is about 100 N/C downward. Under a thundercloud, the electric field can be very large, on the order of 20 000 N/C. How are these electric fields measured? Explanation A device for measuring these fields is called the field mill. Figure 15.17 shows the fundamental components of a field mill: two metal plates parallel to the ground. Each plate is connected to ground with a wire, with an ammeter (a low-resistance device for measuring the flow of charge, to be discussed in Section 19.6) in one path. Consider first just the lower plate. Because it’s connected to ground and the ground carries a negative charge, the plate is negatively charged. The electric field lines are therefore directed downward, ending on the plate as in Figure 15.17a. Now imagine that the upper plate

A

is suddenly moved over the lower plate, as in Figure 15.17b. This plate is also connected to ground and is also negatively charged, so the field lines now end on the upper plate. The negative charges in the lower plate are repelled by those on the upper plate and must pass through the ammeter, registering a flow of charge. The amount of charge that was on the lower plate is related to the strength of the electric field. In this way, the flow of charge through the ammeter can be calibrated to measure the electric field. The plates are normally designed like the blades of a fan, with the upper plate rotating so that the lower plate is alternately covered and uncovered. As a result, charges flow back and forth continually through the ammeter, and the reading can be related to the electric field strength.

A

(a)

(b)

FIGURE 15.17 (Applying Physics 15.1) In (a) electric field lines end on negative charges on the lower plate. In (b) the second plate is moved above the lower plate. Electric field lines now end on the upper plate, and the negative charges in the lower plate are repelled through the ammeter.

15.6

Conductors in Electrostatic Equilibrium

513

15.6 CONDUCTORS IN ELECTROSTATIC EQUILIBRIUM A good electric conductor like copper, although electrically neutral, contains charges (electrons) that aren’t bound to any atom and are free to move about within the material. When no net motion of charge occurs within a conductor, the conductor is said to be in electrostatic equilibrium. An isolated conductor (one that is insulated from ground) has the following properties: 1. The electric field is zero everywhere inside the conducting material. 2. Any excess charge on an isolated conductor resides entirely on its surface. 3. The electric field just outside a charged conductor is perpendicular to the conductor’s surface. 4. On an irregularly shaped conductor, the charge accumulates at sharp points, where the radius of curvature of the surface is smallest. The first property can be understood by examining what would happen if it were not true. If there were an electric field inside a conductor, the free charge there would move and a flow of charge, or current, would be created. If there were a net movement of charge, however, the conductor would no longer be in electrostatic equilibrium. Property 2 is a direct result of the 1/r 2 repulsion between like charges described by Coulomb’s law. If by some means an excess of charge is placed inside a conductor, the repulsive forces between the like charges push them as far apart as possible, causing them to quickly migrate to the surface. (We won’t prove it here, but the excess charge resides on the surface because Coulomb’s law is an inversesquare law. With any other power law, an excess of charge would exist on the surface, but there would be a distribution of charge, of either the same or opposite sign, inside the conductor.) Property 3 can be understood by again considering what would happen if it were not true. If the electric field in Figure 15.18a were not perpendicular to the surface, it would have a component along the surface, which would cause the free charges of the conductor to move (to the left in the figure). If the charges moved, however, a current would be created and the conductor would no longer be in elecS trostatic equilibrium. Therefore, E must be perpendicular to the surface. To see why property 4 must be true, consider Figure 15.19a (page 514), which shows a conductor that is fairly flat at one end and relatively pointed at the other. Any excess charge placed on the object moves to its surface. Figure 15.19b shows the forces between two such charges at the flatter end of the object. These forces are predominantly directed parallel to the surface, so the charges move apart until repulsive forces from other nearby charges establish an equilibrium. At the sharp end, however, the forces of repulsion between two charges are directed predominantly away from the surface, as in Figure 15.19c. As a result, there is less tendency for the charges to move apart along the surface here, and the amount of charge

Image not available due to copyright restrictions

– – (a)

conductor

Image not available due to copyright restrictions

FIGURE 15.18 (a) Negative charges at the surface of a conductor. If the electric field were at an angle to the surface, as shown, an electric force would be exerted on the charges along the surface and they would move to the left. Because the conductor is assumed to be in electrostatic equiS librium, E cannot have a component along the surface and hence must be perpendicular to it.

E

F

O Properties of an isolated

514

Chapter 15

Electric Forces and Electric Fields

FIGURE 15.19 (a) A conductor with a flatter end A and a relatively sharp end B. Excess charge placed on this conductor resides entirely at its surface and is distributed so that (b) there is less charge per unit area on the flatter end and (c) there is a large charge per unit area on the sharper end.

– – – – B A

B (a)

– –– – –– 0

(a)

– – + + –+ –+ –+ –+ –+ + – (b)

– – – – –

– – – – (d)

+ + – –

–

–

(c)

–

– –– – ––

– + – + +– +– +– +– + + – –

– – – –

–

–

–

–

–

–

– – – –

0

– – – –

0

–

–

– – – –

0

FIGURE 15.20 An experiment showing that any charge transferred to a conductor resides on its surface in electrostatic equilibrium. The hollow conductor is insulated from ground, and the small metal ball is supported by an insulating thread.

APPLICATION Lightning Rods

A (b)

(c)

per unit area is greater than at the flat end. The cumulative effect of many such outward forces from nearby charges at the sharp end produces a large resultant force directed away from the surface that can be great enough to cause charges to leap from the surface into the surrounding air. Many experiments have shown that the net charge on a conductor resides on its surface. One such experiment was first performed by Michael Faraday and is referred to as Faraday’s ice-pail experiment. Faraday lowered a metal ball having a negative charge at the end of a silk thread (an insulator) into an uncharged hollow conductor insulated from ground, a metal ice-pail as in Figure 15.20a. As the ball entered the pail, the needle on an electrometer attached to the outer surface of the pail was observed to deflect. (An electrometer is a device used to measure charge.) The needle deflected because the charged ball induced a positive charge on the inner wall of the pail, which left an equal negative charge on the outer wall (Fig. 15.20b). Faraday next touched the inner surface of the pail with the ball and noted that the deflection of the needle did not change, either when the ball touched the inner surface of the pail (Fig. 15.20c) or when it was removed (Fig. 15.20d). Further, he found that the ball was now uncharged because when it touched the inside of the pail, the excess negative charge on the ball had been drawn off, neutralizing the induced positive charge on the inner surface of the pail. In this way Faraday discovered the useful result that all the excess charge on an object can be transferred to an already charged metal shell if the object is touched to the inside of the shell. As we will see, this result is the principle of operation of the Van de Graaff generator. Faraday concluded that because the deflection of the needle in the electrometer didn’t change when the charged ball touched the inside of the pail, the positive charge induced on the inside surface of the pail was just enough to neutralize the negative charge on the ball. As a result of his investigations, he concluded that a charged object suspended inside a metal container rearranged the charge on the container so that the sign of the charge on its inside surface was opposite the sign of the charge on the suspended object. This produced a charge on the outside surface of the container of the same sign as that on the suspended object. Faraday also found that if the electrometer was connected to the inside surface of the pail after the experiment had been run, the needle showed no deflection. Thus, the excess charge acquired by the pail when contact was made between ball and pail appeared on the outer surface of the pail. If a metal rod having sharp points is attached to a house, most of any charge on the house passes through these points, eliminating the induced charge on the house produced by storm clouds. In addition, a lightning discharge striking the house passes through the metal rod and is safely carried to the ground through wires leading from the rod to the Earth. Lightning rods using this principle were first developed by Benjamin Franklin. Some European countries couldn’t accept the fact that such a worthwhile idea could have originated in the New World, so they “improved” the design by eliminating the sharp points!

15.7

APPLYING PHYSICS 15.2

515

CONDUCTORS AND FIELD LINES

Suppose a point charge Q is in empty space. Wearing rubber gloves, you proceed to surround the charge with a concentric spherical conducting shell. What effect does that have on the field lines from the charge?

conductor, so the electric field inside the conductor becomes zero. This means the field lines originating on the Q charge now terminate on the negative charges. The movement of the negative charges to the inner surface of the sphere leaves a net charge of Q on the outer surface of the sphere. Then the field lines outside the sphere look just as before: the only change, overall, is the absence of field lines within the conductor.

Explanation When the spherical shell is placed around the charge, the charges in the shell rearrange to satisfy the rules for a conductor in equilibrium. A net charge of
Q moves to the interior surface of the

APPLYING PHYSICS 15.3

The Millikan Oil-Drop Experiment

DRIVER SAFET Y DURING ELECTRICAL STORMS

Why is it safe to stay inside an automobile during a lightning storm?

ber. The safety of remaining in the car is due to the fact that charges on the metal shell of the car will reside on the outer surface of the car, as noted in property 2 discussed earlier. As a result, an occupant in the automobile touching the inner surfaces is not in danger.

Explanation Many people believe that staying inside the car is safe because of the insulating characteristics of the rubber tires, but in fact that isn’t true. Lightning can travel through several kilometers of air, so it can certainly penetrate a centimeter of rub-

15.7 THE MILLIKAN OIL-DROP EXPERIMENT From 1909 to 1913, Robert Andrews Millikan (1868–1953) performed a brilliant set of experiments at the University of Chicago in which he measured the elementary charge e of the electron and demonstrated the quantized nature of the electronic charge. The apparatus he used, diagrammed in Active Figure 15.21, contains two parallel metal plates. Oil droplets that have been charged by friction in an atomizer are allowed to pass through a small hole in the upper plate. A horizontal light beam is used to illuminate the droplets, which are viewed by a telescope with axis at right angles to the beam. The droplets then appear as shining stars against a dark background, and the rate of fall of individual drops can be determined. We assume a single drop having a mass of m and carrying a charge of q is being viewed and its charge is negative. If no electric field is present between the plates, S the two forces acting on the chargeS are the force of gravity, mg , acting downward, and an upward viscous drag force D (Fig. 15.22a, page 516). The drag force is proportional to the speed of the drop. When the drop reaches its terminal speed, v, the two forces balance each other (mg  D). ACTIVE FIGURE 15.21 A schematic view of Millikan’s oildrop apparatus.

Oil droplets Pinhole +

d

q

–

v Illumination

Telescope with scale in eyepiece

516

Chapter 15

Electric Forces and Electric Fields FIGURE 15.22 The forces on a negatively charged oil droplet in Millikan’s experiment.

qE

D

E

v


v

–

– q

mg (a) Field off

mg

D


(b) Field on

Now suppose an electric field is set up between the plates by a battery connected S so that the upper plate is positively charged. In this case a third force, qE, acts on S the charged drop. Because q is negative and E is downward, the electric force is upward as in FigureS15.22b. If this force is great enough, the drop moves upward S and the drag force D r acts downward. When the upward electric force, qE, balances the sum of the force of gravity and the drag force, both acting downward, the drop reaches a new terminal speed v . With the field turned on, a drop moves slowly upward, typically at a rate of hundredths of a centimeter per second. The rate of fall in the absence of a field is comparable. Hence, a single droplet with constant mass and radius can be followed for hours as it alternately rises and falls, simply by turning the electric field on and off. After making measurements on thousands of droplets, Millikan and his coworkers found that, to within about 1% precision, every drop had a charge equal to some positive or negative integer multiple of the elementary charge e, q  ne n  0, 1, 2, 3, . . .

[15.7]

10
19

Metal dome +

+

+

+

where e  1.60  C. It was later established that positive integer multiples of e would arise when an oil droplet had lost one or more electrons. Likewise, negative integer multiples of e would arise when a drop had gained one or more electrons. Gains or losses in integral numbers provide conclusive evidence that charge is quantized. In 1923 Millikan was awarded the Nobel Prize in Physics for this work.

+

+ + B + + +

+ +

+ + + + + + + + + + + + + + A

Belt

P

Ground Insulator FIGURE 15.23 A diagram of a Van de Graaff generator. Charge is transferred to the dome by means of a rotating belt. The charge is deposited on the belt at point A and transferred to the dome at point B.

15.8

THE VAN DE GRAAFF GENERATOR

In 1929 Robert J. Van de Graaff (1901–1967) designed and built an electrostatic generator that has been used extensively in nuclear physics research. The principles of its operation can be understood with knowledge of the properties of electric fields and charges already presented in this chapter. Figure 15.23 shows the basic construction of this device. A motor-driven pulley P moves a belt past positively charged comb-like metallic needles positioned at A. Negative charges are attracted to these needles from the belt, leaving the left side of the belt with a net positive charge. The positive charges attract electrons onto the belt as it moves past a second comb of needles at B, increasing the excess positive charge on the dome. Because the electric field inside the metal dome is negligible, the positive charge on it can easily be increased regardless of how much charge is already present. The result is that the dome is left with a large amount of positive charge. This accumulation of charge on the dome can’t continue indefinitely. As more and more charge appears on the surface of the dome, the magnitude of the electric field at that surface is also increasing. Finally, the strength of the field becomes great enough to partially ionize the air near the surface, increasing the conductivity of the air. Charges on the dome now have a pathway to leak off into the air, producing some spectacular “lightning bolts” as the discharge occurs. As noted earlier, charges find it easier to leap off a surface at points where the curvature is great. As a result, one way to inhibit the electric discharge, and to increase the amount of charge that can be stored on the dome, is to increase its radius. Another method

15.9

Electric Flux and Gauss’s Law

517

for inhibiting discharge is to place the entire system in a container filled with a high-pressure gas, which is significantly more difficult to ionize than air at atmospheric pressure. If protons (or other charged particles) are introduced into a tube attached to the dome, the large electric field of the dome exerts a repulsive force on the protons, causing them to accelerate to energies high enough to initiate nuclear reactions between the protons and various target nuclei.

15.9 ELECTRIC FLUX AND GAUSS’S LAW Gauss’s law is essentially a technique for calculating the average electric field on a closed surface, developed by Karl Friedrich Gauss (1777–1855). When the electric field, because of its symmetry, is constant everywhere on that surface and perpendicular to it, the exact electric field can be found. In such special cases, Gauss’s law is far easier to apply than Coulomb’s law. Gauss’s law relates the electric flux through a closed surface and the total charge inside that surface. A closed surface has an inside and an outside: an example is a sphere. Electric flux is a measure of how much the electric field vectors penetrate through a given surface. If the electric field vectors are tangent to the surface at all points, for example, they don’t penetrate the surface and the electric flux through the surface is zero. These concepts will be discussed more fully in the next two subsections. As we’ll see, Gauss’s law states that the electric flux through a closed surface is proportional to the charge contained inside the surface.

Area = A

E

FIGURE 15.24 Field lines of a uniform electric field penetrating a plane of area A perpendicular to the field. The electric flux E through this area is equal to EA.

Electric Flux Consider an electric field that is uniform in both magnitude and direction, as in Figure 15.24. The electric field lines penetrate a surface of area A, which is perpendicular to the field. The technique used for drawing a figure such as Figure 15.24 is that the number of lines per unit area, N/A, is proportional to the magnitude of the electric field, or E N/A. We can rewrite this proportion as N EA, which means that the number of field lines is proportional to the product of E and A, called the electric flux and represented by the symbol E : E  EA

[15.8]

Note that E has SI units of N  m2/C and is proportional to the number of field lines that pass through some area A oriented perpendicular to the field. (It’s called flux by analogy with the term flux in fluid flow, which is the volume of liquid flowing through a perpendicular area per second.) If the surface under consideration is not perpendicular to the field, as in Figure 15.25, the expression for the electric flux is FE 5 EA cos u

[15.9]

where a vector perpendicular to the area A is at an angle u with respect to the field. This vector is often said to be normal to the surface, and we will refer to it as “the normal vector to the surface.” The number of lines that cross this area is equal to the number that cross the projected area A , which is perpendicular to the field. We see that the two areas are related by A  A cos u. From Equation 15.9, we see that the flux through a surface of fixed area has the maximum value EA when the surface is perpendicular to the field (when u  0°) and that the flux is zero when the surface is parallel to the field (when u  90°). By convention, for a closed surface, the flux lines passing into the interior of the volume are negative and those passing out of the interior of the volume are positive. This convention is equivalent to requiring the normal vector of the surface to point outward when computing the flux through a closed surface.

A

Normal

θ

θ E A = A cos θ FIGURE 15.25 Field lines for a uniform electric field through an area A that is at an angle of (90°
u) to the field. Because the number of lines that go through the shaded area A is the same as the number that go through A, we conclude that the flux through A is equal to the flux through A and is given by E  EA cos u.

O Electric flux

518

Chapter 15

Electric Forces and Electric Fields

QUICK QUIZ 15.7 Calculate the magnitude of the flux of a constant electric field of 5.00 N/C in the z-direction through a rectangle with area 4.00 m2 in the xy-plane. (a) 0 (b) 10.0 N  m2/C (c) 20.0 N  m2/C (d) More information is needed QUICK QUIZ 15.8 Suppose the electric field of Quick Quiz 15.7 is tilted 60° away from the positive z-direction. Calculate the magnitude of the flux through the same area. (a) 0 (b) 10.0 N  m2/C (c) 20.0 N  m2/C (d) More information is needed

EXAMPLE 15.6 Goal

Flux Through a Cube

Calculate the electric flux through a closed surface.

y

Problem Consider a uniform electric field oriented in the x-direction. Find the electric flux through each surface of a cube with edges L oriented as shown in Figure 15.26, and the net flux. Strategy This problem involves substituting into the definition of electric flux given by Equation 15.9. In each case E and A  L2 are the same; the only difference is the angle u that the electric field makes with respect to a vector perpendicular to a given surface and pointing outward (the normal vector to the surface). The angles can be determined by inspection. The flux through a surface parallel to the xy-plane will be labeled xy and further designated by position (front, back); others will be labeled similarly: xz top or bottom, and yz left or right.

Solution The normal vector to the xy-plane points in the negaS tive z-direction. This, in turn, is perpendicular to E, so u  90°. (The opposite side works similarly.)

쩸 E L

z

L

쩹

x

FIGURE 15.26 (Example 15.6) A hypothetical surface in the shape of a cube in a uniform electric field parallel to the x -axis. The net flux through the surface is zero when the net charge inside the cube is zero.

xy  EA cos (90°)  0 (back and front)

The normal vector to the xz-plane points in the negaS tive y-direction. This, in turn, is perpendicular to E, so again u  90°. (The opposite side works similarly.)

xz  EA cos (90°)  0 (top and bottom)

The normal vector to surface 쩸 (the yz-plane) points S in the negative x-direction. This is antiparallel to E, so u  180°.

yz  EA cos (180°)  2EL2 (surface 쩸)

Surface 쩹 has normal vector pointing in the positive x-direction, so u  0°.

yz  EA cos (0°)  EL2 (surface 쩹)

We calculate the net flux by summing:

net  0  0  0  0
EL2  EL2  0

Remarks In doing this calculation, it is necessary to remember that the angle in the definition of flux is measured from the normal vector to the surface and that this vector must point outwards for a closed surface. As a result, the normal vector for the yz-plane on the left points in the negative x-direction, and the normal vector to the plane parallel to the yz-plane on the right points in the positive x-direction. Notice that there aren’t any charges in the box. The net electric flux is always zero for closed surfaces that don’t contain net charge. QUESTION 15.6 If the surface in Figure 15.26 were spherical, would the answer be (a) greater than, (b) less than, or (c) the same as the net electric flux found for the cubical surface?

15.9

Electric Flux and Gauss’s Law

519

EXERCISE 15.6 Suppose the constant electric field in Example 15.6 points in the positive y-direction instead. Calculate the flux through the xz-plane and the surface parallel to it. What’s the net electric flux through the surface of the cube? Answers xz 
EL2 (bottom), xz  EL2 (top). The net flux is still zero.

Gaussian surface

Gauss’s Law Consider a point charge q surrounded by a spherical surface of radius r centered on the charge, as in Figure 15.27a. The magnitude of the electric field everywhere on the surface of the sphere is E 5 ke

r + q

q r2

(a)

Note that the electric field is perpendicular to the spherical surface at all points on the surface. The electric flux through the surface is therefore EA, where A  4pr 2 is the surface area of the sphere: FE 5 EA 5 k e

q r2

+ q

1 4pr 2 2 5 4pk eq

It’s sometimes convenient to express ke in terms of another constant, P0, as ke  1/(4pP0). The constant P0 is called the permittivity of free space and has the value 1 P0 5 5 8.85 3 10212 C2 /N # m2 [15.10] 4pke The use of ke or P0 is strictly a matter of taste. The electric flux through the closed spherical surface that surrounds the charge q can now be expressed as FE 5 4pkeq 5

(b) FIGURE 15.27 (a) The flux through a spherical surface of radius r surrounding a point charge q is E  q/P0. (b) The flux through any arbitrary surface surrounding the charge is also equal to q/P0.

q P0

This result says that the electric flux through a sphere that surrounds a charge q is equal to the charge divided by the constant P0. Using calculus, this result can be proven for any closed surface that surrounds the charge q. For example, if the surface surrounding q is irregular, as in Figure 15.27b, the flux through that surface is also q/P0. This leads to the following general result, known as Gauss’s law: The electric flux E through any closed surface is equal to the net charge inside the surface, Q inside, divided by P0: FE 5

Q inside P0

[15.11]

Although it’s not obvious, Gauss’s law describes how charges create electric fields. In principle it can always be used to calculate the electric field of a system of charges or a continuous distribution of charge. In practice, the technique is useful only in a limited number of cases in which there is a high degree of symmetry, such as spheres, cylinders, or planes. With the symmetry of these special shapes, the charges can be surrounded by an imaginary surface, called a Gaussian surface. This imaginary surface is used strictly for mathematical calculation, and need not be an actual, physical surface. If the imaginary surface is chosen so that the electric field is constant everywhere on it, the electric field can be computed with EA 5 FE 5

Q inside P0

O Gauss’s Law

[15.12]

TIP 15.2 Gaussian Surfaces Aren’t Real A Gaussian surface is an imaginary surface, created solely to facilitate a mathematical calculation. It doesn’t necessarily coincide with the surface of a physical object.

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as will be seen in the examples. Although Gauss’s law in this form can be used to obtain the electric field only for problems with a lot of symmetry, it can always be used to obtain the average electric field on any surface.



2 C

3 C

+

2 C

+

QUICK QUIZ 15.9 Find the electric flux through the surface in Active Figure 15.28. (a)
(3 C)/P0 (b) (3 C)/P0 (c) 0 (d)
(6 C)/P0

4C



5 C

+ 1C

ACTIVE FIGURE 15.28 (Quick Quiz 15.9)

QUICK QUIZ 15.10 For a closed surface through which the net flux is zero, each of the following four statements could be true. Which of the statements must be true? (There may be more than one.) (a) There are no charges inside the surface. (b) The net charge inside the surface is zero. (c) The electric field is zero everywhere on the surface. (d) The number of electric field lines entering the surface equals the number leaving the surface.

EXAMPLE 15.7 The Electric Field of a Charged Spherical Shell

+ + + + b a + r + + + + + +

+

+

(a)

+ + + +

+

Strategy For each part, draw a spherical Gaussian surface in the region of interest. Add up the charge inside the Gaussian surface, substitute it and the area into Gauss’s law, and solve for the electric field. To find the distribution of charge in part (c), use Gauss’s law in reverse: the charge distribution must be such that the electrostatic field is zero inside a conductor.

E + + + + b a + + + + Ein = 0 + +

Gaussian surface + + r b a + +

+ +

(b)

(c)

FIGURE 15.29 (Example 15.7) (a) The electric field inside a uniformly charged spherical shell is zero. It is also zero for the conducting material in the region a r b. The field outside is the same as that of a point charge having a total charge Q located at the center of the shell. (b) The construction of a Gaussian surface for calculating the electric field inside a spherical shell. (c) The construction of a Gaussian surface for calculating the electric field outside a spherical shell.

Solution (a) Find the electric field for r a. Apply Gauss’s law, Equation 15.12, to the Gaussian surface illustrated in Figure 15.29b (note that there isn’t any charge inside this surface):

EA 5 E 1 4pr 2 2 5

Q

EA 5 E 1 4pr 2 2 5

Q inside Q 5 P0 P0

inside

P0

50

(b) Find the electric field for r b. Apply Gauss’s law, Equation 15.12, to the Gaussian surface illustrated in Figure 15.29c:

Divide by the area:

+

Problem A spherical conducting shell of inner radius a and outer radius b carries a total charge Q distributed on the surface of a conducting shell (Fig. 15.29a). The quantity Q is taken to be positive. (a) Find the electric field in the interior of the conducting shell, for r a, and (b) the electric field outside the shell, for r b. (c) If an additional charge of
2Q is placed at the center, find the electric field for r b. (d) What is the distribution of charge on the sphere in part (c)?

Gaussian surface

+

Goal Use Gauss’s law to determine electric fields when the symmetry is spherical.

E5

Q 4pP0r 2

S

E50

15.9

Electric Flux and Gauss’s Law

521

(c) Now an additional charge of
2Q is placed at the center of the sphere. Compute the new electric field outside the sphere, for r b. Apply Gauss’s law as in part (b), including the new charge in Q inside:

EA 5 E 1 4pr 2 2 5

E52

1Q 2 2Q Q inside 5 P0 P0

Q 4pP0r 2

(d) Find the charge distribution on the sphere for part (c). Q inside Q center 1 Q inner surface 5 P0 P0

Write Gauss’s law for the interior of the shell:

EA 5

Find the charge on the inner surface of the shell, noting that the electric field in the conductor is zero:

Q center  Q inner surface  0

Find the charge on the outer surface, noting that the inner and outer surface charges must sum to Q:

Q inner surface 
Q center  2Q Q outer surface  Q inner surface  Q Q outer surface 
Q inner surface  Q 
Q

Remarks The important thing to notice is that in each case, the charge is spread out over a region with spherical symmetry or is located at the exact center. That is what allows the computation of a value for the electric field. QUESTION 15.7 If the charge at the center of the sphere is made positive, how is the charge on the inner surface of the sphere affected? EXERCISE 15.7 Suppose the charge at the center is now increased to 2Q, while the surface of the conductor still retains a charge of Q. (a) Find the electric field exterior to the sphere, for r b. (b) What’s the electric field inside the conductor, for a r b? (c) Find the charge distribution on the conductor. Answers (a) E  3Q/4pP0r 2 (b) E  0, which is always the case when charges are not moving in a conductor. (c) Inner surface:
2Q; outer surface: 3Q

In Example 15.7, not much was said about the distribution of charge on the conductor. Whenever there is a net nonzero charge, the individual charges will try to get as far away from each other as possible. Hence, charge will reside either on the inside surface or on the outside surface. Because the electric field in the conductor is zero, there will always be enough charge on the inner surface to cancel whatever charge is at the center. In part (b) there is no charge on the inner surface and a charge of Q on the outer surface. In part (c), with a
Q charge at the center, Q is on the inner surface and 0 C is on the outer surface. Finally, in the exercise, with
2Q in the center, there must be 2Q on the inner surface and
Q on the outer surface. In each case the total charge on the conductor remains the same, Q ; it’s just arranged differently. Problems like Example 15.7 are often said to have “thin, nonconducting shells” carrying a uniformly distributed charge. In these cases no distinction need be made between the outer surface and inner surface of the shell. The next example makes that implicit assumption.

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Electric Forces and Electric Fields

EXAMPLE 15.8 A Nonconducting Plane Sheet of Charge Goal

Apply Gauss’s law to a problem with plane symmetry.

Problem Find the electric field above and below a nonconducting infinite plane sheet of charge with uniform positive charge per unit area s (Fig. 15.30a). Strategy By symmetry, the electric field must be perpendicular to the plane and directed away from it on either side, as shown in Figure 15.30b. For the Gaussian surface, choose a small cylinder with axis perpendicular to the plane, each end having area A0. No electric field lines pass through the curved surface of the cylinder, only through the two ends, which have total area 2A0. Apply Gauss’s law, using Figure 15.30b.

E

A0 E E  EA0

E for z > 0

+ +

Q  σA0 + + + + + + + + + + + + + +

–

–

–

–

–

–

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

Gaussian surface

+

Gaussian surface E (a)

E  EA0

E (b)

E for z < 0 (c)

FIGURE 15.30 (Example 15.8) (a) A cylindrical Gaussian surface penetrating an infinite sheet of charge. (b) A cross section of the same Gaussian cylinder. The flux through each end of the Gaussian surface is EA0. There is no flux through the cylindrical surface. (c) (Exercise 15.8).

Solution (a) Find the electric field above and below a plane of uniform charge. Q inside P0

Apply Gauss’s law, Equation 15.12:

EA 5

The total charge inside the Gaussian cylinder is the charge density times the cross-sectional area:

Q inside  sA0

The electric flux comes entirely from the two ends, each having area A0. Substitute A  2A0 and Q inside and solve for E.

E5

This is the magnitude of the electric field. Find the z-component of the field above and below the plane. The electric field points away from the plane, so it’s positive above the plane and negative below the plane.

Ez 5

sA0 s 5 1 2A0 2 P0 2P0

s z.0 2P0

Ez 5 2

s z,0 2P0

Remarks Notice here that the plate was taken to be a thin, nonconducting shell. If it’s made of metal, of course, the electric field inside it is zero, with half the charge on the upper surface and half on the lower surface. QUESTION 15.8 In reality, the sheet carrying charge would likely be metallic and have a small but nonzero thickness. If it carries the same charge per unit area, what is the electric field inside the sheet between the two surfaces?

Summary

523

EXERCISE 15.8 Suppose an infinite nonconducting plane of charge as in Example 15.8 has a uniform negative charge density of
s. Find the electric field above and below the plate. Sketch the field. Answers

Ez 5

s 2s , z . 0; E z 5 , z , 0. See Figure 15.30c for the sketch. 2P0 2P0 E0 –

An important circuit element that will be studied extensively in the next chapter is the parallel-plate capacitor. The device consists of a plate of positive charge, as in Example 15.8, with the negative plate of Exercise 15.8 placed above it. The sum of these two fields is illustrated in Figure 15.31. The result is an electric field with double the magnitude in between the two plates: s E5 P0

–

–

–

–

–

 E–  0

+

+

+

+

+

+

E0 FIGURE 15.31 Cross section of an idealized parallel-plate capacitor. Electric field vector contributions sum together in between the plates, but cancel outside.

[15.13]

Outside the plates, the electric fields cancel.

SUMMARY 15.1

Properties of Electric Charges

The magnitude of the electric field due to a point charge q at a distance r from the point charge is

Electric charges have the following properties:

1. Unlike charges attract one another and like charges repel one another. 2. Electric charge is always conserved. 3. Charge comes in discrete packets that are integral multiples of the basic electric charge e  1.6  10
19 C. 4. The force between two charged particles is proportional to the inverse square of the distance between them. 15.2

Insulators and Conductors

Conductors are materials in which charges move freely in response to an electric field. All other materials are called insulators.

15.3 Coulomb’s Law Coulomb’s law states that the electric force between two stationary charged particles separated by a distance r has the magnitude 0 q1 0 0 q2 0 F 5 ke [15.1] r2 where
q 1
and
q 2
are the magnitudes of the charges on the particles in coulombs and [15.2] ke  8.99  109 N  m2/C2 is the Coulomb constant.

15.4 The Electric Field S

An electric field E exists at some point in space if a small test charge q placed at that point is acted upon by an elecS 0 tric force F . The electric field is defined as S

E;

15.5

[15.3]

The direction of the electric field at a point in space is defined to be the direction of the electric force that would be exerted on a small positive charge placed at that point.

0q0 r2

[15.6]

Electric Field Lines

Electric field lines are useful for visualizing the electric S field in any region of space. The electric field vector E is tangent to the electric field lines at every point. Further, the number of electric field lines per unit area through a surface perpendicular to the lines is proportional to the strength of the electric field at that surface.

15.6 Conductors in Electrostatic Equilibrium A conductor in electrostatic equilibrium has the following properties:

1. The electric field is zero everywhere inside the conducting material. 2. Any excess charge on an isolated conductor must reside entirely on its surface. 3. The electric field just outside a charged conductor is perpendicular to the conductor’s surface. 4. On an irregularly shaped conductor, charge accumulates where the radius of curvature of the surface is smallest, at sharp points. 15.9

Electric Flux and Gauss’s Law

Gauss’s law states that the electric flux through any closed surface is equal to the net charge Q inside the surface divided by the permittivity of free space, P0 : EA 5 F E 5

S

F q0

E 5 ke

Q inside P0

[15.12]

For highly symmetric distributions of charge, Gauss’s law can be used to calculate electric fields.

524

Chapter 15

Electric Forces and Electric Fields FOR ADDITIONAL STUDENT RESOURCES, GO TO W W W.SERWAYPHYSICS.COM

MULTIPLE-CHOICE QUESTIONS 1. The magnitude of the electric force between two protons is 2.3  10
26 N. How far apart are they? (a) 0.10 m (b) 0.022 m (c) 3.1 m (d) 0.005 7 m (e) 0.48 m 2. Estimate the magnitude of the electric field strength due to the proton in a hydrogen atom at a distance of 5.29  10
11 m, the Bohr radius. (a) 10
11 N/C (b) 108 N/C (c) 1014 N/C (d) 106 N/C (e) 1012 N/C 3. A very small ball has a mass of 5.0  10
3 kg and a charge of 4.0 mC. What magnitude electric field directed upward will balance the weight of the ball? (a) 8.2  102 N/C (b) 1.2  104 N/C (c) 2.0  10
2 N/C (d) 5.1  106 N/C (e) 3.7  103 N/C 4. An electron with a speed of 3.00  106 m/s moves into a uniform electric field of magnitude 1.00  103 N/C. The field is parallel to the electron’s motion. How far does the electron travel before it is brought to rest? (a) 2.56 cm (b) 5.12 cm (c) 11.2 cm (d) 3.34 m (e) 4.24 m 5. Charges of 3.0 nC,
2.0 nC,
7.0 nC, and 1.0 nC are contained inside a rectangular box with length 1.0 m, width 2.0 m, and height 2.5 m. Outside the box are charges of 1.0 nC and 4.0 nC. What is the electric flux through the surface of the box? (a) 0 (b)
560 N  m2/C (c)
340 N  m2/C (d) 260 N  m2/C (e) 170 N  m2/C 6. A uniform electric field of 1.0 N/C is set up by a uniform distribution of charge in the xy-plane. What is the electric field inside a metal ball placed 0.50 m above the xy-plane? (a) 1.0 N/C (b)
1.0 N/C (c) 0 (d) 0.25 N/C (e) It varies depending on the position inside the ball. 7. A charge of
4.00 nC is located at (0, 1.00) m. What is the x-component of the electric field at (4.00,
2.00) m? (a) 1.15 N/C (b)
2.24 N/C (c) 3.91 N/C (d)
1.15 N/C (e) 0.863 N/C 8. Two point charges attract each other with an electric force of magnitude F. If one charge is reduced to onethird its original value and the distance between the charges is doubled, what is the resulting magnitude of the electric force between them? (a) F/12 (b) F/3 (c) F/6 (d) 3F/4 (e) 3F/2 9. What happens when a charged insulator is placed near an uncharged metallic object? (a) They repel each other. (b) They attract each other. (c) They may attract or repel each other, depending on whether the charge on the insulator is positive or negative. (d) They exert no electrostatic force on each other. (e) The charged insulator always spontaneously discharges.

10. In which of the following contexts can Gauss’s law not be readily applied to find the electric field? (a) near a long, uniformly charged wire (b) above a large uniformly charged plane (c) inside a uniformly charged ball (d) outside a uniformly charged sphere (e) Gauss’s law can be readily applied to find the electric field in all these contexts. 11. What prevents gravity from pulling you through the ground to the center of Earth? Choose the best answer. (a) The density of matter is too great. (b) The positive nuclei of your body’s atoms repel the positive nuclei of the atoms of the ground. (c) The density of the ground is greater than the density of your body. (d) Atoms are bound together by chemical bonds. (e) Electrons on the ground’s surface and the surface of your feet repel one another. 12. A metallic coin is given a positive electric charge. Does its mass (a) increase measurably, (b) increase by an amount too small to measure directly, (c) stay unchanged, (d) decrease by an amount too small to measure directly, or (e) decrease measurably? 13. Three charged particles are arranged on corners of a square as shown in Figure MCQ15.13, with charge
Q on both the particle at the upper left corner and the particle at the lower right corner, and charge 2Q on the particle at the lower left corner. What is the direction of the electric field at the upper right corner, which is a point in empty space? (a) upward and to the right (b) to the right (c) downward (d) downward and to the left (e) The field is exactly zero at that point. (a) –Q

(b) (d) (c)

+2Q

–Q

FIGURE MCQ15.13

14. Suppose the 2Q charge at the lower left corner of Figure MCQ15.13 is removed. Which statement is true about the magnitude of the electric field at the upper right corner? (a) It becomes larger. (b) It becomes smaller. (c) It stays the same. (d) It changes unpredictably. (e) It is zero.

Problems

525

CONCEPTUAL QUESTIONS 1. A glass object is charged to 3 nC by rubbing it with a silk cloth. In the rubbing process, have protons been added to the object or have electrons been removed from it?

9. In fair weather there is an electric field at the surface of the Earth, pointing down into the ground. What is the electric charge on the ground in this situation?

2. Why must hospital personnel wear special conducting shoes while working around oxygen in an operating room? What might happen if the personnel wore shoes with rubber soles?

10. A student stands on a thick piece of insulating material, places her hand on top of a Van de Graaff generator, and then turns on the generator. Does she receive a shock?

3. A person is placed in a large, hollow metallic sphere that is insulated from ground. If a large charge is placed on the sphere, will the person be harmed upon touching the inside of the sphere?

11. There are great similarities between electric and gravitational fields. A room can be electrically shielded so that there are no electric fields in the room by surrounding it with a conductor. Can a room be gravitationally shielded? Explain.

4. Explain from an atomic viewpoint why charge is usually transferred by electrons. 5. Explain how a positively charged object can be used to leave another metallic object with a net negative charge. Discuss the motion of charges during the process. 6. If a suspended object A is attracted to a charged object B, can we conclude that A is charged? Explain. 7. If a metal object receives a positive charge, does its mass increase, decrease, or stay the same? What happens to its mass if the object receives a negative charge? 8. Consider point A in Figure CQ15.8. Does charge exist at this point? Does a force exist at this point? Does a field exist at this point? Explain. q1

q2

12. Why should a ground wire be connected to the metal support rod for a television antenna? 13. A charged comb often attracts small bits of dry paper that then fly away when they touch the comb. Explain. 14. A spherical surface surrounds a point charge q. Describe what happens to the total flux through the surface if (a) the charge is tripled, (b) the volume of the sphere is doubled, (c) the surface is changed to a cube, (d) the charge is moved to another location inside the surface, and (e) the charge is moved outside the surface. 15. If more electric field lines leave a Gaussian surface than enter it, what can you conclude about the net charge enclosed by that surface?

A FIGURE CQ15.8

PROBLEMS The Problems for this chapter may be assigned online at WebAssign. 1, 2, 3  straightforward, intermediate, challenging GP  denotes guided problem ecp  denotes enhanced content problem  biomedical application 䡺  denotes full solution available in Student Solutions Manual/ Study Guide

SECTION 15.3 COULOMB’S LAW 1. A 7.5-nC charge is located 1.8 m from a 4.2-nC charge. Find the magnitude of the electrostatic force that one charge exerts on the other. Is the force attractive or repulsive? 2. A charged particle A exerts a force of 2.62 mN to the right on charged particle B when the particles are 13.7 mm apart. Particle B moves straight away from A to make the distance between them 17.7 mm. What vector force does particle B then exert on A?

3. ecp Two metal balls A and B of negligible radius are floating at rest on Space Station Freedom between two metal bulkheads, connected by a taut nonconducting thread of length 2.00 m. Ball A carries charge q, and ball B carries charge 2q. Each ball is 1.00 m away from a bulkhead. (a) If the tension in the string is 2.50 N, what is the magnitude of q? (b) What happens to the system as time passes? Explain. 4. ecp (a) Find the electrostatic force between a Na ion and a Cl
ion separated by 0.50 nm. (b) Would the answer change if the sodium ion were replaced by Li and the Chloride ion by Br
? Explain. 5. The nucleus of 8Be, which consists of 4 protons and 4 neutrons, is very unstable and spontaneously breaks into two alpha particles (helium nuclei, each consisting of 2 protons and 2 neutrons). (a) What is the force between the two alpha particles when they are 5.00  10
15 m apart, and (b) what is the initial magnitude of the acceleration of the alpha particles due to this force? Note that the mass of an alpha particle is 4.0026 u.

526

6.

Chapter 15

Electric Forces and Electric Fields

7. Suppose 1.00 g of hydrogen is separated into electrons and protons. Suppose also the protons are placed at the Earth’s North Pole and the electrons are placed at the South Pole. What is the resulting compressional force on the Earth? 8.

3.00 nC

A molecule of DNA (deoxyribonucleic acid) is 2.17 mm long. The ends of the molecule become singly ionized: negative on one end, positive on the other. The helical molecule acts like a spring and compresses 1.00% upon becoming charged. Determine the effective spring constant of the molecule.

ecp

Four point charges are at the corners of a square of side a as shown in Figure P15.8. Determine the magnitude and direction of the resultant electric force on q, with ke , q, and a left in symbolic form.

0.500 m 0.500 m

2.00 nC FIGURE P15.12

13. Three point charges are located at the corners of an equilateral triangle as in Figure P15.13. Find the magnitude and direction of the net electric force on the 2.00 mC charge. y

a 2q

6.00 nC

0.500 m

7.00 µ C +

q 0.500 m

a

60.0° + 2.00 µ C

a

3q

2q

FIGURE P15.13

a FIGURE P15.8

9. Two small identical conducting spheres are placed with their centers 0.30 m apart. One is given a charge of 12  10
9 C, the other a charge of
18  10
9 C. (a) Find the electrostatic force exerted on one sphere by the other. (b) The spheres are connected by a conducting wire. Find the electrostatic force between the two after equilibrium is reached. 10. Calculate the magnitude and direction of the Coulomb force on each of the three charges shown in Figure P15.10. 6.00 µC

1.50 µC

FIGURE P15.10

15. Two small metallic spheres, each of mass 0.20 g, are suspended as pendulums by light strings from a common point as shown in Figure P15.15. The spheres are given the same electric charge, and it is found that they come to equilibrium when each string is at an angle of 5.0° with the vertical. If each string is 30.0 cm long, what is the magnitude of the charge on each sphere?

–2.00 µC

2.00 cm

θ

(Problems 10 and 18)

0.20 g

11. Three charges are arranged as shown in Figure P15.11. Find the magnitude and direction of the electrostatic force on the charge at the origin.

0.300 m

6.00 nC x

0.100 m –3.00 nC FIGURE P15.11

12. Three charges are arranged as shown in Figure P15.12. Find the magnitude and direction of the electrostatic force on the 6.00-nC charge.

0.20 g

FIGURE P15.15

16. y 5.00 nC

(Problems 13 and 24)

14. A charge of
3.00 nC and a charge of
5.80 nC are separated by a distance of 50.0 cm. Find the position at which a third charge of 7.50 nC can be placed so that the net electrostatic force on it is zero.

30.0 cm 3.00 cm


x –4.00 µ C

GP Particle A of charge 3.00  10
4 C is at the origin, particle B of charge
6.00  10
4 C is at (4.00 m, 0) and particle C of charge 1.00  10
4 C is at (0, 3.00 m). (a) What is the x-component of the electric force exerted by A on C ? (b) What is the y-component of the force exerted by A on C ? (c) Find the magnitude of the force exerted by B on C. (d) Calculate the x-component of the force exerted by B on C. (e) Calculate the y-component of the force exerted by B on C. (f) Sum the two x-components to obtain the resultant x-component of the electric force acting on C. (g) Repeat part (f) for the y-component. (h) Find the magnitude and direction of the resultant electric force acting on C.

Problems

SECTION 15.4 THE ELECTRIC FIELD 17. A small object of mass 3.80 g and charge
18 mC “floats” in a uniform electric field. What is the magnitude and direction of the electric field?

28. Three charges are at the corners of an equilateral triangle, as shown in Figure P15.28. Calculate the electric field at a point midway between the two charges on the x-axis. y

3.00 nC

18. (a) Determine the electric field strength at a point 1.00 cm to the left of the middle charge shown in Figure P15.10. (b) If a charge of
2.00 mC is placed at this point, what are the magnitude and direction of the force on it?

0.500 m 60.0°

19. An airplane is flying through a thundercloud at a height of 2 000 m. (Flying at this height is very dangerous because of updrafts, turbulence, and the possibility of electric discharge.) If there are charge concentrations of 40.0 C at a height of 3 000 m within the cloud and
40.0 C at a height S of 1 000 m, what is the electric field E at the aircraft? 20. An electron is accelerated by a constant electric field of magnitude 300 N/C. (a) Find the acceleration of the electron. (b) Use the equations of motion with constant acceleration to find the electron’s speed after 1.00  10
8 s, assuming it starts from rest. 21. A charge of
5.0 nC is at the origin and a second charge of 7.0 nC is at x  4.00 m. Find the magnitude and direction of the electric field halfway in between the two charges. 22. Each of the protons in a particle beam has a kinetic energy of 3.25  10
15 J. What are the magnitude and direction of the electric field that will stop these protons in a distance of 1.25 m? 23. A proton accelerates from rest in a uniform electric field of 640 N/C. At some later time, its speed is 1.20  106 m/s. (a) Find the magnitude of the acceleration of the proton. (b) How long does it take the proton to reach this speed? (c) How far has it moved in that interval? (d) What is its kinetic energy at the later time? 24. ecp (a) Find the magnitude and direction of the electric field at the position of the 2.00 mC charge in Figure P15.13. (b) How would the electric field at that point be affected if the charge there were doubled? Would the magnitude of the electric force be affected? 25. An alpha particle (a helium nucleus) is traveling along the positive x-axis at 1 250 m/s when it enters a cylindrical tube of radius 0.500 m centered on the x-axis. Inside the tube is a uniform electric field of 4.50  10
4 N/C pointing in the negative y-direction. How far does the particle travel before hitting the tube wall? Neglect any gravitational forces. Note: m a  6.64  10
27 kg; q a  2e. 26. Two point charges lie along the y-axis. A charge of q 1 
9.0 mC is at y  6.0 m, and a charge of q 2 
8.0 mC is at y 
4.0 m. Locate the point (other than infinity) at which the total electric field is zero. 27. In Figure P15.27 determine the point (other than infinity) at which the total electric field is zero. 1.0 m

– 2.5 µC

6.0 µC FIGURE P15.27

527

x – 5.00 nC

8.00 nC

FIGURE P15.28

29. Three identical charges (q 
5.0 mC) lie along a circle of radius 2.0 m at angles of 30°, 150°, and 270°, as shown in Figure P15.29. What is the resultant electric field at the center of the circle? y

q

q 150° 30°

x

270°

q FIGURE P15.29

SECTION 15.5 ELECTRIC FIELD LINES SECTION 15.6 CONDUCTORS IN ELECTROSTATIC EQUILIBRIUM 30. Figure P15.30 shows the electric field lines for two point charges separated by a small distance. (a) Determine the ratio q 1/q 2. (b) What are the signs of q 1 and q 2?

q2 q1

FIGURE P15.30

31. (a) Sketch the electric field lines around an isolated point charge q 0. (b) Sketch the electric field pattern around an isolated negative point charge of magnitude
2q. 32. (a) Sketch the electric field pattern around two positive point charges of magnitude 1 mC placed close together. (b) Sketch the electric field pattern around two negative point charges of
2 mC, placed close together. (c) Sketch the pattern around two point charges of 1 mC and
2 mC, placed close together.

528

Chapter 15

Electric Forces and Electric Fields

33. Two point charges are a small distance apart. (a) Sketch the electric field lines for the two if one has a charge four times that of the other and both charges are positive. (b) Repeat for the case in which both charges are negative. 34. (a) Sketch the electric field pattern set up by a positively charged hollow sphere. Include regions inside and regions outside the sphere. (b) A conducting cube is given a positive charge. Sketch the electric field pattern both inside and outside the cube. 35. Refer to Figure 15.20. The charge lowered into the center of the hollow conductor has a magnitude of 5 mC. Find the magnitude and sign of the charge on the inside and outside of the hollow conductor when the charge is as shown in (a) Figure 15.20a, (b) Figure 15.20b, (c) Figure 15.20c, and (d) Figure 15.20d.

41. An electric field of intensity 3.50 kN/C is applied along the x-axis. Calculate the electric flux through a rectangular plane 0.350 m wide and 0.700 m long if (a) the plane is parallel to the yz-plane, (b) the plane is parallel to the xy-plane, and (c) the plane contains the y-axis and its normal makes an angle of 40.0° with the x-axis. 42. ecp The electric field everywhere on the surface of a charged sphere of radius 0.230 m has a magnitude of 575 N/C and points radially outward from the center of the sphere. (a) What is the net charge on the sphere? (b) What can you conclude about the nature and distribution of charge inside the sphere? 43. Four closed surfaces, S1 through S4, together with the charges
2Q , Q , and
Q , are sketched in Figure P15.43. (The colored lines are the intersections of the surfaces with the page.) Find the electric flux through each surface.

SECTION 15.8 THE VAN DE GRAAFF GENERATOR 36. The dome of a Van de Graaff generator receives a charge of 2.0  10
4 C. Find the strength of the electric field (a) inside the dome, (b) at the surface of the dome, assuming it has a radius of 1.0 m, and (c) 4.0 m from the center of the dome. (Hint: See Section 15.6 to review properties of conductors in electrostatic equilibrium. Also, use that the points on the surface are outside a spherically symmetric charge distribution; the total charge may be considered to be located at the center of the sphere.) 37. If the electric field strength in air exceeds 3.0  106 N/C, the air becomes a conductor. Using this fact, determine the maximum amount of charge that can be carried by a metal sphere 2.0 m in radius. (See the hint in Problem 36.) 38. In the Millikan oil-drop experiment, an atomizer (a sprayer with a fine nozzle) is used to introduce many tiny droplets of oil between two oppositely charged parallel metal plates. Some of the droplets pick up one or more excess electrons. The charge on the plates is adjusted so that the electric force on the excess electrons exactly balances the weight of the droplet. The idea is to look for a droplet that has the smallest electric force and assume it has only one excess electron. This strategy lets the observer measure the charge on the electron. Suppose we are using an electric field of 3  104 N/C. The charge on one electron is about 1.6  10
19 C. Estimate the radius of an oil drop of density 858 kg/m3 for which its weight could be balanced by the electric force of this field on one electron. (Problem 38 is courtesy of E. F. Redish. For more problems of this type, visit www.physics.umd.edu/perg/.) 39. A Van de Graaff generator is charged so that a proton at its surface accelerates radially outward at 1.52  1012 m/s2. Find (a) the magnitude of the electric force on the proton at that instant and (b) the magnitude and direction of the electric field at the surface of the generator.

SECTION 15.9 ELECTRIC FLUX AND GAUSS’S LAW m2

40. A flat surface having an area of 3.2 is rotated in a uniform electric field of magnitude E  6.2  105 N/C. Determine the electric flux through this area (a) when the electric field is perpendicular to the surface and (b) when the electric field is parallel to the surface.

S1

S4


2Q

S3

Q
Q S2 FIGURE P15.43

44. A vertical electric field of magnitude 1.80  104 N/C exists above Earth’s surface on a stormy day. A car with a rectangular size of 5.50 m by 2.00 m is traveling along a horizontal roadway. Find the magnitude of the electric flux through the bottom of the car. 45. A point charge q is located at the center of a spherical shell of radius a that has a charge
q uniformly distributed on its surface. Find the electric field (a) for all points outside the spherical shell and (b) for a point inside the shell a distance r from the center. 46. ecp A charge of 1.70  102 mC is at the center of a cube of edge 80.0 cm. No other charges are nearby. (a) Find the flux through the whole surface of the cube. (b) Find the flux through each face of the cube. (c) Would your answers to parts (a) or (b) change if the charge were not at the center? Explain. 47. Suppose the conducting spherical shell of Figure 15.29 carries a charge of 3.00 nC and that a charge of
2.00 nC is at the center of the sphere. If a  2.00 m and b  2.40 m, find the electric field at (a) r  1.50 m, (b) r  2.20 m, and (c) r  2.50 m. (d) What is the charge distribution on the sphere? 48. ecp A very large nonconducting plate lying in the xyplane carries a charge per unit area of s. A second such plate located at z  2.00 cm and oriented parallel to the xy-plane carries a charge per unit area of
2s. Find the electric field (a) for z 0, (b) 0 z 2.00 cm, and (c) z 2.00 cm.

Problems

ADDITIONAL PROBLEMS 49. In deep space two spheres each of radius 5.00 m are connected by a 3.00  102 m nonconducting cord. If a uniformly distributed charge of 35.0 mC resides on the surface of each sphere, calculate the tension in the cord. 50. ecp A nonconducting, thin plane sheet of charge carries a uniform charge per unit area of 5.20 mC/m2 as in Figure 15.30. (a) Find the electric field at a distance of 8.70 cm from the plate. (b) Explain whether your result changes as the distance from the sheet is varied. 51. Three point charges are aligned along the x-axis as shown in Figure P15.51. Find the electric field at the position x  2.0 m, y  0. y 0.50 m

529

lowered to reach its equilibrium position? (b) The ball is given a charge of 0.050 0 C. If an electric field directed upward is applied, increasing slowly to a maximum value of 355.0 N/C, how far below the unstretched position is the new equilibrium position of the ball? 56. ecp A 2.00-mC charged 1.00-g cork ball is suspended vertically on a 0.500-m-long light string in the presence of a uniform downward-directed electric field of magnitude E  1.00  105 N/C. If the ball is displaced slightly from the vertical, it oscillates like a simple pendulum. (a) Determine the period of the ball’s oscillation. (b) Should gravity be included in the calculation for part (a)? Explain. 57. Two 2.0-g spheres are suspended by 10.0-cm-long light strings (Fig. P15.57). A uniform electric field is applied in the x-direction. If the spheres have charges of
5.0  10
8 C and 5.0  10
8 C, determine the electric field intensity that enables the spheres to be in equilibrium at u  10°.

0.80 m x

– 4.0 nC

5.0 nC

3.0 nC

θ θ

FIGURE P15.51

52. A small, 2.00-g plastic ball is suspended by a 20.0-cmlong string in a uniform electric field, as shown in Figure P15.52. If the ball is in equilibrium when the string makes a 15.0° angle with the vertical as indicated, what is the net charge on the ball?

y

E = 1.00  103 N/C x 20.0 cm 15.0° m = 2.00 g FIGURE P15.52

53. (a) Two identical point charges q are located on the y-axis at y  a and y 
a. What is the electric field along the x-axis at x  b? (b) A circular ring of charge of radius a has a total positive charge Q distributed uniformly around it. The ring is in the x  0 plane with its center at the origin. What is the electric field along the x-axis at x  b due to the ring of charge? (Hint: Consider the charge Q to consist of many pairs of identical point charges positioned at the ends of diameters of the ring.) 54. ecp The electrons in a particle beam each have a kinetic energy K. Find the magnitude of the electric field that will stop these electrons in a distance d, expressing the answer symbolically in terms of K, e, and d. Should the electric field point in the direction of the motion of the electron, or should it point in the opposite direction? 55. A vertical spring with constant 845 N/m has a ball of mass 4.00 kg attached to the bottom of it, which is held with the spring unstretched. (a) How far must the ball be

–

+ E

FIGURE P15.57

58. ecp A point charge of magnitude 5.00 mC is at the origin of a coordinate system, and a charge of
4.00 mC is at the point x  1.00 m. There is a point on the x-axis, at x less than infinity, where the electric field goes to zero. (a) Show by conceptual arguments that this point cannot be located between the charges. (b) Show by conceptual arguments that the point cannot be at any location between x  0 and negative infinity. (c) Show by conceptual arguments that the point must be between x  1.00 m and x  positive infinity. (d) Use the values given to find the point and show that it is consistent with your conceptual argument. 59. Two hard rubber spheres of mass 15 g are rubbed vigorously with fur on a dry day and are then suspended from a rod with two insulating strings of length 5.0 cm. They are observed to hang at equilibrium as shown in Figure P15.59, each at an angle of 10° with the vertical. Estimate the amount of charge that is found on each sphere. (Problem 59 is courtesy of E. F. Redish. For more problems of this type, visit www.physics.umd.edu/perg/.)

3.0 cm

10°

5.0 cm

FIGURE P15.59

530

Chapter 15

Electric Forces and Electric Fields

60. Two small silver spheres, each with a mass of 100 g, are separated by 1.00 m. Calculate the fraction of the electrons in one sphere that must be transferred to the other to produce an attractive force of 1.00  104 N (about 1 ton) between the spheres. (The number of electrons per atom of silver is 47, and the number of atoms per gram is Avogadro’s number divided by the molar mass of silver, 107.87 g/mol.) 61. A solid conducting sphere of radius 2.00 cm has a charge of 8.00 mC. A conducting spherical shell of inner radius 4.00 cm and outer radius 5.00 cm is concentric with the solid sphere and has a charge of
4.00 mC. Find the electric field at (a) r  1.00 cm, (b) r  3.00 cm, (c) r  4.50 cm, and (d) r  7.00 cm from the center of this charge configuration.

63. Each of the electrons in a particle beam has a kinetic energy of 1.60  10
17 J. (a) What is the magnitude of the uniform electric field (pointing in the direction of the electrons’ movement) that will stop these electrons in a distance of 10.0 cm? (b) How long will it take to stop the electrons? (c) After the electrons stop, what will they do? Explain. 64. Protons are projected with an initial speed v 0  9 550 m/s into a region where a uniform electric field E  720 N/C is present (Fig. P15.64). The protons are to hit a target that lies a horizontal distance of 1.27 mm from the point where the protons are launched. Find (a) the two projection angles u that will result in a hit and (b) the total duration of flight for each of the two trajectories.

62. Three identical point charges, each of mass m  0.100 kg, hang from three strings, as shown in Figure P15.62. If the lengths of the left and right strings are each L  30.0 cm and if the angle u is 45.0°, determine the value of q.

E = 720 N/C v0

θ

 Target

1.27 mm

θ θ L

L

g

Proton beam FIGURE P15.64

+q

+q m

+q m

FIGURE P15.62

m

16 Yoshiki Hase, Courtesy Museum of Science, Boston

The world’s largest air-insulated Van de Graaff generator produces bolts of lightning indoors at the Museum of Science in Boston. The discharges occur when the voltage difference gets large enough to ionize the air, an instance of dielectric breakdown.

ELECTRICAL ENERGY AND CAPACITANCE The concept of potential energy was first introduced in Chapter 5 in connection with the conservative forces of gravity and springs. By using the principle of conservation of energy, we were often able to avoid working directly with forces when solving problems. Here we learn that the potential energy concept is also useful in the study of electricity. Because the Coulomb force is conservative, we can define an electric potential energy corresponding to that force. In addition, we define an electric potential—the potential energy per unit charge—corresponding to the electric field. With the concept of electric potential in hand, we can begin to understand electric circuits, starting with an investigation of common circuit elements called capacitors. These simple devices store electrical energy and have found uses virtually everywhere, from etched circuits on a microchip to the creation of enormous bursts of power in fusion experiments.

16.1

16.1

Potential Difference and Electric Potential

16.2

Electric Potential and Potential Energy Due to Point Charges

16.3

Potentials and Charged Conductors

16.4

Equipotential Surfaces

16.5

Applications

16.6

Capacitance

16.7

The Parallel-Plate Capacitor

16.8

Combinations of Capacitors

16.9

Energy Stored in a Charged Capacitor

16.10 Capacitors with Dielectrics

POTENTIAL DIFFERENCE AND ELECTRIC POTENTIAL

Electric potential energy and electric potential are closely related concepts. The electric potential turns out to be just the electric potential energy per unit charge. This relationship is similar to that between electric force and the electric field, which is the electric force per unit charge.

Work and Electric Potential Energy

S

Recall from Chapter 5 that the work done by a conservative force F on an object depends only on the initial and final positions of the object and not on the path taken between those two points. This, in turn, means that a potential energy function PE exists. As we have seen, potential energy is a scalar quantity with the change in potential energy equal by defi nition to the negative of the work done by the conservative force: PE  PEf
PEi 
W F .

531

532

Chapter 16

High PE +

Electrical Energy and Capacitance Low PE –

E

+ + 0

+ + + +

– A xi q +

∆x

B xf

qE

x  xf
xi

–

x

– – – –

FIGURE 16.1 When a charge q S moves in a uniform electric field E from point A to point B, the work done on the charge by the electric force is qEx x.

Both the Coulomb force law and the universal law of gravity are proportional to 1/r 2. Because they have the same mathematical form and because the gravity force is conservative, it follows that the Coulomb force is also conservative. As with gravity, an electrical potential energy function can be associated with this force. To make these ideas more quantitative, imagine a small positive charge placed S at point A in a uniform electric field E, as in Figure 16.1. For simplicity, we first consider only constant electric fields and charges that move parallel to that field in one dimension (taken to be the x-axis). The electric field between equally and oppositely charged parallel plates is an example of a field that is approximately constant. (See Chapter 15.) AsSthe charge moves from point A to point B under the influence of the electric field E, the work done on the charge by the electric field is S equal to the part of the electric force qE acting parallel to the displacement times the displacement x  xf
xi: WAB  Fx x  qEx(xf
xi) S

In this expression q is the chargeS and Ex is the vector component of E in the S x-direction (not the magnitude of E). Unlike the magnitude Sof E, the component Ex can be positive or negative, depending on the direction of E, although in Figure 16.1 Ex is positive. Finally, note that the displacement, like q and Ex , can also be either positive or negative, depending on the direction of the displacement. The preceding expression for the work done by an electric field on a charge moving in one dimension is valid for both positive and negative charges and for constant electric fields pointing in any direction. When numbers are substituted with correct signs, the overall correct sign automatically results. In some books the expression W  qEd is used, instead, where E is the magnitude of the electric field and d is the distance the particle travels. The weakness of this formulation is that it doesn’t allow, mathematically, for negative electric work on positive charges, nor for positive electric work on negative charges! Nonetheless, the expression is easy to remember and useful for finding magnitudes: the magnitude of the work done by a constant electric field on a charge moving parallel to the field is always given by
W

q
Ed. We can substitute our definition of electric work into the work–energy theorem (assume other forces are absent): W  qEx x  KE The electric force is conservative, so the electric work depends only on the endpoints of the path, A and B, not on the path taken. Therefore, as the charge accelerates to the right in Figure 16.1, it gains kinetic energy and loses an equal amount of potential energy. Recall from Chapter 5 that the work done by a conservative force can be reinterpreted as the negative of the change in a potential energy associated with that force. This interpretation motivates the definition of the change in electric potential energy: Change in electric potential energy R

The change in the electric potential energy, PE, of a system consisting of an objectSof charge q moving through a displacement x in a constant electric field E is given by PE 
WAB 
qEx x

[16.1]

where Ex is the x-component of the electric field and x  xf
xi is the displacement of the charge along the x-axis. SI unit: joule ( J) Although potential energy can be defined for any electric field Equation 16.1 is valid only for the case of a uniform (i.e., constant) electric field, for a particle that undergoes a displacement along a given axis (here called the x-axis). Because the electric field is conservative, the change in potential energy doesn’t depend on

16.1

Potential Difference and Electric Potential

the path. Consequently, it’s unimportant whether or not the charge remains on the axis at all times during the displacement: the change in potential energy will be the same. In subsequent sections we will examine situations in which the electric field is not uniform. Electric and gravitational potential energy can be compared in Figure 16.2. In this figure the electric and gravitational fields are both directed downwards. We see that positive charge in an electric field acts very much like mass in a gravity field: a positive charge at point A falls in the direction of the electric field, just as a positive mass falls in the direction of the gravity field. Let point B be the zero point for potential energy in both Figures 16.2a and 16.2b. From conservation of energy, in falling from point A to point B the positive charge gains kinetic energy equal in magnitude to the loss of electric potential energy: KE  PE el  KE  (0

q
Ed)  0

:

KE 
q
Ed

The absolute-value signs on q are there only to make explicit that the charge is positive in this case. Similarly, the object in Figure 16.2b gains kinetic energy equal in magnitude to the loss of gravitational potential energy: KE  PEg  KE  (0
mgd)  0 :

KE  mgd

A

533

A

d

d q qE

B

m mg

B

g

E (a)

(b)

FIGURE 16.2 (a) When the electric S field E is directed downward, point B is at a lower electric potential than point A. As a positive test charge moves from A to B, the electric potential energy decreases. (b) As an object of mass m moves in the direction of S the gravitational field g , the gravitational potential energy decreases.

So for positive charges, electric potential energy works very much like gravitational potential energy. In both cases moving an object opposite the direction of the field results in a gain of potential energy, and upon release, the potential energy is converted to the object’s kinetic energy. Electric potential energy differs significantly from gravitational potential energy, however, in that there are two kinds of electrical charge—positive and negative—whereas gravity has only positive “gravitational charge” (i.e. mass). A negatively charged particle at rest at point A in Figure 16.2a would have to be pushed down to point B. To see why, apply the work–energy theorem to a negative charge at rest at point A and assumed to have some speed v on arriving at point B: W 5 DKE 1 DPE el 5 1 12mv 2 2 0 2 1 3 0 2 1 2 0 q 0 Ed 2 4

W 5 12mv 2 1 0 q 0 Ed

Notice that the negative charge,

q
, unlike the positive charge, had a positive change in electric potential energy in moving from point A to point B. If the negative charge has any speed at point B, the kinetic energy corresponding to that speed is also positive. Because both terms on the right-hand side of the work–energy equation are positive, there is no way of getting the negative charge from point A to point B without doing positive work W on it. In fact, if the negative charge is simply released at point A, it will “fall” upwards against the direction of the field! QUICK QUIZ 16.1 If an electron is released from rest in a uniform electric field, does the electric potential energy of the charge–field system (a) increase, (b) decrease, or (c) remain the same?

EXAMPLE 16.1 Potential Energy Differences in an Electric Field Goal

Illustrate the concept of electric potential energy.

Problem A proton is released from rest at x 
2.00 cm in a constant electric field with magnitude 1.50  103 N/C, pointing in the positive x-direction. (a) Calculate the change in the electric potential energy associated with the proton when it reaches x  5.00 cm. (b) An electron is now fired in the same direction from the same position. What is its change in electric potential energy associated with the electron if it reaches x  12.0 cm? (c) If the direction of

the electric field is reversed and an electron is released from rest at x  3.00 cm, by how much has the electric potential energy changed when the electron reaches x  7.00 cm? Strategy This problem requires a straightforward substitution of given values into the definition of electric potential energy, Equation 16.1.

534

Chapter 16

Electrical Energy and Capacitance

Solution (a) Calculate the change in the electric potential energy associated with the proton. Apply Equation 16.1:

PE 
qEx x 
qEx(xf
xi) 
(1.60  10
19 C)(1.50  103 N/C)  [0.050 0 m
(
0.020 0 m)]  21.68 3 10217 J

(b) Find the change in electric potential energy associated with an electron fired from x 
0.020 0 m and reaching x  0.120 m. Apply Equation 16.1, but in this case note that the electric charge q is negative:

PE 
qEx x 
qEx(xf
xi) 
(
1.60  10
19 C)(1.50  103 N/C)  [(0.120 m
(
0.020 0 m)]  13.36 3 10217 J

(c) Find the change in potential energy associated with an electron traveling from x  3.00 cm to x  7.00 cm if the direction of the electric field is reversed. Substitute, but now the electric field points in the negative x-direction, hence carries a minus sign:

PE 
qEx x 
qEx (xf
xi) 
(
1.60  10
19 C)(
1.50  103 N/C)  (0.070 m
0.030 m)  29.60 3 10218 J

Remarks Notice that the proton (actually the proton–field system) lost potential energy when it moved in the positive x-direction, whereas the electron gained potential energy when it moved in the same direction. Finding changes in potential energy with the field reversed was only a matter of supplying a minus sign, bringing the total number in this case to three! It’s important not to drop any of the signs. QUESTION 16.1 True or False: When an electron is released from rest in a constant electric field, the change in the electric potential energy associated with the electron becomes more negative with time. EXERCISE 16.1 Find the change in electric potential energy associated with the electron in part (b) as it goes on from x  0.120 m to x 
0.180 m. (Note that the electron must turn around and go back at some point. The location of the turning point is unimportant because changes in potential energy depend only on the endpoints of the path.) Answer
7.20  10
17 J

EXAMPLE 16.2 Dynamics of Charged Particles Goal

Use electric potential energy in conservation of energy problems.

Problem (a) Find the speed of the proton at x  0.050 0 m in part (a) of Example 16.1. (b) Find the initial speed of the electron (at x 
2.00 cm) in part (b) of Example 16.1 given that its speed has fallen by half when it reaches x  0.120 m.

16.1

Potential Difference and Electric Potential

535

Strategy Apply conservation of energy, solving for the unknown speeds. Part (b) involves two equations: the conservation of energy equation and the condition v f 5 12v i for the unknown initial and final speeds. The changes in electric potential energy have already been calculated in Example 16.1. Solution (a) Calculate the proton’s speed at x  0.050 m. Use conservation of energy, with an initial speed of zero:

DKE 1 DPE 5 0

Solve for v and substitute the change in potential energy found in Example 16.1a:

v2 5 2

v5

5

S

1 12m pv 2 2 0 2 1 DPE 5 0

2 DPE mp

Å

2

2 DPE mp

Å

2

2 1 21.68 3 10217 J 2 1 1.67 3 10227 kg 2

 1.42 3 105 m/s (b) Find the electron’s initial speed given that its speed has fallen by half at x  0.120 m. Apply conservation of energy once again, substituting expressions for the initial and final kinetic energies:

1 12m ev f 2

KE  PE  0

2

1 1 2 2m e 1 2v i 2

Substitute the condition v f 5 12v i and subtract the change in potential energy from both sides:

1 2 2 m ev i 2

1 DPE 5 0

2 12m ev i2 5 2DPE

238m ev i2 5 2DPE

Combine terms and solve for vi , the initial speed, and substitute the change in potential energy found in Example 16.1b:

vi 5

8 1 3.36 3 10217 J 2 8 DPE 5 Å 3m e Å 3 1 9.11 3 10231 kg 2

 9.92 3 106 m/s Remarks Although the changes in potential energy associated with the proton and electron were similar in magnitude, the effect on their speeds differed dramatically. The change in potential energy had a proportionately much greater effect on the much lighter electron than on the proton. QUESTION 16.2 True or False: If a proton and electron both move through the same displacement in an electric field, the change in potential energy associated with the proton must be equal in magnitude and opposite in sign to the change in potential energy associated with the electron. EXERCISE 16.2 Refer to Exercise 16.1. Find the electron’s speed at x 
0.180 m. Answer 1.35  107 m/s

Electric Potential

The answer is 4.5% of the speed of light.

S

In Chapter 15 itS was convenient to define an electric field E related to the elecS tric force F 5 qE. In this way the properties of fixed collections of charges could be easily studied, and the force on any particle in the electric field could be obtained simply by multiplying by the particle’s charge q. For the same reasons, it’s

536

Chapter 16

Electrical Energy and Capacitance

useful to define an electric potential difference V related to the potential energy by PE  q V: Potential difference between two points R

The electric potential difference V between points A and B is the change in electric potential energy as a charge q moves from A to B divided by the charge q: DPE [16.2] DV 5 VB 2 VA 5 q SI unit: joule per coulomb, or volt (J/C, or V) This definition is completely general, although in many cases calculus would be required to compute the change in potential energy of the system. Because electric potential energy is a scalar quantity, electric potential is also a scalar quantity. From Equation 16.2, we see that electric potential difference is a measure of the change in electric potential energy per unit charge. Alternately, the electric potential difference is the work per unit charge that would have to be done by some force to move a charge from point A to point B in the electric field. The SI unit of electric potential is the joule per coulomb, called the volt (V). From the definition of that unit, 1 J of work must be done to move a 1-C charge between two points that are at a potential difference of 1 V. In the process of moving through a potential difference of 1 V, the 1-C charge gains 1 J of energy. For the special case of a uniform electric field such as that between charged parallel plates, dividing Equation 16.1 by q gives DPE 5 2E x Dx q Comparing this equation with Equation 16.2, we find that

TIP 16.1 Potential and Potential Energy

DV 5 2E x Dx

Electric potential is characteristic of the field only, independent of a test charge that may be placed in that field. On the other hand, potential energy is a characteristic of the charge-field system due to an interaction between the field and a charge placed in the field.

Equation 16.3 shows that potential difference also has units of electric field times distance. It then follows that the SI unit of the electric field, the newton per coulomb, can also be expressed as volts per meter: 1 N/C  1 V/m Because Equation 16.3 is directly related to Equation 16.1, remember that it’s valid only for the system consisting of a uniform electric field and a charge moving in one dimension. Released from rest, positive charges accelerate spontaneously from regions of high potential to low potential. If a positive charge is given some initial velocity in the direction of high potential, it can move in that direction, but will slow and finally turn around, just like a ball tossed upwards in a gravity field. Negative charges do exactly the opposite: released from rest, they accelerate from regions of low potential toward regions of high potential. Work must be done on negative charges to make them go in the direction of lower electric potential. QUICK QUIZ 16.2 If a negatively charged particle is placed at rest in an electric potential field that increases in the positive x-direction, will the particle (a) accelerate in the positive x-direction, (b) accelerate in the negative x-direction, or (c) remain at rest?

V

A FIGURE 16.3 & 16.4)

[16.3]

B

(Quick Quizzes 16.3

x

QUICK QUIZ 16.3 Figure 16.3 is a graph of an electric potential as a function of position. If a positively charged particle is placed at point A, what will its subsequent motion be? Will it (a) go to the right, (b) go to the left, (c) remain at point A, or (d) oscillate around point B?

16.1

Potential Difference and Electric Potential

537

QUICK QUIZ 16.4 If a negatively charged particle is placed at point B in Figure 16.3 and given a very small kick to the right, what will its subsequent motion be? Will it (a) go to the right and not return, (b) go to the left, (c) remain at point B, or (d) oscillate around point B? An application of potential difference is the 12-V battery found in an automobile. Such a battery maintains a potential difference across its terminals, with the positive terminal 12 V higher in potential than the negative terminal. In practice the negative terminal is usually connected to the metal body of the car, which can be considered to be at a potential of zero volts. The battery provides the electrical current necessary to operate headlights, a radio, power windows, motors, and so forth. Now consider a charge of 1 C, to be moved around a circuit that contains the battery connected to some of these external devices. As the charge is moved inside the battery from the negative terminal (at 0 V) to the positive terminal (at 12 V), the work done on the charge by the battery is 12 J. Every coulomb of positive charge that leaves the positive terminal of the battery carries an energy of 12 J. As the charge moves through the external circuit toward the negative terminal, it gives up its 12 J of electrical energy to the external devices. When the charge reaches the negative terminal, its electrical energy is zero again. At this point, the battery takes over and restores 12 J of energy to the charge as it is moved from the negative to the positive terminal, enabling it to make another transit of the circuit. The actual amount of charge that leaves the battery each second and traverses the circuit depends on the properties of the external devices, as seen in the next chapter.

APPLICATION Automobile Batteries

EXAMPLE 16.3 TV Tubes and Atom Smashers Goal

Relate electric potential to an electric field and conservation of energy.

High potential +

Problem In atom smashers (also known as cyclotrons and linear accelerators) charged particles are accelerated in much the same way they are accelerated in TV tubes: through potential differences. Suppose a proton is injected at a speed of 1.00  106 m/s between two plates 5.00 cm apart, as shown in Figure 16.4. The proton subsequently accelerates across the gap and exits through the opening. (a) What must the electric potential difference be if the exit speed is to be 3.00  106 m/s? (b) What is the magnitude of the electric field between the plates, assuming it’s constant? Strategy Use conservation of energy, writing the change in potential energy in terms of the change in electric potential, V, and solve for V. For part (b), solve Equation 16.3 for the electric field.

+ + +

v

+ +

+

–

E –

– +

–

+ +

– –

– Low potential 5.00 cm

FIGURE 16.4 (Example 16.3) A proton enters a cavity and accelerates from one charged plate toward the S other in an electric field E .

Solution (a) Find the electric potential yielding the desired exit speed of the proton. Apply conservation of energy, writing the potential energy in terms of the electric potential: Solve the energy equation for the change in potential:

Substitute the given values, obtaining the necessary potential difference:

KE  PE  KE  q V  0 1 1 2 2 mp 2 m pv f 2 2 m pv i DKE 1v 2 2 vi 22 DV 5 2 52 52 q q 2q f

DV 5 2

1 1.67 3 10227 kg 2

2 1 1.60 3 10219 C 2

3 1 3.00 3 106 m/s 2 2


(1.00  106 m/s)24

 V  24.18 3 104 V

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(b) What electric field must exist between the plates? Solve Equation 16.3 for the electric field and substitute:

E52

4.18 3 104 V DV 5 5 8.36 3 105 N/C Dx 0.050 0 m

Remarks Systems of such cavities, consisting of alternating positive and negative plates, are used to accelerate charged particles to high speed before smashing them into targets. To prevent a slowing of, say, a positively charged particle after it passes through the negative plate of one cavity and enters the next, the charges on the plates are reversed. Otherwise, the particle would be traveling from the negative plate to a positive plate in the second cavity, and the kinetic energy gained in the previous cavity would be lost in the second.

EXERCISE 16.3 Suppose electrons in a TV tube are accelerated through a potential difference of 2.00  104 V from the heated cathode (negative electrode), where they are produced, toward the screen, which also serves as the anode (positive electrode), 25.0 cm away. (a) At what speed would the electrons impact the phosphors on the screen? Assume they accelerate from rest and ignore relativistic effects (Chapter 26). (b) What’s the magnitude of the electric field, if it is assumed constant?

QUESTION 16.3 True or False: A more massive particle gains less energy in traversing a given potential difference than does a lighter particle.

Answers (a) 8.38  107 m/s (b) 8.00  104 V/m

E in volts/m V in volts

1.00

16.2

0.800 0.600

keq V= r

0.400 0.200

E =

0.0

keq r2 2.00

4.00

r (m) 6.00

FIGURE 16.5 Electric field and electric potential versus distance from a point charge of 1.11  10
10 C. Note that V is proportional to 1/r, whereas E is proportional to 1/r 2.

Electric potential created by a point charge R

Superposition principle R

ELECTRIC POTENTIAL AND POTENTIAL ENERGY DUE TO POINT CHARGES

In electric circuits a point of zero electric potential is often defined by grounding (connecting to the Earth) some point in the circuit. For example, if the negative terminal of a 12-V battery were connected to ground, it would be considered to have a potential of zero, whereas the positive terminal would have a potential of 12 V. The potential difference created by the battery, however, is only locally defined. In this section we describe the electric potential of a point charge, which is defined throughout space. The electric field of a point charge extends throughout space, so its electric potential does, also. The zero point of electric potential could be taken anywhere, but is usually taken to be an infinite distance from the charge, far from its influence and the influence of any other charges. With this choice, the methods of calculus can be used to show that the electric potential created by a point charge q at any distance r from the charge is given by V 5 ke

q r

[16.4]

Equation 16.4 shows that the electric potential, or work per unit charge, required to move a test charge in from infinity to a distance r from a positive point charge q increases as the positive test charge moves closer to q. A plot of Equation 16.4 in Figure 16.5 shows that the potential associated with a point charge decreases as 1/r with increasing r, in contrast to the magnitude of the charge’s electric field, which decreases as 1/r 2. The electric potential of two or more charges is obtained by applying the superposition principle: the total electric potential at some point P due to several point charges is the algebraic sum of the electric potentials due to the individual charges. This method is similar to the one used in Chapter 15 to find the resultant electric field at a point in space. Unlike electric field superposition, which involves a sum of vectors, the superposition of electric potentials requires evaluating a sum of scalars. As a result, it’s much easier to evaluate the electric potential at some point due to several charges than to evaluate the electric field, which is a vector quantity.

16.2

Electric Potential and Potential Energy Due to Point Charges

FIGURE 16.6 The electric potential (in arbitrary units) in the plane containing an electric dipole. Potential is plotted in the vertical dimension.

2.0

Electric potential

539

1.0

0

–1.0

–2.0

Figure 16.6 is a computer-generated plot of the electric potential associated with an electric dipole, which consists of two charges of equal magnitude but opposite in sign. The charges lie in a horizontal plane at the center of the potential spikes. The value of the potential is plotted in the vertical dimension. The computer program has added the potential of each charge to arrive at total values of the potential. Just as in the case of constant electric fields, there is a relationship between electric potential and electric potential energy. If V1 is the electric potential due to charge q 1 at a point P (Active Figure 16.7a) the work required to bring charge q 2 from infinity to P without acceleration is q 2V1. By definition, this work equals the potential energy PE of the two-particle system when the particles are separated by a distance r (Active Fig. 16.7b). We can therefore express the electrical potential energy of the pair of charges as PE 5 q 2V1 5 k e

q 1q 2 r

[16.5]

O Potential energy of a pair of charges

If the charges are of the same sign, PE is positive. Because like charges repel, positive work must be done on the system by an external agent to force the two charges near each other. Conversely, if the charges are of opposite sign, the force is attractive and PE is negative. This means that negative work must be done to prevent unlike charges from accelerating toward each other as they are brought close together. QUICK QUIZ 16.5 Consider a collection of charges in a given region and suppose all other charges are distant and have a negligible effect. Further, the electric potential is taken to be zero at infinity. If the electric potential at a given point in the region is zero, which of the following statements must be true? (a) The electric field is zero at that point. (b) The electric potential energy is a minimum at that point. (c) There is no net charge in the region. (d) Some charges in the region are positive, and some are negative. (e) The charges have the same sign and are symmetrically arranged around the given point. QUICK QUIZ 16.6 A spherical balloon contains a positively charged particle at its center. As the balloon is inflated to a larger volume while the charged particle remains at the center, which of the following are true? (a) The electric potential at the surface of the balloon increases. (b) The magnitude of the electric field at the surface of the balloon increases. (c) The electric flux through the balloon remains the same. (d) none of these.

P V = ke q1 1 r r

q1 (a)

q2

r PE =

ke q1q2 r

q1 (b) ACTIVE FIGURE 16.7 (a) The electric potential V1 at P due to the point charge q 1 is V1  keq 1/r. (b) If a second charge, q 2, is brought from infinity to P, the potential energy of the pair is PE  keq 1q 2/r.

540

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PROBLEM -SOLVING STRATEGY ELECTRIC POTENTIAL

1. Draw a diagram of all charges and circle the point of interest. 2. Calculate the distance from each charge to the point of interest, labeling it on the diagram. k eq 3. For each charge q, calculate the scalar quantity V 5 . The sign of each r charge must be included in your calculations! 4. Sum all the numbers found in the previous step, obtaining the electric potential at the point of interest.

EXAMPLE 16.4

Finding the Electric Potential

Goal Calculate the electric potential due to a collection of point charges. Problem A 5.00-mC point charge is at the origin, and a point charge q 2 
2.00 mC is on the x-axis at (3.00, 0) m, as in Figure 16.8. (a) If the electric potential is taken to be zero at infinity, find the total electric potential due to these charges at point P with coordinates (0, 4.00) m. (b) How much work is required to bring a third point charge of 4.00 mC from infinity to P ?

y (m) (0, 4.00) P

r1

r2

q1 0

q2 

x (m)
(3.00, 0)

FIGURE 16.8 (Example 16.4) The electric potential at point P due to the point charges q 1 and q 2 is the algebraic sum of the potentials due to the individual charges.

Strategy For part (a), the electric potential at P due to each charge can be calculated from V  keq/r. The total electric potential at P is the sum of these two numbers. For part (b), use the work–energy theorem, together with Equation 16.5, recalling that the potential at infinity is taken to be zero. Solution (a) Find the electric potential at point P. Calculate the electric potential at P due to the 5.00-mC charge:

V1 5 k e

q1 N # m2 5.00 3 1026 C 5 a8.99 3 109 ba b r1 4.00 m C2

5 1.12 3 104 V Find the electric potential at P due to the
2.00-mC charge:

V2 5 k e

q2 N # m2 22.00 3 1026 C 5 a8.99 3 109 ba b r2 5.00 m C2

5 20.360 3 104 V Sum the two numbers to find the total electric potential at P :

V P  V1  V2  1.12  104 V  (
0.360  104 V)  7.6 3 103 V

(b) Find the work needed to bring the 4.00-mC charge from infinity to P. Apply the work-energy theorem, with Equation 16.5:

W  PE  q 3 V  q 3(V P
V)  (4.00  10
6 C)(7.6  103 V
0) W  3.0 3 1022 J

16.2

Electric Potential and Potential Energy Due to Point Charges

Remarks Unlike the electric field, where vector addition is required, the electric potential due to more than one charge can be found with ordinary addition of scalars. Further, notice that the work required to move the charge is equal to the change in electric potential energy. The sum of the work done moving the particle plus the work done by the electric field is zero (Wother  Welectric  0) because the particle starts and ends at rest. Therefore, Wother 
Welectric  Uelectric  q V.

541

increase. (b) It would decrease. (c) It would remain the same. EXERCISE 16.4 Suppose a charge of
2.00 mC is at the origin and a charge of 3.00 mC is at the point (0, 3.00) m. (a) Find the electric potential at (4.00, 0) m, assuming the electric potential is zero at infinity, and (b) find the work necessary to bring a 4.00 mC charge from infinity to the point (4.00, 0) m.

QUESTION 16.4 If q 2 were moved to the right, what would happen to the electric potential Vp at point P? (a) It would

Answers (a) 8.99  102 V

(b) 3.60  10
3 J

EXAMPLE 16.5 Electric Potential Energy and Dynamics Goal Apply conservation of energy and electrical potential energy to a configuration of charges. Problem Suppose three protons lie on the x-axis, at rest relative to one another at a given instant of time, as in Figure 16.9. If proton q 3 on the right is released while the others are held fixed in place, find a symbolic expression for the proton’s speed at infinity and evaluate this speed when r 0  2.00 fm. (Note: 1 fm  10
15 m.)

q1

q2

q3







r0

FIGURE 16.9

r0

x

(Example 16.5)

Strategy First calculate the initial electric potential energy associated with the system of three particles. There will be three terms, one for each interacting pair. Then calculate the final electric potential energy associated with the system when the proton on the right is arbitrarily far away. Because the electric potential energy falls off as 1/r, two of the terms will vanish. Using conservation of energy then yields the speed of the particle in question. Solution Calculate the electric potential energy associated with the initial configuration of charges:

PE i 5

k e q 1q 2 k e q 1q 3 k e q 2q 3 kee 2 kee 2 kee 2 1 1 5 1 1 r 12 r 13 r 23 r0 r0 2r 0

Calculate the electric potential energy associated with the final configuration of charges:

PE f 5

k e q 1q 2 kee 2 5 r 12 r0

Write the conservation of energy equation:

KE  PE  KEf
KEi  PEf
PEi  0

Substitute appropriate terms:

1 2 2 m 3v 3

201

1 2 2 m 3v 3

2a

Solve for v 3 after combining the two remaining potential energy terms:

v3 

Evaluate taking r 0  2.00 fm:

v3 5

kee 2 kee 2 kee 2 kee 2 2a 1 1 b50 r0 r0 r0 2r 0

kee 2 kee 2 1 b50 r0 2r 0

3k e e 2 Å m 3r 0 3 1 8.99 3 109 N # m2 /C2 2 1 1.60 3 10219 C 2 2 5 1.44 3 107 m/s 1 1.67 3 10 227 kg 2 1 2.00 3 10215 m 2 Å

542

Chapter 16

Electrical Energy and Capacitance

Remarks The difference in the initial and final kinetic energies yields the energy available for motion. This calculation is somewhat contrived because it would be difficult, although not impossible, to arrange such a configuration of protons; it could conceivably occur by chance inside a star. QUESTION 16.5 If a fourth proton were placed to the right of q 3, how many additional potential energy terms would have to be calculated in the initial configuration? EXERCISE 16.5 Starting from the initial configuration of three protons, suppose the end two particles are released simultaneously and the middle particle is fi xed. Obtain a numerical answer for the speed of the two particles at infinity. (Note that their speeds, by symmetry, must be the same.) Answer 1.31  107 m/s

16.3

POTENTIALS AND CHARGED CONDUCTORS

The electric potential at all points on a charged conductor can be determined by combining Equations 16.1 and 16.2. From Equation 16.1, we see that the work done on a charge by electric forces is related to the change in electrical potential energy of the charge by W 
PE From Equation 16.2, we see that the change in electric potential energy between two points A and B is related to the potential difference between those points by PE  q(V B
VA) Combining these two equations, we find that

++ ++

W 5 2q 1 VB 2 VA 2

+ + + +

+ + +

+ +

+ + + + + + + + +

+ + + B + + + + A E

FIGURE 16.10 An arbitrarily shaped conductor with an excess positive charge. When the conductor is in electrostatic equilibrium, all the S charge resides at the surface, E 5 0, inside the conductor, and the electric field just outside the conductor is perpendicular to the surface. The potential is constant inside the conductor and is equal to the potential at the surface.

[16.6]

Using this equation, we obtain the following general result: No net work is required to move a charge between two points that are at the same electric potential. In mathematical terms this result says that W  0 whenever V B  VA . In Chapter 15 we found that when a conductor is in electrostatic equilibrium, a net charge placed on it resides entirely on its surface. Further, we showed that the electric field just outside the surface of a charged conductor in electrostatic equilibrium is perpendicular to the surface and that the field inside the conductor is zero. We now show that all points on the surface of a charged conductor in electrostatic equilibrium are at the same potential. Consider a surface path connecting any points A and B on a charged conductor, as in Figure 16.10. The charges on the conductor are assumed to be in equilibS rium with each other, so none are moving. In this case the electric field E is always perpendicular to the displacement along this path. This must be so, for otherwise the part Sof the electric field tangent to the surface would move the charges. Because E is perpendicular to the path, no work is done by the electric field if a charge is moved between the given two points. From Equation 16.6 we see that if the work done is zero, the difference in electric potential, V B
VA , is also zero. It follows that the electric potential is a constant everywhere on the surface of a charged conductor in equilibrium. Further, because the electric field inside a conductor is zero, no work is required to move a charge between two points inside the conductor. Again, Equation 16.6 shows that if the work done is zero, the difference in electric potential between any two points inside a conductor must also be zero. We conclude that the electric potential is constant everywhere inside a conductor. Finally, because one of the points inside the conductor could be arbitrarily close to the surface of the conductor, we conclude that the electric potential is constant

16.4

Equipotential Surfaces

543

everywhere inside a conductor and equal to that same value at the surface. As a consequence, no work is required to move a charge from the interior of a charged conductor to its surface. (It’s important to realize that the potential inside a conductor is not necessarily zero, even though the interior electric field is zero.)

The Electron Volt An appropriately sized unit of energy commonly used in atomic and nuclear physics is the electron volt (eV). For example, electrons in normal atoms typically have energies of tens of eV’s, excited electrons in atoms emitting x-rays have energies of thousands of eV’s, and high-energy gamma rays (electromagnetic waves) emitted by the nucleus have energies of millions of eV’s. The electron volt is defined as the kinetic energy that an electron gains when accelerated through a potential difference of 1 V. Because 1 V  1 J/C and because the magnitude of the charge on the electron is 1.60  10
19 C, we see that the electron volt is related to the joule by 1 eV  1.60  10
19 C  V  1.60  10
19 J

[16.7]

QUICK QUIZ 16.7 An electron initially at rest accelerates through a potential difference of 1 V, gaining kinetic energy KEe , whereas a proton, also initially at rest, accelerates through a potential difference of
1 V, gaining kinetic energy KEp. Which of the following relationships holds? (a) KEe  KEp (b) KEe KEp (c) KEe KEp (d) The answer can’t be determined from the given information.

16.4

EQUIPOTENTIAL SURFACES

A surface on which all points are at the same potential is called an equipotential surface. The potential difference between any two points on an equipotential surface is zero. Hence, no work is required to move a charge at constant speed on an equipotential surface. Equipotential surfaces have a simple relationship to the electric field: The electric field at every point of an equipotential surface is perpendicular to the surS face. If the electric field E had a component parallel to the surface, that component would produce an electric force on a charge placed on the surface. This force would do work on the charge as it moved from one point to another, in contradiction to the definition of an equipotential surface. Equipotential surfaces can be represented on a diagram by drawing equipotential contours, which are two-dimensional views of the intersections of the equipotential surfaces with the plane of the drawing. These equipotential contours are generally referred to simply as equipotentials. Figure 16.11a (page 544) shows the equipotentials (in blue) associated with a positive point charge. Note that the equipotentials are perpendicular to the electric field lines (in red) at all points. Recall that the electric potential created by a point charge q is given by V  ke q/r. This relation shows that, for a single point charge, the potential is constant on any surface on which r is constant. It follows that the equipotentials of a point charge are a family of spheres centered on the point charge. Figure 16.11b shows the equipotentials associated with two charges of equal magnitude but opposite sign.

O Definition of the electron volt

544

Chapter 16

Electrical Energy and Capacitance

FIGURE 16.11 Equipotentials (dashed blue lines) and electric field lines (red lines) for (a) a positive point charge and (b) two point charges of equal magnitude and opposite sign. In all cases the equipotentials are perpendicular to the electric field lines at every point.

Electric field line




+ +

q

Equipotential (a)

16.5

(b)

APPLICATIONS

The Electrostatic Precipitator APPLICATION

One important application of electric discharge in gases is a device called an electrostatic precipitator. This device removes particulate matter from combustion gases, thereby reducing air pollution. It’s especially useful in coal-burning power plants and in industrial operations that generate large quantities of smoke. Systems currently in use can eliminate approximately 90% by mass of the ash and dust from the smoke. Unfortunately, a very high percentage of the lighter particles still escape, and they contribute significantly to smog and haze. Figure 16.12 illustrates the basic idea of the electrostatic precipitator. A high voltage (typically 40 kV to 100 kV) is maintained between a wire running down the center of a duct and the outer wall, which is grounded. The wire is maintained at a negative electric potential with respect to the wall, so the electric field is directed toward the wire. The electric field near the wire reaches a high enough value to cause a discharge around the wire and the formation of positive ions, electrons, and negative ions, such as O2
. As the electrons and negative ions are acceler-

The Electrostatic Precipitator

b, Riei O’Harra/Black Star/PNI; c, Greig Cranna/Stock, Boston/PNI

Insulator

Clean air out

Dirty air in

Weight

Dirt out (a)

(b)

(c)

FIGURE 16.12 (a) A schematic diagram of an electrostatic precipitator. The high voltage maintained on the central wires creates an electric discharge in the vicinity of the wire. Compare the air pollution when the precipitator is (b) operating and (c) turned off.

16.5

ated toward the outer wall by the nonuniform electric field, the dirt particles in the streaming gas become charged by collisions and ion capture. Because most of the charged dirt particles are negative, they are also drawn to the outer wall by the electric field. When the duct is shaken, the particles fall loose and are collected at the bottom. In addition to reducing the amounts of harmful gases and particulate matter in the atmosphere, the electrostatic precipitator recovers valuable metal oxides from the stack. A similar device called an electrostatic air cleaner is used in homes to relieve the discomfort of allergy sufferers. Air laden with dust and pollen is drawn into the device across a positively charged mesh screen. The airborne particles become positively charged when they make contact with the screen, and then they pass through a second, negatively charged mesh screen. The electrostatic force of attraction between the positively charged particles in the air and the negatively charged screen causes the particles to precipitate out on the surface of the screen, removing a very high percentage of contaminants from the air stream.

Applications

545

APPLICATION The Electrostatic Air Cleaner

Xerography and Laser Printers

APPLICATION

Xerography is widely used to make photocopies of printed materials. The basic idea behind the process was developed by Chester Carlson, who was granted a patent for his invention in 1940. In 1947 the Xerox Corporation launched a full-scale program to develop automated duplicating machines using Carlson’s process. The huge success of that development is evident: today, practically all offices and libraries have one or more duplicating machines, and the capabilities of these machines continue to evolve. Some features of the xerographic process involve simple concepts from electrostatics and optics. The one idea that makes the process unique, however, is the use of photoconductive material to form an image. A photoconductor is a material that is a poor conductor of electricity in the dark, but a reasonably good conductor when exposed to light. Figure 16.13 illustrates the steps in the xerographic process. First, the surface of a plate or drum is coated with a thin film of the photoconductive material (usually selenium or some compound of selenium), and the photoconductive surface is given a positive electrostatic charge in the dark (Fig. 16.13a). The page to be copied is then projected onto the charged surface (Fig. 16.13b). The photoconducting

Xerographic Copiers

Lens

Interlaced pattern of laser lines Laser beam

Selenium-coated drum (a) Charging the drum

Negatively charged toner (b) Imaging the document

(c) Applying the toner

(d) Transferring the toner to the paper

FIGURE 16.13 The xerographic process. (a) The photoconductive surface is positively charged. (b) Through the use of a light source and a lens, a hidden image is formed on the charged surface in the form of positive charges. (c) The surface containing the image is covered with a negatively charged powder, which adheres only to the image area. (d) A piece of paper is placed over the surface and given a charge. This transfers the image to the paper, which is then heated to “fi x” the powder to the paper. (e) The image on the drum of a laser printer is produced by turning a laser beam on and off as it sweeps across the selenium-coated drum.

(e) Laser printer drum

546

Chapter 16

Electrical Energy and Capacitance

surface becomes conducting only in areas where light strikes; there the light produces charge carriers in the photoconductor that neutralize the positively charged surface. The charges remain on those areas of the photoconductor not exposed to light, however, leaving a hidden image of the object in the form of a positive distribution of surface charge. Next, a negatively charged powder called a toner is dusted onto the photoconducting surface (Fig. 16.13c). The charged powder adheres only to the areas that contain the positively charged image. At this point, the image becomes visible. It is then transferred to the surface of a sheet of positively charged paper. Finally, the toner is “fixed” to the surface of the paper by heat (Fig. 16.13d), resulting in a permanent copy of the original. The steps for producing a document on a laser printer are similar to those used in a photocopy machine in that parts (a), (c), and (d) of Figure 16.13 remain essentially the same. The difference between the two techniques lies in the way the image is formed on the selenium-coated drum. In a laser printer the command to print the letter O, for instance, is sent to a laser from the memory of a computer. A rotating mirror inside the printer causes the beam of the laser to sweep across the selenium-coated drum in an interlaced pattern (Fig. 16.13e). Electrical signals generated by the printer turn the laser beam on and off in a pattern that traces out the letter O in the form of positive charges on the selenium. Toner is then applied to the drum, and the transfer to paper is accomplished as in a photocopy machine.

APPLICATION Laser Printers

16.6 –Q +Q

Area = A

d +

–

FIGURE 16.14 A parallel-plate capacitor consists of two parallel plates, each of area A, separated by a distance d. The plates carry equal and opposite charges.

Capacitance of a pair of conductors R

TIP 16.2 Potential Difference Is V, Not V Use the symbol V for the potential difference across a circuit element or a device (many other books use simply V for potential difference). The dual use of V to represent potential in one place and a potential difference in another can lead to unnecessary confusion.

CAPACITANCE

A capacitor is a device used in a variety of electric circuits, such as to tune the frequency of radio receivers, eliminate sparking in automobile ignition systems, or store short-term energy for rapid release in electronic flash units. Figure 16.14 shows a typical design for a capacitor. It consists of two parallel metal plates separated by a distance d. Used in an electric circuit, the plates are connected to the positive and negative terminals of a battery or some other voltage source. When this connection is made, electrons are pulled off one of the plates, leaving it with a charge of Q, and are transferred through the battery to the other plate, leaving it with a charge of
Q, as shown in the figure. The transfer of charge stops when the potential difference across the plates equals the potential difference of the battery. A charged capacitor is a device that stores energy that can be reclaimed when needed for a specific application. The capacitance C of a capacitor is the ratio of the magnitude of the charge on either conductor (plate) to the magnitude of the potential difference between the conductors (plates): C;

Q DV

[16.8]

SI unit: farad (F)  coulomb per volt (C/V) The quantities Q and V are always taken to be positive when used in Equation 16.8. For example, if a 3.0-mF capacitor is connected to a 12-V battery, the magnitude of the charge on each plate of the capacitor is Q  C V  (3.0  10
6 F)(12 V)  36 mC From Equation 16.8, we see that a large capacitance is needed to store a large amount of charge for a given applied voltage. The farad is a very large unit of capacitance. In practice, most typical capacitors have capacitances ranging from microfarads (1 mF  1  10
6 F) to picofarads (1 pF  1  10
12 F).

16.7

The Parallel-Plate Capacitor

547

FIGURE 16.15 (a) The electric field between the plates of a parallel-plate capacitor is uniform near the center, but nonuniform near the edges.

+Q Image not available due to copyright restrictions

–Q

(a)

16.7

THE PARALLEL-PLATE CAPACITOR

The capacitance of a device depends on the geometric arrangement of the conductors. The capacitance of a parallel-plate capacitor with plates separated by air (see Fig. 16.14) can be easily calculated from three facts. First, recall from Chapter 15 that the magnitude of the electric field between two plates is given by E  s/0, where s is the magnitude of the charge per unit area on each plate. Second, we found earlier in this chapter that the potential difference between two plates is V  Ed, where d is the distance between the plates. Third, the charge on one plate is given by q  sA, where A is the area of the plate. Substituting these three facts into the definition of capacitance gives the desired result: C5

q DV

5

sA sA 5 1 Ed s/P0 2 d

Key

B

Movable plate Dielectric Fixed plate FIGURE 16.16 When the key of one type of keyboard is pressed, the capacitance of a parallel-plate capacitor increases as the plate spacing decreases. The substance labeled “dielectric” is an insulating material, as described in Section 16.10.

Canceling the charge per unit area, s, yields C 5 P0

A d

[16.9]

where A is the area of one of the plates, d is the distance between the plates, and P0 is the permittivity of free space. From Equation 16.9, we see that plates with larger area can store more charge. The same is true for a small plate separation d because then the positive charges on one plate exert a stronger force on the negative charges on the other plate, allowing more charge to be held on the plates. Figure 16.15 shows the electric field lines of a more realistic parallel-plate capacitor. The electric field is very nearly constant in the center between the plates, but becomes less so when approaching the edges. For most purposes, however, the field may be taken as constant throughout the region between the plates. One practical device that uses a capacitor is the flash attachment on a camera. A battery is used to charge the capacitor, and the stored charge is then released when the shutter-release button is pressed to take a picture. The stored charge is delivered to a flash tube very quickly, illuminating the subject at the instant more light is needed. Computers make use of capacitors in many ways. For example, one type of computer keyboard has capacitors at the bases of its keys, as in Figure 16.16. Each key is connected to a movable plate, which represents one side of the capacitor; the fixed plate on the bottom of the keyboard represents the other side of the capacitor. When a key is pressed, the capacitor spacing decreases, causing an increase in capacitance. External electronic circuits recognize each key by the change in its capacitance when it is pressed. Capacitors are useful for storing a large amount of charge that needs to be delivered quickly. A good example on the forefront of fusion research is electrostatic confinement. In this role capacitors discharge their electrons through a grid. The negatively charged electrons in the grid draw positively charged particles to them and therefore to each other, causing some particles to fuse and release energy in the process.

O Capacitance of a parallelplate capacitor

APPLICATION Camera Flash Attachments

APPLICATION Computer Keyboards

APPLICATION Electrostatic Confinement

548

Chapter 16

EXAMPLE 16.6 Goal

Electrical Energy and Capacitance

A Parallel-Plate Capacitor

Calculate fundamental physical properties of a parallel-plate capacitor.

Problem A parallel-plate capacitor has an area A  2.00  10
4 m2 and a plate separation d  1.00  10
3 m. (a) Find its capacitance. (b) How much charge is on the positive plate if the capacitor is connected to a 3.00-V battery? Calculate (c) the charge density on the positive plate, assuming the density is uniform, and (d) the magnitude of the electric field between the plates. Strategy Parts (a) and (b) can be solved by substituting into the basic equations for capacitance. In part (c) use the definition of charge density, and in part (d) use the fact that the voltage difference equals the electric field times the distance. Solution (a) Find the capacitance. C 5 P0

Substitute into Equation 16.9:

A 2.00 3 1024 m2 5 1 8.85 3 10212 C2 /N # m2 2 a b d 1.00 3 1023 m

C  1.77 3 10212 F 5 1.77 pF (b) Find the charge on the positive plate after the capacitor is connected to a 3.00-V battery. Substitute into Equation 16.8:

C5

Q DV

S

Q 5 C DV 5 1 1.77 3 10212 F 2 1 3.00 V 2 5 5.31 3 10212 C

(c) Calculate the charge density on the positive plate. Charge density is charge divided by area:

s5

Q A

5

5.31 3 10212 C 5 2.66 3 1028 C/m2 2.00 3 1024 m2

(d) Calculate the magnitude of the electric field between the plates. Apply V  Ed:

E5

3.00 V DV 5 3.00 3 103 V/m 5 d 1.00 3 1023 m

Remarks The answer to part (d) could also have been obtained from the electric field derived for a parallel plate capacitor, Equation 15.13, E  s/P0. QUESTION 16.6 How do the answers change if the distance between the plates is doubled? EXERCISE 16.6 Two plates, each of area 3.00  10
4 m2, are used to construct a parallel-plate capacitor with capacitance 1.00 pF. (a) Find the necessary separation distance. (b) If the positive plate is to hold a charge of 5.00  10
12 C, find the charge density. (c) Find the electric field between the plates. (d) What voltage battery should be attached to the plate to obtain the preceding results? Answers (a) 2.66  10
3 m

(b) 1.67  10
8 C/m2 (c) 1.89  103 N/C (d) 5.00 V

Symbols for Circuit Elements and Circuits The symbol that is commonly used to represent a capacitor in a circuit is or sometimes . Don’t confuse either of these symbols with the circuit symbol,

which is used to designate a battery (or any other source of + – direct current). The positive terminal of the battery is at the higher potential and is represented by the longer vertical line in the battery symbol. In the next chapter

16.8

we discuss another circuit element, called a resistor, represented by the symbol . When wires in a circuit don’t have appreciable resistance compared with the resistance of other elements in the circuit, the wires are represented by straight lines. It’s important to realize that a circuit is a collection of real objects, usually containing a source of electrical energy (such as a battery) connected to elements that convert electrical energy to other forms (light, heat, sound) or store the energy in electric or magnetic fields for later retrieval. A real circuit and its schematic diagram are sketched side by side in Figure 16.17. The circuit symbol for a lightbulb shown in Figure 16.17b is . If you are not familiar with circuit diagrams, trace the path of the real circuit with your finger to see that it is equivalent to the geometrically regular schematic diagram.

16.8

Combinations of Capacitors

+

12 V

549

–

Resistor

(a) 12 V + –

COMBINATIONS OF CAPACITORS

Two or more capacitors can be combined in circuits in several ways, but most reduce to two simple configurations, called parallel and series. The idea, then, is to find the single equivalent capacitance due to a combination of several different capacitors that are in parallel or in series with each other. Capacitors are manufactured with a number of different standard capacitances, and by combining them in different ways, any desired value of the capacitance can be obtained.

Capacitors in Parallel Two capacitors connected as shown in Active Figure 16.18a are said to be in parallel. The left plate of each capacitor is connected to the positive terminal of the battery by a conducting wire, so the left plates are at the same potential. In the same way, the right plates, both connected to the negative terminal of the battery, are also at the same potential. This means that capacitors in parallel both have the same potential difference V across them. Capacitors in parallel are illustrated in Active Figure 16.18b. When the capacitors are first connected in the circuit, electrons are transferred from the left plates through the battery to the right plates, leaving the left plates positively charged and the right plates negatively charged. The energy source for this transfer of charge is the internal chemical energy stored in the battery, which is converted to electrical energy. The flow of charge stops when the voltage across the capacitors equals the voltage of the battery, at which time the capacitors have their maximum charges. If the maximum charges on the two capacitors are Q 1 and Q 2, respectively, the total charge, Q , stored by the two capacitors is Q  Q1  Q 2 V1  V2  V

C1 

C1




Q1

C eq  C 1  C 2

C2 

C2




Q2 




 V

(a)









V

V

(b)

(c)

[16.10]

ACTIVE FIGURE 16.18 (a) A parallel connection of two capacitors. (b) The circuit diagram for the parallel combination. (c) The potential differences across the capacitors are the same, and the equivalent capacitance is C eq  C 1  C 2.

(b) FIGURE 16.17 (a) A real circuit and (b) its equivalent circuit diagram.

550

Chapter 16

Electrical Energy and Capacitance

We can replace these two capacitors with one equivalent capacitor having a capacitance of C eq. This equivalent capacitor must have exactly the same external effect on the circuit as the original two, so it must store Q units of charge and have the same potential difference across it. The respective charges on each capacitor are TIP 16.3 Voltage Is the Same as Potential Difference A voltage across a device, such as a capacitor, has the same meaning as the potential difference across the device. For example, if we say that the voltage across a capacitor is 12 V, we mean that the potential difference between its plates is 12 V.

Q 1  C 1 V

and Q 2  C 2 V

The charge on the equivalent capacitor is Q  C eq V Substituting these relationships into Equation 16.10 gives C eq V  C 1 V  C 2 V or C eq 5 C 1 1 C 2

a

parallel b combination

[16.11]

If we extend this treatment to three or more capacitors connected in parallel, the equivalent capacitance is found to be C eq 5 C 1 1 C 2 1 C 3 1 # # #

a

parallel b combination

[16.12]

We see that the equivalent capacitance of a parallel combination of capacitors is larger than any of the individual capacitances.

EXAMPLE 16.7 Four Capacitors Connected in Parallel Goal

Analyze a circuit with several capacitors in parallel.

Problem (a) Determine the capacitance of the single capacitor that is equivalent to the parallel combination of capacitors shown in Figure 16.19. Find (b) the charge on the 12.0-mF capacitor and (c) the total charge contained in the configuration. (d) Derive a symbolic expression for the fraction of the total charge contained on one of the capacitors. Strategy For part (a), add the individual capacitances. For part (b), apply the formula C  Q/V to the 12.0-mF capacitor. The voltage difference is the same as the difference across the battery. To find the total charge contained in all four capacitors, use the equivalent capacitance in the same formula.

3.00 µF 6.00 µF 12.0 µF 24.0 µF

+

– 18.0 V

FIGURE 16.19 (Example 16.7) Four capacitors connected in parallel.

Solution (a) Find the equivalent capacitance. Apply Equation 16.12:

C eq  C 1  C 2  C 3  C 4  3.00 mF  6.00 mF  12.0 mF  24.0 mF  45.0 mF

(b) Find the charge on the 12-mF capacitor (designated C 3). Solve the capacitance equation for Q and substitute:

Q  C 3 V  (12.0  10
6 F)(18.0 V)  216  10
6 C  216 mC

16.8

(c) Find the total charge contained in the configuration. Use the equivalent capacitance:

C eq 5

Q DV

Combinations of Capacitors

551

Q 5 C eq DV 5 1 45.0 mF 2 1 18.0 V 2 5 8.10 3 102 mC

S

(d) Derive a symbolic expression for the fraction of the total charge contained in one of the capacitors. Write a symbolic expression for the fractional charge in the ith capacitor and use the capacitor definition:

Qi Q tot

5

C i DV Ci 5 C eq DV C eq

Remarks The charge on any one of the parallel capacitors can be found as in part (b) because the potential difference is the same. Notice that finding the total charge does not require finding the charge on each individual capacitor and adding. It’s easier to use the equivalent capacitance in the capacitance definition. QUESTION 16.7 If all four capacitors had the same capacitance, what fraction of the total charge would be held by each? EXERCISE 16.7 Find the charge on the 24.0-mF capacitor. Answer 432 mC

Capacitors in Series Now consider two capacitors connected in series, as illustrated in Active Figure 16.20a. For a series combination of capacitors, the magnitude of the charge must be the same on all the plates. To understand this principle, consider the charge transfer process in some detail. When a battery is connected to the circuit, electrons with total charge
Q are transferred from the left plate of C 1 to the right plate of C 2 through the battery, leaving the left plate of C 1 with a charge of Q. As a consequence, the magnitudes of the charges on the left plate of C 1 and the right plate of C 2 must be the same. Now consider the right plate of C 1 and the left plate of C 2, in the middle. These plates are not connected to the battery (because of the gap across the plates) and, taken together, are electrically neutral. The charge of Q on the left plate of C 1, however, attracts negative charges to the right plate of C 1. These charges will continue to accumulate until the left and right plates of C 1, taken together, become electrically neutral, which means that the charge on the right plate of C 1 is
Q. This negative charge could only have come from the left plate of C 2, so C 2 has a charge of Q. Therefore, regardless of how many capacitors are in series or what their capacitances are, all the right plates gain charges of Q and all the left plates have charges of Q (a consequence of the conservation of charge).

V 1

+Q

C1

V 2

–Q

+Q –

+

C2

–Q – V

V (a)

(b)

connected in series

ACTIVE FIGURE 16.20 A series combination of two capacitors. The charges on the capacitors are the same, and the equivalent capacitance can be calculated from the reciprocal relationship 1/C eq  (1/C 1)  (1/C 2).

C eq

+

O Q is the same for all capacitors

552

Chapter 16

Electrical Energy and Capacitance

After an equivalent capacitor for a series of capacitors is fully charged, the equivalent capacitor must end up with a charge of Q on its right plate and a charge of Q on its left plate. Applying the definition of capacitance to the circuit in Active Figure 16.20b, we have DV 5

Q C eq

where V is the potential difference between the terminals of the battery and C eq is the equivalent capacitance. Because Q  C V can be applied to each capacitor, the potential differences across them are given by DV1 5

Q

DV2 5

C1

Q C2

From Active Figure 16.20a, we see that V  V1  V2

[16.13]

where V1 and V2 are the potential differences across capacitors C 1 and C 2 (a consequence of the conservation of energy). The potential difference across any number of capacitors (or other circuit elements) in series equals the sum of the potential differences across the individual capacitors. Substituting these expressions into Equation 16.13 and noting that V  Q/C eq, we have Q C eq

5

Q C1

1

Q C2

Canceling Q, we arrive at the following relationship: 1 1 1 5 1 C eq C1 C2

a

series b combination

[16.14]

If this analysis is applied to three or more capacitors connected in series, the equivalent capacitance is found to be 1 1 1 1 5 1 1 1 ### C eq C1 C2 C3

a

series b combination

[16.15]

As we will show in Example 16.7, Equation 16.15 implies that the equivalent capacitance of a series combination is always smaller than any individual capacitance in the combination. QUICK QUIZ 16.8 A capacitor is designed so that one plate is large and the other is small. If the plates are connected to a battery, (a) the large plate has a greater charge than the small plate, (b) the large plate has less charge than the small plate, or (c) the plates have equal, but opposite, charge.

EXAMPLE 16.8 Four Capacitors Connected in Series Goal Find an equivalent capacitance of capacitors in series, and the charge and voltage on each capacitor.

3.0 µF

6.0 µF

12 µF

24 µF

Problem Four capacitors are connected in series with a battery, as in Figure 16.21. (a) Calculate the capacitance of the equivalent capacitor. (b) Compute the charge on the 12-mF capacitor. (c) Find the voltage drop across the 12-mF capacitor. Strategy Combine all the capacitors into a single, equivalent capacitor using Equation 16.15. Find the charge on this equivalent capacitor using C  Q/V. This charge is the same as on the individual capacitors. Use this same equation again to find the voltage drop across the 12-mF capacitor.

+

– 18 V

FIGURE 16.21 (Example 16.8) Four capacitors connected in series.

16.8

Solution (a) Calculate the equivalent capacitance of the series. Apply Equation 16.15:

Combinations of Capacitors

553

1 1 1 1 1 5 1 1 1 C eq 3.0 mF 6.0 mF 12 mF 24 mF C eq  1.6 mF

(b) Compute the charge on the 12-mF capacitor. The desired charge equals the charge on the equivalent capacitor:

Q  C eq V  (1.6  10
6 F)(18 V)  29 mC

(c) Find the voltage drop across the 12-mF capacitor. Apply the basic capacitance equation:

C5

Q DV

S

DV 5

Q C

5

29 mC 5 2.4 V 12 mF

Remarks Notice that the equivalent capacitance is less than that of any of the individual capacitors. The relationship C  Q/V can be used to find the voltage drops on the other capacitors, just as in part (c). QUESTION 16.8 Over which capacitor is the voltage drop the smallest? The largest? EXERCISE 16.8 The 24-mF capacitor is removed from the circuit, leaving only three capacitors in series. Find (a) the equivalent capacitance, (b) the charge on the 6-mF capacitor, and (c) the voltage drop across the 6-mF capacitor. Answers (a) 1.7 mF

(b) 31 mC

(c) 5.2 V

PROBLEM -SOLVING STRATEGY COMPLEX CAPACITOR COMBINATIONS

1. Combine capacitors that are in series or in parallel, following the derived formulas. 2. Redraw the circuit after every combination. 3. Repeat the first two steps until there is only a single equivalent capacitor. 4. Find the charge on the single equivalent capacitor, using C  Q/V. 5. Work backwards through the diagrams to the original one, finding the charge and voltage drop across each capacitor along the way. To do this, use the following collection of facts: A. The capacitor equation: C  Q/V B. Capacitors in parallel: C eq  C 1  C 2 C. Capacitors in parallel all have the same voltage difference, V, as does their equivalent capacitor. 1 1 1 D. Capacitors in series: 5 1 C eq C1 C2 E. Capacitors in series all have the same charge, Q, as does their equivalent capacitor.

554

Chapter 16

Electrical Energy and Capacitance

EXAMPLE 16.9 Equivalent Capacitance Goal

Solve a complex combination of series and parallel capacitors.

Problem (a) Calculate the equivalent capacitance between a and b for the combination of capacitors shown in Figure 16.22a. All capacitances are in microfarads. (b) If a 12-V battery is connected across the system between points a and b, find the charge on the 4.0-mF capacitor in the first diagram and the voltage drop across it.

1.0 4.0 4.0

2.0

4.0 3.0

a

6.0

2.0

b

a

8.0

b

8.0

a

8.0

b

a 6.0 b

4.0

Strategy For part (a), use Equations 16.12 (a) (b) (c) (d) and 16.15 to reduce the combination step by step, as indicated in the figure. For part (b), FIGURE 16.22 (Example 16.9) To find the equivalent capacitance of the circuit in use the series and parallel rules described in the text to successively reduce the to find the charge on the 4.0-mF capacitor, (a), circuit as indicated in (b), (c), and (d). start with Figure 16.22c, finding the charge on the 2.0-mF capacitor. This same charge is on each of the 4.0-mF capacitors in the second diagram, by fact 5E of the Problem-Solving Strategy. One of these 4.0-mF capacitors in the second diagram is simply the original 4.0-mF capacitor in the first diagram. Solution (a) Calculate the equivalent capacitance. Find the equivalent capacitance of the parallel 1.0-mF and 3.0-mF capacitors in Figure 16.22a:

C eq  C 1  C 2  1.0 mF  3.0 mF  4.0 mF

Find the equivalent capacitance of the parallel 2.0-mF and 6.0-mF capacitors in Figure 16.22a:

C eq  C 1  C 2  2.0 mF  6.0 mF  8.0 mF

Combine the two series 4.0-mF capacitors in Figure 16.22b:

1 1 1 1 1 5 1 5 1 C eq C1 C2 4.0 mF 4.0 mF 5

Combine the two series 8.0-mF capacitors in Figure 16.22b:

S

C eq 5 2.0 mF

1 1 1 1 1 5 1 5 1 C eq C1 C2 8.0 mF 8.0 mF 5

Finally, combine the two parallel capacitors in Figure 16.22c to find the equivalent capacitance between a and b:

1 2.0 mF

1 4.0 mF

S

C eq 5 4.0 mF

C eq  C 1  C 2  2.0 mF  4.0 mF  6.0 mF

(b) Find the charge on the 4.0-mF capacitor and the voltage drop across it. Compute the charge on the 2.0-mF capacitor in Figure 16.22c, which is the same as the charge on the 4.0-mF capacitor in Figure 16.22a:

C5

Use the basic capacitance equation to find the voltage drop across the 4.0-mF capacitor in Figure 16.22a:

C5

Q DV

Q DV

S

Q 5 C DV 5 1 2.0 mF 2 1 12 V 2 5 24 mC

S

DV 5

Q C

5

24 mC 5 6.0 V 4.0 mF

Remarks To find the rest of the charges and voltage drops, it’s just a matter of using C  Q/V repeatedly, together with facts 5C and 5E in the Problem-Solving Strategy. The voltage drop across the 4.0-mF capacitor could also have been found by noticing, in Figure 16.22b, that both capacitors had the same value and so by symmetry would split the total drop of 12 volts between them.

16.9

Energy Stored in a Charged Capacitor

555

QUESTION 16.9 Which capacitor holds more charge, the 1.0-mF capacitor or the 3.0-mF capacitor? EXERCISE 16.9 (a) In Example 16.9 find the charge on the 8.0-mF capacitor in Figure 16.22a and the voltage drop across it. (b) Do the same for the 6.0-mF capacitor in Figure 16.22a. Answers (a) 48 mC, 6.0 V

16.9

(b) 36 mC, 6.0 V

ENERGY STORED IN A CHARGED CAPACITOR

Almost everyone who works with electronic equipment has at some time verified that a capacitor can store energy. If the plates of a charged capacitor are connected by a conductor such as a wire, charge transfers from one plate to the other until the two are uncharged. The discharge can often be observed as a visible spark. If you accidentally touched the opposite plates of a charged capacitor, your fingers would act as a pathway by which the capacitor could discharge, inflicting an electric shock. The degree of shock would depend on the capacitance and voltage applied to the capacitor. Where high voltages and large quantities of charge are present, as in the power supply of a television set, such a shock can be fatal. Capacitors store electrical energy, and that energy is the same as the work required to move charge onto the plates. If a capacitor is initially uncharged (both plates are neutral) so that the plates are at the same potential, very little work is required to transfer a small amount of charge Q from one plate to the other. Once this charge has been transferred, however, a small potential difference V  Q/C appears between the plates, so work must be done to transfer additional charge against this potential difference. From Equation 16.6, if the potential difference at any instant during the charging process is V, the work W required to move more charge Q through this potential difference is given by W  V Q We know that V  Q/C for a capacitor that has a total charge of Q. Therefore, a plot of voltage versus total charge gives a straight line with a slope of 1/C, as shown in Figure 16.23. The work W, for a particular V, is the area of the blue rectangle. Adding up all the rectangles gives an approximation of the total work needed to fill the capacitor. In the limit as Q is taken to be infinitesimally small, the total work needed to charge the capacitor to a final charge Q and voltage V is the area under the line. This is just the area of a triangle, one-half the base times the height, so it follows that W 5 12 Q DV

V

[16.16]

As previously stated, W is also the energy stored in the capacitor. From the definition of capacitance, we have Q  C V; hence, we can express the energy stored three different ways: Energy stored 5

1 2Q

1 2C

DV 5

1 DV 2 2 5

Q2 2C

For example, the amount of energy stored in a 5.0-mF capacitor when it is connected across a 120-V battery is Energy stored 5

1 2C

1 DV 2 5 2

1 2 1 5.0

26

3 10

F 2 1 120 V 2 5 3.6 3 10 2

Q

[16.17]

22

J

In practice, there is a limit to the maximum energy (or charge) that can be stored in a capacitor. At some point, the Coulomb forces between the charges on the plates become so strong that electrons jump across the gap, discharging the capacitor. For this reason, capacitors are usually labeled with a maximum operating voltage. (This physical fact can actually be exploited to yield a circuit with a regularly blinking light).

Q FIGURE 16.23 A plot of voltage vs. charge for a capacitor is a straight line with slope 1/C. The work required to move a charge of Q through a potential difference of V across the capacitor plates is W  V Q, which equals the area of the blue rectangle. The total work required to charge the capacitor to a final charge of Q is the area under the straight line, which equals Q V/2.

556

Chapter 16

Electrical Energy and Capacitance

APPLICATION Defibrillators

Large capacitors can store enough electrical energy to cause severe burns or even death if they are discharged so that the flow of charge can pass through the heart. Under the proper conditions, however, they can be used to sustain life by stopping cardiac fibrillation in heart attack victims. When fibrillation occurs, the heart produces a rapid, irregular pattern of beats. A fast discharge of electrical energy through the heart can return the organ to its normal beat pattern. Emergency medical teams use portable defibrillators that contain batteries capable of charging a capacitor to a high voltage. (The circuitry actually permits the capacitor to be charged to a much higher voltage than the battery.) In this case and others (camera flash units and lasers used for fusion experiments), capacitors serve as energy reservoirs that can be slowly charged and then quickly discharged to provide large amounts of energy in a short pulse. The stored electrical energy is released through the heart by conducting electrodes, called paddles, placed on both sides of the victim’s chest. The paramedics must wait between applications of electrical energy because of the time it takes the capacitors to become fully charged. The high voltage on the capacitor can be obtained from a low-voltage battery in a portable machine through the phenomenon of electromagnetic induction, to be studied in Chapter 20.

EXAMPLE 16.10 Typical Voltage, Energy, and Discharge Time for a Defibrillator Goal

Apply energy and power concepts to a capacitor.

Problem A fully charged defibrillator contains 1.20 kJ of energy stored in a 1.10  10
4 F capacitor. In a discharge through a patient, 6.00  102 J of electrical energy are delivered in 2.50 ms. (a) Find the voltage needed to store 1.20 kJ in the unit. (b) What average power is delivered to the patient? Strategy Because we know the energy stored and the capacitance, we can use Equation 16.17 to find the required voltage in part (a). For part (b), dividing the energy delivered by the time gives the average power. Solution (a) Find the voltage needed to store 1.20 kJ in the unit. Solve Equation 16.17 for V:

Energy stored 5 12C DV 2 DV 5 5

Å

2 3 1 energy stored 2 C

2 1 1.20 3 103 J 2

Å 1.10 3 10 24 F

 4.67 3 103 V (b) What average power is delivered to the patient? Divide the energy delivered by the time:


av 5

energy delivered Dt

5

6.00 3 102 J 2.50 3 1023 s

 2.40 3 105 W Remarks The power delivered by a draining capacitor isn’t constant, as we’ll find in the study of RC circuits in Chapter 18. For that reason, we were able to find only an average power. Capacitors are necessary in defibrillators because they can deliver energy far more quickly than batteries. Batteries provide current through relatively slow chemical reactions, whereas capacitors can quickly release charge that has already been produced and stored. QUESTION 16.10 If the voltage across the capacitor were doubled, would the energy stored be (a) halved, (b) doubled, or (c) quadrupled?

16.10

Capacitors with Dielectrics

557

EXERCISE 16.10 (a) Find the energy contained in a 2.50  10
5 F parallel-plate capacitor if it holds 1.75  10
3 C of charge. (b) What’s the voltage between the plates? (c) What new voltage will result in a doubling of the stored energy? Answers (a) 6.13  10
2 J

(b) 70.0 V

APPLYING PHYSICS 16.1

(c) 99.0 V

MAXIMUM ENERGY DESIGN

How should three capacitors and two batteries be connected so that the capacitors will store the maximum possible energy?

potential difference, so we would like to maximize each of these quantities. If the three capacitors are connected in parallel, their capacitances add, and if the batteries are in series, their potential differences, similarly, also add together.

Explanation The energy stored in the capacitor is proportional to the capacitance and the square of the

QUICK QUIZ 16.9 A parallel-plate capacitor is disconnected from a battery, and the plates are pulled a small distance farther apart. Do the following quantities increase, decrease, or stay the same? (a) C (b) Q (c) E between the plates (d) V (e) energy stored in the capacitor

16.10

CAPACITORS WITH DIELECTRICS

A dielectric is an insulating material, such as rubber, plastic, or waxed paper. When a dielectric is inserted between the plates of a capacitor, the capacitance increases. If the dielectric completely fills the space between the plates, the capacitance is multiplied by the factor k, called the dielectric constant. The following experiment illustrates the effect of a dielectric in a capacitor. Consider a parallel-plate capacitor of charge Q 0 and capacitance C 0 in the absence of a dielectric. The potential difference across the capacitor plates can be measured, and is given by V0  Q 0/C 0 (Fig. 16.24a). Because the capacitor is not connected to an external circuit, there is no pathway for charge to leave or be added to the plates. If a dielectric is now inserted between the plates as in Figure 16.24b, the voltage across the plates is reduced by the factor k to the value DV 5

DV0 k

Dielectric C0

Q0




C

Q0




V

V0

(a)

(b)

FIGURE 16.24 (a) With air between the plates, the voltage across the capacitor is V0, the capacitance is C 0, and the charge is Q 0. (b) With a dielectric between the plates, the charge remains at Q 0, but the voltage and capacitance change.

558

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Because k 1, V is less than V0. Because the charge Q 0 on the capacitor doesn’t change, we conclude that the capacitance in the presence of the dielectric must change to the value C5

Q0 DV

5

Q0 DV0 /k

5

kQ 0 DV0

or C  kC 0

[16.18]

According to this result, the capacitance is multiplied by the factor k when the dielectric fills the region between the plates. For a parallel-plate capacitor, where the capacitance in the absence of a dielectric is C 0  P0A/d, we can express the capacitance in the presence of a dielectric as

© Loren Winters/Visuals Unlimited

C 5 kP0

FIGURE 16.25 Dielectric breakdown in air. Sparks are produced when a large alternating voltage is applied across the wires by a highvoltage induction coil power supply.

A d

[16.19]

From this result, it appears that the capacitance could be made very large by decreasing d, the separation between the plates. In practice the lowest value of d is limited by the electric discharge that can occur through the dielectric material separating the plates. For any given plate separation, there is a maximum electric field that can be produced in the dielectric before it breaks down and begins to conduct. This maximum electric field is called the dielectric strength, and for air its value is about 3  106 V/m. Most insulating materials have dielectric strengths greater than that of air, as indicated by the values listed in Table 16.1. Figure 16.25 shows an instance of dielectric breakdown in air. Commercial capacitors are often made by using metal foil interlaced with thin sheets of paraffin-impregnated paper or Mylar®, which serves as the dielectric material. These alternate layers of metal foil and dielectric are rolled into a small cylinder (Fig. 16.26a). One type of a high-voltage capacitor consists of a number of interwoven metal plates immersed in silicone oil (Fig. 16.26b). Small capacitors are often constructed from ceramic materials. Variable capacitors (typically 10 pF to 500 pF) usually consist of two interwoven sets of metal plates, one fixed and the other movable, with air as the dielectric. An electrolytic capacitor (Fig. 16.26c) is often used to store large amounts of charge at relatively low voltages. It consists of a metal foil in contact with an elecTABLE 16.1 Dielectric Constants and Dielectric Strengths of Various Materials at Room Temperature Material

Dielectric Constant K

Dielectric Strength (V/m)

Air Bakelite® Fused quartz Neoprene rubber Nylon Paper Polystyrene Pyrex® glass Silicone oil Strontium titanate Teflon® Vacuum Water

1.000 59 4.9 3.78 6.7 3.4 3.7 2.56 5.6 2.5 233 2.1 1.000 00 80

3  106 24  106 8  10 6 12  106 14  106 16  106 24  106 14  106 15  106 8  106 60  106 — —

16.10 Metal foil

Capacitors with Dielectrics

FIGURE 16.26 Three commercial capacitor designs. (a) A tubular capacitor whose plates are separated by paper and then rolled into a cylinder. (b) A high-voltage capacitor consisting of many parallel plates separated by oil. (c) An electrolytic capacitor.

Case

Plates

Electrolyte

Contacts Oil

559

Metallic foil  oxide layer

Paper (a)

(b)

(c)

FIGURE 16.27 (a) A collection of capacitors used in a variety of applications. (b) A variable capacitor. When one set of metal plates is rotated so as to lie between a fi xed set of plates, the capacitance of the device changes.

© Cengage Learning/George Semple

Paul Silverman/Fundamental Photographs

trolyte—a solution that conducts charge by virtue of the motion of the ions contained in it. When a voltage is applied between the foil and the electrolyte, a thin layer of metal oxide (an insulator) is formed on the foil, and this layer serves as the dielectric. Enormous capacitances can be attained because the dielectric layer is very thin. Figure 16.27 shows a variety of commercially available capacitors. Variable capacitors are used in radios to adjust the frequency. When electrolytic capacitors are used in circuits, the polarity (the plus and minus signs on the device) must be observed. If the polarity of the applied voltage is opposite that intended, the oxide layer will be removed and the capacitor will conduct rather than store charge. Further, reversing the polarity can result in such a large current that the capacitor may either burn or produce steam and explode.

(a)

APPLYING PHYSICS 16.2

(b)

STUD FINDERS

If you have ever tried to hang a picture on a wall securely, you know that it can be difficult to locate a wooden stud in which to anchor your nail or screw. The principles discussed in this section can be used to detect a stud electronically. The primary element of an electronic stud finder is a capacitor with its plates arranged side by side instead of facing one another, as in Figure 16.28. How does this device work? Explanation As the detector is moved along a wall, its capacitance changes when it passes across a stud because the dielectric constant of the material “between” the plates changes. The change in capacitance can be used to cause a light to come on, signaling the presence of the stud.

Capacitor plates

Stud finder

Stud

Wallboard (a)

(b)

FIGURE 16.28 (Applying Physics 16.2) A stud finder. (a) The materials between the plates of the capacitor are the drywall and the air behind it. (b) The materials become drywall and wood when the detector moves across a stud in the wall. The change in the dielectric constant causes a signal light to illuminate.

560

Chapter 16

Electrical Energy and Capacitance

QUICK QUIZ 16.10 A fully charged parallel-plate capacitor remains connected to a battery while a dielectric is slid between the plates. Do the following quantities increase, decrease, or stay the same? (a) C (b) Q (c) E between the plates (d) V (e) energy stored in the capacitor

EXAMPLE 16.11 A Paper-Filled Capacitor Goal

Calculate fundamental physical properties of a parallel-plate capacitor with a dielectric.

Problem A parallel-plate capacitor has plates 2.0 cm by 3.0 cm. The plates are separated by a 1.0-mm thickness of paper. Find (a) the capacitance of this device and (b) the maximum charge that can be placed on the capacitor. (c) After the fully charged capacitor is disconnected from the battery, the dielectric is subsequently removed. Find the new electric field across the capacitor. Does the capacitor discharge? Strategy For part (a), obtain the dielectric constant for paper from Table 16.1 and substitute, with other given quantities, into Equation 16.19. For part (b), note that Table 16.1 also gives the dielectric strength of paper, which is the maximum electric field that can be applied before electrical breakdown occurs. Use Equation 16.3, V  Ed, to obtain the maximum voltage and substitute into the basic capacitance equation. For part (c), remember that disconnecting the battery traps the extra charge on the plates, which must remain even after the dielectric is removed. Find the charge density on the plates and use Gauss’s law to find the new electric field between the plates. Solution (a) Find the capacitance of this device. Substitute into Equation 16.19:

C 5 kP0

A d

5 3.7 a8.85 3 10212

6.0 3 1024 m2 C2 b b a N # m2 1.0 3 1023 m

 2.0 3 10211 F (b) Find the maximum charge that can be placed on the capacitor. Calculate the maximum applied voltage, using the dielectric strength of paper, E max:

Vmax  E maxd  (16  106 V/m)(1.0  10
3 m)

Solve the basic capacitance equation for Q max and substitute Vmax and C:

Q max  C Vmax  (2.0  10
11 F)(1.6  104 V)

 1.6  104 V

 0.32 mC

(c) Suppose the fully charged capacitor is disconnected from the battery and the dielectric is subsequently removed. Find the new electric field between the plates of the capacitor. Does the capacitor discharge? Compute the charge density on the plates:

s5

Calculate the electric field from the charge density:

E5

Because the electric field without the dielectric exceeds the value of the dielectric strength of air, the capacitor discharges across the gap.

Q max A

5

3.2 3 1027 C 5 5.3 3 1024 C/m2 6.0 3 1024 m2

s 5.3 3 1024 C/m2 5 6.0 3 107 N/C 5 P0 8.85 3 10212 C2 /m2 # N

16.10

Capacitors with Dielectrics

561

Remarks Dielectrics allow k times as much charge to be stored on a capacitor for a given voltage. They also allow an increase in the applied voltage by increasing the threshold of electrical breakdown. QUESTION 16.11 Subsequent to part (c), the capacitor is reconnected to the battery. Is the charge on the plates (a) larger than, (b) smaller than, or (c) the same as found in part (b)? EXERCISE 16.11 A parallel-plate capacitor has plate area of 2.50  10
3 m2 and distance between the plates of 2.00 mm. (a) Find the maximum charge that can be placed on the capacitor if air is between the plates. (b) Find the maximum charge if the air is replaced by polystyrene. Answers (a) 7  10
8 C

(b) 1.4  10
6 C

An Atomic Description of Dielectrics

O

The explanation of why a dielectric increases the capacitance of a capacitor is based on an atomic description of the material, which in turn involves a property of some molecules called polarization. A molecule is said to be polarized when there is a separation between the average positions of its negative charge and its positive charge. In some molecules, such as water, this condition is always present. To see why, consider the geometry of a water molecule (Fig. 16.29). The molecule is arranged so that the negative oxygen atom is bonded to the positively charged hydrogen atoms with a 105° angle between the two bonds. The center of negative charge is at the oxygen atom, and the center of positive charge lies at a point midway along the line joining the hydrogen atoms (point x in the diagram). Materials composed of molecules that are permanently polarized in this way have large dielectric constants, and indeed, Table 16.1 shows that the dielectric constant of water is large (k  80) compared with other common substances. A symmetric molecule (Fig. 16.30a) can have no permanent polarization, but a polarization can be induced in it by an external electric field. A field directed to the left, as in Figure 16.30b, would cause the center of positive charge to shift to the left from its initial position and the center of negative charge to shift to the right. This induced polarization is the effect that predominates in most materials used as dielectrics in capacitors. To understand why the polarization of a dielectric can affect capacitance, consider the slab of dielectric shown in Figure 16.31. Before placing the slab between the plates of the capacitor, the polar molecules are randomly oriented (Fig. 16.31a). The polar molecules are dipoles, and each creates a dipole electric field, but because of their random orientation, this field averages to zero. S After insertion of the dielectric slab into the electric field E0 between the plates (Fig. 16.31b), the positive plate attracts the negative ends of the dipoles and the negative plate attracts the positive ends of the dipoles. These forces exert a torque on the molecules making up the dielectric, reorienting them so that on average the negative pole is more inclined toward the positive plate and the positive pole is more aligned toward the negative plate. The positive and negative charges in the

– –

– +

– +

– +

+

– +

–

+ –

+

– +

+ – – +

–

+

–

+

– +

– + – +

+

– + – + – +

– +

– + – + – +

+

– +

– +

– +

+ +

+ (a)

–

+

–

+

–

+

–

+

–

+

–

+

–

+ – – E0

– E ind

– –

E0 (b)

–

(c)

H

105°

H



x•



FIGURE 16.29 The water molecule, H2O, has a permanent polarization resulting from its bent geometry. The point labeled x is the center of positive charge.

+

–

+

(a) E

+

–

+

(b) FIGURE 16.30 (a) A symmetric molecule has no permanent polarization. (b) An external electric field induces a polarization in the molecule.

FIGURE 16.31 (a) In the absence of an external electric field, polar molecules are randomly oriented. (b) When an external electric field is applied, the molecules partially align with the field. (c) The charged edges of the dielectric can be modeled as an additional pair of parallel plates S establishing an electricSfield E ind in the direction opposite E 0.

562

Chapter 16

Electrical Energy and Capacitance

middle still cancel each other, but there is a net accumulation of negative charge in the dielectric next to the positive plate and a net accumulation of positive charge next to the negative plate. This configuration can be modeled as an additional S pair of charged plates, as in Figure 16.31c, creating an induced electric field Eind S that partly cancels the original electric field E0 . If the battery is not connected when the dielectric is inserted, the potential difference V0 across the plates is reduced to V0/k. If the capacitor is still connected to the battery, however, the negative poles push more electrons off the positive plate, making it more positive. Meanwhile, the positive poles attract more electrons onto the negative plate. This situation continues until the potential difference across the battery reaches its original magnitude, equal to the potential gain across the battery. The net effect is an increase in the amount of charge stored on the capacitor. Because the plates can store more charge for a given voltage, it follows from C  Q V that the capacitance must increase. QUICK QUIZ 16.11 Consider a parallel-plate capacitor with a dielectric material between the plates. If the temperature of the dielectric increases, does the capacitance (a) decrease, (b) increase, or (c) remain the same?

SUMMARY 16.1

Potential Difference and Electric Potential

These equations can be used in the solution of conservation of energy problems and in the work–energy theorem.

The change in the electric potential energy of a system consisting of an object of charge q moving through a disS placement x in a constant electric field E is given by

16.3

Potentials and Charged Conductors

PE 
WAB 
qEx x

16.4

Equipotential Surfaces

[16.1]

where Ex is the component of the electric field in the xdirection and x  xf
xi . The difference in electric potential between two points A and B is DV 5 VB 2 VA 5

DPE q

[16.2]

where PE is the change in electrical potential energy as a charge q moves between A and B. The units of potential difference are joules per coulomb, or volts; 1 J/C  1 V. The electric potential difference between two points A S and B in a uniform electric field E is DV 5 2E x Dx

16.2 Electric Potential and Potential Energy Due to Point Charges The electric potential due to a point charge q at distance r from the point charge is [16.4]

q 1q 2 r

[16.7]

Any surface on which the potential is the same at every point is called an equipotential surface. The electric field is always oriented perpendicular to an equipotential surface.

16.6 Capacitance A capacitor consists of two metal plates with charges that are equal in magnitude but opposite in sign. The capacitance C of any capacitor is the ratio of the magnitude of the charge Q on either plate to the magnitude of potential difference V between them: C;

[16.5]

Q DV

[16.8]

Capacitance has the units coulombs per volt, or farads; 1 C/V  1 F.

16.7

The electric potential energy of a pair of point charges separated by distance r is PE 5 k e

1 eV  1.60  10
19 C  V  1.60  10
19 J

[16.3]

where x  xf
xi is the displacement between A and B and Ex is the x-component of the electric field in that region.

q V 5 ke r

Every point on the surface of a charged conductor in electrostatic equilibrium is at the same potential. Further, the potential is constant everywhere inside the conductor and equals its value on the surface. The electron volt is defined as the energy that an electron (or proton) gains when accelerated through a potential difference of 1 V. The conversion between electron volts and joules is

The Parallel-Plate Capacitor

The capacitance of two parallel metal plates of area A separated by distance d is C 5 P0

A d

[16.9]

Multiple-Choice Questions

563

where 0  8.85  10
12 C2/N  m2 is a constant called the permittivity of free space.

same voltage drop, and that series capacitors have the same charge.

16.8

16.9

Combinations of Capacitors

The equivalent capacitance of a parallel combination of capacitors is C eq  C 1  C 2  C 3    

[16.12]

If two or more capacitors are connected in series, the equivalent capacitance of the series combination is 1 1 1 1 5 1 1 1 ### C eq C1 C2 C3

[16.15]

Problems involving a combination of capacitors can be solved by applying Equations 16.12 and 16.13 repeatedly to a circuit diagram, simplifying it as much as possible. This step is followed by working backwards to the original diagram, applying C  Q/V, that parallel capacitors have the

Energy Stored in a Charged Capacitor

Three equivalent expressions for calculating the energy stored in a charged capacitor are Energy stored 5 12 Q DV 5 12 C 1 DV 2 2 5

16.10

Q2 2C

[16.17]

Capacitors with Dielectrics

When a nonconducting material, called a dielectric, is placed between the plates of a capacitor, the capacitance is multiplied by the factor k, which is called the dielectric constant, a property of the dielectric material. The capacitance of a parallel-plate capacitor filled with a dielectric is C 5 kP0

A d

[16.19]

FOR ADDITIONAL STUDENT RESOURCES, GO TO W W W.SERWAYPHYSICS.COM

MULTIPLE-CHOICE QUESTIONS 1. A proton is released at the origin in a constant electric field of 850 N/C acting in the positive x-direction. Find the change in the electric potential energy associated with the proton after it travels to x  2.5 m. (a) 3.4  10
16 J (b)
3.4  10
16 J (c) 2.5  10
16 J (d)
2.5  10
16 J (e)
1.6  10
19 J 2. An electron in a TV picture tube is accelerated through a potential difference of 1.0  104 V before it hits the screen. What is the kinetic energy of the electron in electron volts? (a) 1.0  104 eV (b) 1.6  10 –15 eV (c) 1.6  10 –22 eV (d) 6.25  1022 eV (e) 1.6  10 –19 eV 10
27

kg) 3. A helium nucleus (charge  2e, mass  6.63  traveling at a speed of 6.20  105 m/s enters an electric field, traveling from point 훽, at a potential of 1.50  103 V, to point 훾, at 4.00  103 V. What is its speed at point 훾? (a) 7.91  105 m/s (b) 3.78  105 m/s (c) 2.13  105 m/s (d) 2.52  106 m/s (e) 3.01  108 m/s 4. The electric potential at x  3.0 m is 120 V, and the electric potential at x  5.0 m is 190 V. What is the electric field in this region, assuming it’s constant? (a) 140 N/C (b)
140 N/C (c) 35 N/C (d)
35 N/C (e) 75 N/C 5. An electronics technician wishes to construct a parallel-plate capacitor using rutile (k  1.00  102) as the dielectric. If the cross-sectional area of the plates is 1.0 cm2, what is the capacitance if the rutile thickness is 1.0 mm? (a) 88.5 pF (b) 177.0 pF (c) 8.85 mF (d) 100.0 mF (e) 354 mF 6. Four point charges are positioned on the rim of a circle. The charge on each of the four is 0.5 mC, 1.5 mC,
1.0 mC, and
0.5 mC. If the electrical potential at the center of the circle due to the 0.5 mC charge alone is 4.5  104 V, what is the total electric potential at the center due to the four charges? (a) 18.0  104 V (b) 4.5  104 V (c) 0 (d)
4.5  104 V (e) 9.0  104 V

7. If three unequal capacitors, initially uncharged, are connected in series across a battery, which of the following statements is true? (a) The equivalent capacitance is greater than any of the individual capacitances. (b) The largest voltage appears across the capacitor with the smallest capacitance. (c) The largest voltage appears across the capacitor with the largest capacitance. (d) The capacitor with the largest capacitance has the greatest charge. (e) The capacitor with the smallest capacitance has the smallest charge. 8. A parallel-plate capacitor is connected to a battery. What happens if the plate separation is doubled while the capacitor remains connected to the battery? (a) The stored energy remains the same. (b) The stored energy is doubled. (c) The stored energy decreases by a factor of 2. (d) The stored energy decreases by a factor of 4. (e) The stored energy increases by a factor of 4. 9. A parallel-plate capacitor filled with air carries a charge Q. The battery is disconnected, and a slab of material with dielectric constant k  2 is inserted between the plates. Which of the following statements is correct? (a) The voltage across the capacitor decreases by a factor of 2. (b) The voltage across the capacitor is doubled. (c) The charge on the plates is doubled. (d) The charge on the plates decreases by a factor of 2. (e) The electric field is doubled. 10. After a parallel-plate capacitor is charged by a battery, it is disconnected from the battery and its plate separation is increased. Which of the following statements is correct? (a) The energy stored in the capacitor decreases. (b) The energy stored in the capacitor increases. (c) The electric field between the plates decreases. (d) The potential difference between the plates decreases. (e) The charge on the plates decreases.

564

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11. A battery is attached to several different capacitors connected in parallel. Which of the following statements is true? (a) All the capacitors have the same charge, and the equivalent capacitance is greater than the capacitance of any of the capacitors in the group. (b) The capacitor with the largest capacitance carries the smallest charge. (c) The potential difference across each capacitor is the same, and the equivalent capacitance is greater than any of the capacitors in the group. (d) The capacitor with the smallest capacitance carries the largest charge. (e) The potential differences across the capacitors are the same only if the capacitances are the same.

12. A battery is attached across several different capacitors connected in series. Which of the following statements are true? (a) All the capacitors have the same charge, and the equivalent capacitance is less than the capacitance of any of the individual capacitors in the group. (b) All the capacitors have the same charge, and the equivalent capacitance is greater than any of the individual capacitors in the group. (c) The capacitor with the largest capacitance carries the largest charge. (c) The potential difference across each capacitor must be the same. (d) The largest potential difference appears across the capacitor having the largest capacitance. (e) The largest potential difference appears across the capacitor with the smallest capacitance.

CONCEPTUAL QUESTIONS 1. (a) Describe the motion of a proton after it is released from rest in a uniform electric field. (b) Describe the changes (if any) in its kinetic energy and the electric potential energy associated with the proton. 2. Describe how you can increase the maximum operating voltage of a parallel-plate capacitor for a fixed plate separation. 3. A parallel-plate capacitor is charged by a battery, and the battery is then disconnected from the capacitor. Because the charges on the capacitor plates are opposite in sign, they attract each other. Hence, it takes positive work to increase the plate separation. Show that the external work done when the plate separation is increased leads to an increase in the energy stored in the capacitor. 4. Distinguish between electric potential and electrical potential energy. 5. Suppose you are sitting in a car and a 20-kV power line drops across the car. Should you stay in the car or get out? The power line potential is 20 kV compared to the potential of the ground. 6. Why is it important to avoid sharp edges or points on conductors used in high-voltage equipment? 7. Explain why, under static conditions, all points in a conductor must be at the same electric potential.

8. If you are given three different capacitors C 1, C 2, and C 3, how many different combinations of capacitance can you produce, using all capacitors in your circuits? 9. Why is it dangerous to touch the terminals of a highvoltage capacitor even after the voltage source that charged the battery is disconnected from the capacitor? What can be done to make the capacitor safe to handle after the voltage source has been removed? 10. The plates of a capacitor are connected to a battery. What happens to the charge on the plates if the connecting wires are removed from the battery? What happens to the charge if the wires are removed from the battery and connected to each other? 11. Can electric field lines ever cross? Why or why not? Can equipotentials ever cross? Why or why not? 12. Is it always possible to reduce a combination of capacitors to one equivalent capacitor with the rules developed in this chapter? Explain. 13. If you were asked to design a capacitor for which a small size and a large capacitance were required, what factors would be important in your design? 14. Explain why a dielectric increases the maximum operating voltage of a capacitor even though the physical size of the capacitor doesn’t change.

PROBLEMS The Problems for this chapter may be assigned online at WebAssign. 1, 2, 3  straightforward, intermediate, challenging GP  denotes guided problem ecp  denotes enhanced content problem  biomedical application 䡺  denotes full solution available in Student Solutions Manual/ Study Guide

SECTION 16.1 POTENTIAL DIFFERENCE AND ELECTRIC POTENTIAL 1. A uniform electric field of magnitude 375 N/C pointing in the positive x-direction acts on an electron, which is initially at rest. After the electron has moved 3.20 cm, what is (a) the work done by the field on the electron, (b) the change in potential energy associated with the electron, and (c) the velocity of the electron?

Problems

2. A uniform electric field of magnitude 327 N/C is directed along the y-axis. A 5.40-mC charge moves from the origin to the point (x, y)  (
15.0 cm,
32.0 cm). (a) What is the change in the potential energy associated with this charge? (b) Through what potential difference did the charge move? 3.

A potential difference of 90 mV exists between the inner and outer surfaces of a cell membrane. The inner surface is negative relative to the outer surface. How much work is required to eject a positive sodium ion (Na) from the interior of the cell?

4. An ion accelerated through a potential difference of 60.0 V has its potential energy decreased by 1.92  10
17 J. Calculate the charge on the ion. 5. The potential difference between the accelerating plates of a TV set is about 25 kV. If the distance between the plates is 1.5 cm, find the magnitude of the uniform electric field in the region between the plates. 6. To recharge a 12-V battery, a battery charger must move 3.6  105 C of charge from the negative terminal to the positive terminal. How much work is done by the charger? Express your answer in joules. 7. Oppositely charged parallel plates are separated by 5.33 mm. A potential difference of 600 V exists between the plates. (a) What is the magnitude of the electric field between the plates? (b) What is the magnitude of the force on an electron between the plates? (c) How much work must be done on the electron to move it to the negative plate if it is initially positioned 2.90 mm from the positive plate? 8. ecp (a) Find the potential difference Ve required to stop an electron (called a “stopping potential”) moving with an initial speed of 2.85  107 m/s. (b) Would a proton traveling at the same speed require a greater or lesser magnitude potential difference? Explain. (c) Find a symbolic expression for the ratio of the proton stopping potential and the electron stopping potential, Vp /Ve . The answer should be in terms of the proton mass mp and electron mass me . 9. ecp A 74.0-g block carrying a charge Q  35.0 mC is connected to a spring for which k  78.0 N/m. The block lies on a frictionless, horizontal surface and is immersed in a uniform electric field of magnitude E  4.86  104 N/C directed as shown in Figure P16.9. If the block is released from rest when the spring is unstretched (x  0), (a) by what maximum distance does the block move from its initial position? (b) Find the subsequent equilibrium position of the block and the amplitude of its motion. (c) Using conservation of energy, find a symbolic rela-

m, Q k

E x x=0 FIGURE P16.9

565

tionship giving the potential difference between its initial position and the point of maximum extension in terms of the spring constant k, the amplitude A, and the charge Q. 10. On planet Tehar, the free-fall acceleration is the same as that on the Earth, but there is also a strong downward electric field that is uniform close to the planet’s surface. A 2.00-kg ball having a charge of 5.00 mC is thrown upward at a speed of 20.1 m/s. It hits the ground after an interval of 4.10 s. What is the potential difference between the starting point and the top point of the trajectory?

SECTION 16.2 ELECTRIC POTENTIAL AND POTENTIAL ENERGY DUE TO POINT CHARGES SECTION 16.3 POTENTIALS AND CHARGED CONDUCTORS SECTION 16.4 EQUIPOTENTIAL SURFACES 11. ecp An electron is at the origin. (a) Calculate the electric potential VA at point A, x  0.250 cm. (b) Calculate the electric potential V B at point B, x  0.750 cm. What is the potential difference V B
VA? (c) Would a negatively charged particle placed at point A necessarily go through this same potential difference upon reaching point B? Explain. 12. Two point charges are on the y-axis. A 4.50- mC charge is located at y  1.25 cm, and a
2.24- mC charge is located at y 
1.80 cm. Find the total electric potential at (a) the origin and (b) the point having coordinates (1.50 cm, 0). 13. (a) Find the electric potential, taking zero at infinity, at the upper right corner (the corner without a charge) of the rectangle in Figure P16.13. (b) Repeat if the 2.00-mC charge is replaced with a charge of
2.00 mC. 8.00 µC

6.00 cm

3.00 cm

2.00 µC FIGURE P16.13

4.00 µC (Problems 13 and 14)

14. Three charges are situated at corners of a rectangle as in Figure P16.13. How much energy would be expended in moving the 8.00-mC charge to infinity? 15. ecp Two point charges Q 1  5.00 nC and Q 2 
3.00 nC are separated by 35.0 cm. (a) What is the electric potential at a point midway between the charges? (b) What is the potential energy of the pair of charges? What is the significance of the algebraic sign of your answer? 16. A point charge of 9.00  10
9 C is located at the origin. How much work is required to bring a positive charge of 3.00  10
9 C from infinity to the location x  30.0 cm? 17. The three charges in Figure P16.17 are at the vertices of an isosceles triangle. Let q  7.00 nC and calculate the electric potential at the midpoint of the base.

566

Chapter 16

Electrical Energy and Capacitance

P16.23. How close does the alpha particle get to the gold nucleus before turning around? Assume the gold nucleus remains stationary.

q

4.00 cm

–q

–q

2.00 cm

SECTION 16.6 CAPACITANCE

FIGURE P16.17

SECTION 16.7 THE PARALLEL-PLATE CAPACITOR

18. An electron starts from rest 3.00 cm from the center of a uniformly charged sphere of radius 2.00 cm. If the sphere carries a total charge of 1.00  10
9 C, how fast will the electron be moving when it reaches the surface of the sphere? 19.

GP

A proton is located at the origin, and a second proton is located on the x-axis at x  6.00 fm (1 fm  10
15 m). (a) Calculate the electric potential energy associated with this configuration. (b) An alpha particle (charge  2e, mass  6.64  10
27 kg) is now placed at (x, y)  (3.00, 3.00) fm. Calculate the electric potential energy associated with this configuration. (c) Starting with the threeparticle system, find the change in electric potential energy if the alpha particle is allowed to escape to infinity while the two protons remain fi xed in place. (Throughout, neglect any radiation effects.) (d) Use conservation of energy to calculate the speed of the alpha particle at infinity. (e) If the two protons are released from rest and the alpha particle remains fi xed, calculate the speed of the protons at infinity.

20. ecp A proton and an alpha particle (charge  2e, mass  6.64  10
27 kg) are initially at rest, separated by 4.00  10
15 m. (a) If they are both released simultaneously, explain why you can’t find their velocities at infinity using only conservation of energy. (b) What other conservation law can be applied in this case? (c) Find the speeds of the proton and alpha particle, respectively, at infinity. 21. A small spherical object carries a charge of 8.00 nC. At what distance from the center of the object is the potential equal to 100 V? 50.0 V? 25.0 V? Is the spacing of the equipotentials proportional to the change in voltage? 22. ecp Starting with the definition of work, prove that the local electric field must be everywhere perpendicular to a surface having the same potential at every point. 23. In Rutherford’s famous scattering experiments that led to the planetary model of the atom, alpha particles (having charges of 2e and masses of 6.64  10
27 kg) were fired toward a gold nucleus with charge 79e. An alpha particle, initially very far from the gold nucleus, is fired at 2.00  107 m/s directly toward the nucleus, as in Figure 2e + +

24. ecp Four point charges each having charge Q are located at the corners of a square having sides of length a. Find symbolic expressions for (a) the total electric potential at the center of the square due to the four charges and (b) the work required to bring a fi fth charge q from infinity to the center of the square.

79e + + + + + + + + +

v=0 + +

d FIGURE P16.23

25. Consider the Earth and a cloud layer 800 m above the planet to be the plates of a parallel-plate capacitor. (a) If the cloud layer has an area of 1.0 km2  1.0  106 m2, what is the capacitance? (b) If an electric field strength greater than 3.0  106 N/C causes the air to break down and conduct charge (lightning), what is the maximum charge the cloud can hold? 26. (a) When a 9.00-V battery is connected to the plates of a capacitor, it stores a charge of 27.0 mC. What is the value of the capacitance? (b) If the same capacitor is connected to a 12.0-V battery, what charge is stored? 27. An air-filled parallel-plate capacitor has plates of area 2.30 cm2 separated by 1.50 mm. The capacitor is connected to a 12.0-V battery. (a) Find the value of its capacitance. (b) What is the charge on the capacitor? (c) What is the magnitude of the uniform electric field between the plates? 28. (a) How much charge is on each plate of a 4.00-mF capacitor when it is connected to a 12.0-V battery? (b) If this same capacitor is connected to a 1.50-V battery, what charge is stored? 29. An air-filled capacitor consists of two parallel plates, each with an area of 7.60 cm2 and separated by a distance of 1.80 mm. If a 20.0-V potential difference is applied to these plates, calculate (a) the electric field between the plates, (b) the capacitance, and (c) the charge on each plate. 30. A 1-megabit computer memory chip contains many 60.0  10
15 -F capacitors. Each capacitor has a plate area of 21.0  10
12 m2. Determine the plate separation of such a capacitor. (Assume a parallel-plate configuration.) The diameter of an atom is on the order of 10
10 m  1 Å. Express the plate separation in angstroms. 31. ecp A parallel-plate capacitor with area 0.200 m2 and plate separation of 3.00 mm is connected to a 6.00-V battery. (a) What is the capacitance? (b) How much charge is stored on the plates? (c) What is the electric field between the plates? (d) Find the magnitude of the charge density on each plate. (e) Without disconnecting the battery, the plates are moved farther apart. Qualitatively, what happens to each of the previous answers? 32. A small object with a mass of 350 mg carries a charge of 30.0 nC and is suspended by a thread between the vertical plates of a parallel-plate capacitor. The plates are separated by 4.00 cm. If the thread makes an angle of 15.0° with the vertical, what is the potential difference between the plates?

Problems

SECTION 16.8 COMBINATIONS OF CAPACITORS 33. Given a 2.50-mF capacitor, a 6.25- mF capacitor, and a 6.00-V battery, find the charge on each capacitor if you connect them (a) in series across the battery and (b) in parallel across the battery.

39. Find the charge on each of the capacitors in Figure P16.39. + 24.0 V

6.00 µF

8.00 µF

2.00 µF

8.00 µF

9.00 V FIGURE P16.35

36. Two capacitors give an equivalent capacitance of 9.00 pF when connected in parallel and an equivalent capacitance of 2.00 pF when connected in series. What is the capacitance of each capacitor? 37. For the system of capacitors shown in Figure P16.37, find (a) the equivalent capacitance of the system, (b) the charge on each capacitor, and (c) the potential difference across each capacitor. 3.00 µF

6.00 µF

2.00 µF

4.00 µF

1.00 µF

5.00 µF

8.00 µF

4.00 µF

–

34. Find the equivalent capacitance of a 4.20- mF capacitor and an 8.50- mF capacitor when they are connected (a) in series and (b) in parallel. 35. Find (a) the equivalent capacitance of the capacitors in Figure P16.35, (b) the charge on each capacitor, and (c) the potential difference across each capacitor.

567

FIGURE P16.39

40. A 10.0-mF capacitor is fully charged across a 12.0-V battery. The capacitor is then disconnected from the battery and connected across an initially uncharged capacitor with capacitance C. The resulting voltage across each capacitor is 3.00 V. What is the value of C? 41. A 25.0-mF capacitor and a 40.0-mF capacitor are charged by being connected across separate 50.0-V batteries. (a) Determine the resulting charge on each capacitor. (b) The capacitors are then disconnected from their batteries and connected to each other, with each negative plate connected to the other positive plate. What is the final charge of each capacitor, and what is the final potential difference across the 40.0-mF capacitor? 42. (a) Find the equivalent capacitance between points a and b for the group of capacitors connected as shown in Figure P16.42 if C 1  5.00 mF, C 2  10.00 mF, and C 3  2.00 mF. (b) If the potential between points a and b is 60.0 V, what charge is stored on C 3? a

C1 C2

C1

C3 C2

C2

C2 b

38.

90.0 V

FIGURE P16.42

FIGURE P16.37

43. A 1.00-mF capacitor is charged by being connected across a 10.0-V battery. It is then disconnected from the battery and connected across an uncharged 2.00-mF capacitor. Determine the resulting charge on each capacitor.

GP Consider the combination of capacitors in Figure P16.38. (a) Find the equivalent single capacitance of the two capacitors in series and redraw the diagram (called diagram 1) with this equivalent capacitance. (b) In diagram 1 find the equivalent capacitance of the three capacitors in parallel and redraw the diagram as a single battery and single capacitor in a loop. (c) Compute the charge on the single equivalent capacitor. (d) Returning to diagram 1, compute the charge on each individual capacitor. Does the sum agree with the value found in part (c)? (e) What is the charge on the 24.0-mF capacitor and on the 8.00-mF capacitor? (f) Compute the voltage drop across the 24.0-mF capacitor and (g) the 8.00- mF capacitor.

44. Find the equivalent capacitance between points a and b in the combination of capacitors shown in Figure P16.44. 4.0 µF 7.0 µF a

b

5.0 µF 6.0 µF FIGURE P16.44

36.0 V

4.00 µF

24.0 µF

SECTION 16.9 ENERGY STORED IN A CHARGED CAPACITOR

8.00 µF

45. A 12.0-V battery is connected to a 4.50-mF capacitor. How much energy is stored in the capacitor?

2.00 µF

FIGURE P16.38

568

Chapter 16

Electrical Energy and Capacitance

46. ecp Two capacitors, C 1  18.0 mF and C 2  36.0 mF, are connected in series, and a 12.0-V battery is connected across them. (a) Find the equivalent capacitance, and the energy contained in this equivalent capacitor. (b) Find the energy stored in each individual capacitor. Show that the sum of these two energies is the same as the energy found in part (a). Will this equality always be true, or does it depend on the number of capacitors and their capacitances? (c) If the same capacitors were connected in parallel, what potential difference would be required across them so that the combination stores the same energy as in part (a)? Which capacitor stores more energy in this situation, C 1 or C 2? 47. A parallel-plate capacitor has capacitance 3.00 mF. (a) How much energy is stored in the capacitor if it is connected to a 6.00-V battery? (b) If the battery is disconnected and the distance between the charged plates doubled, what is the energy stored? (c) The battery is subsequently reattached to the capacitor, but the plate separation remains as in part (b). How much energy is stored? (Answer each part in microjoules.) 48. A certain storm cloud has a potential difference of 1.00  108 V relative to a tree. If, during a lightning storm, 50.0 C of charge is transferred through this potential difference and 1.00% of the energy is absorbed by the tree, how much water (sap in the tree) initially at 30.0°C can be boiled away? Water has a specific heat of 4 186 J/kg°C, a boiling point of 100°C, and a heat of vaporization of 2.26  106 J/kg.

SECTION 16.10 CAPACITORS WITH DIELECTRICS 49. ecp The voltage across an air-filled parallel-plate capacitor is measured to be 85.0 V. When a dielectric is inserted and completely fills the space between the plates as in Figure 16.24, the voltage drops to 25.0 V. (a) What is the dielectric constant of the inserted material? Can you identify the dielectric? (b) If the dielectric doesn’t completely fill the space between the plates, what could you conclude about the voltage across the plates? 50. A parallel-plate capacitor in air has a plate separation of 1.50 cm and a plate area of 25.0 cm2. The plates are charged to a potential difference of 2.50  102 V and disconnected from the source. The capacitor is then immersed in distilled water. Determine (a) the charge on the plates before and after immersion, (b) the capacitance and potential difference after immersion, and (c) the change in energy stored in the capacitor due to immersion. Assume the distilled water is an insulator. 51. Determine (a) the capacitance and (b) the maximum voltage that can be applied to a Teflon-filled parallel-plate capacitor having a plate area of 175 cm2 and an insulation thickness of 0.040 0 mm. 52. A commercial capacitor is constructed as in Figure 16.26a. This particular capacitor is made from a strip of aluminum foil separated by two strips of paraffi n-coated paper. Each strip of foil and paper is 7.00 cm wide. The foil is 0.004 00 mm thick, and the paper is 0.025 0 mm thick and has a dielectric constant of 3.70. What length should the strips be if a capacitance of 9.50  10
8 F is desired

before the capacitor is rolled up? (Use the parallel-plate formula. Adding a second strip of paper and rolling up the capacitor doubles its capacitance by allowing both surfaces of each strip of foil to store charge.) 53.

A model of a red blood cell portrays the cell as a shperical capacitor, a positively charged liquid sphere of surface area A separated from the surrounding negatively charged fluid by a membrane of thickness t. Tiny electrodes introduced into the interior of the cell show a potential difference of 100 mV across the membrane. The membrane’s thickness is estimated to be 100 nm and has a dielectric constant of 5.00. (a) If an average red blood cell has a mass of 1.00  10
12 kg, estimate the volume of the cell and thus find its surface area. The density of blood is 1 100 kg/m3. (b) Estimate the capacitance of the cell by assuming the membrane surfaces act as parallel plates. (c) Calculate the charge on the surface of the membrane. How many electronic charges does the surface charge represent?

ADDITIONAL PROBLEMS 54. ecp Three parallel-plate capacitors are constructed, each having the same plate spacing d and with C 1 having plate area A1, C 2 having area A 2, and C 3 having area A 3. Show that the total capacitance C of the three capacitors connected in parallel is the same as that of a capacitor having plate spacing d and plate area A  A1  A 2  A3. 55. ecp Three parallel-plate capacitors are constructed, each having the same plate area A and with C 1 having plate spacing d1, C 2 having plate spacing d 2, and C 3 having plate spacing d 3. Show that the total capacitance C of the three capacitors connected in series is the same as a capacitor of plate area A and with plate spacing d  d1  d 2  d 3. 56. ecp For the system of four capacitors shown in Figure P16.37, find (a) the total energy stored in the system and (b) the energy stored by each capacitor. (c) Compare the sum of the answers in part (b) with your result to part (a) and explain your observation. 57. ecp A parallel-plate capacitor with a plate separation d has a capacitance C 0 in the absence of a dielectric. A slab of dielectric material of dielectric constant k and thickness d/3 is then inserted between the plates as in Figure P16.57. Show that the capacitance of this partially filled capacitor is given by C5a

3k bC 2k 1 1 0

(Hint: Treat the system as two capacitors connected in series, one with dielectric in it and the other one empty.)

κ

1d 3 2d 3

1d 3

2d 3

(a)

κ

C1

d

(b) FIGURE P16.57

C2

Problems

58. ecp Two capacitors give an equivalent capacitance of Cp when connected in parallel and an equivalent capacitance of Cs when connected in series. What is the capacitance of each capacitor? 59. An isolated capacitor of unknown capacitance has been charged to a potential difference of 100 V. When the charged capacitor is disconnected from the battery and then connected in parallel to an uncharged 10.0-mF capacitor, the voltage across the combination is measured to be 30.0 V. Calculate the unknown capacitance. 60. Two charges of 1.0 mC and
2.0 mC are 0.50 m apart at two vertices of an equilateral triangle as in Figure P16.60. (a) What is the electric potential due to the 1.0- mC charge at the third vertex, point P ? (b) What is the electric potential due to the
2.0- mC charge at P ? (c) Find the total electric potential at P. (d) What is the work required to move a 3.0-mC charge from infinity to P. P 0.50 m

1.0 µC

0.50 m

0.50 m


2.0 µC

FIGURE P16.60

61. Find the equivalent capacitance of the group of capacitors shown in Figure P16.61. 5.00 µF 3.00 µF 2.00 µF

4.00 µF

3.00 µF 6.00 µF

7.00 µF

48.0 V FIGURE P16.61

tric shock to the chest over a time of a few milliseconds. The device contains a capacitor of a few microfarads, charged to several thousand volts. Electrodes called paddles, about 8 cm across and coated with conducting paste, are held against the chest on both sides of the heart. Their handles are insulated to prevent injury to the operator, who calls “Clear!” and pushes a button on one paddle to discharge the capacitor through the patient’s chest. Assume an energy of 300 W  s is to be delivered from a 30.0-mF capacitor. To what potential difference must it be charged? 64. When a certain air-filled parallel-plate capacitor is connected across a battery, it acquires a charge of 150 mC on each plate. While the battery connection is maintained, a dielectric slab is inserted into, and fills, the region between the plates. This results in the accumulation of an additional charge of 200 mC on each plate. What is the dielectric constant of the slab? 65. Capacitors C 1  6.0 mF and C 2  2.0 mF are charged as a parallel combination across a 250-V battery. The capacitors are disconnected from the battery and from each other. They are then connected positive plate to negative plate and negative plate to positive plate. Calculate the resulting charge on each capacitor. 66. The energy stored in a 52.0-mF capacitor is used to melt a 6.00-mg sample of lead. To what voltage must the capacitor be initially charged, assuming the initial temperature of the lead is 20.0°C? Lead has a specific heat of 128 J/kg°C, a melting point of 327.3°C, and a latent heat of fusion of 24.5 kJ/kg. 67. Metal sphere A of radius 12.0 cm carries 6.00 mC of charge, and metal sphere B of radius 18.0 cm carries
4.00 mC of charge. If the two spheres are attached by a very long conducting thread, what is the final distribution of charge on the two spheres? 68. An electron is fired at a speed v 0  5.6  106 m/s and at an angle u0 
45° between two parallel conducting plates that are D  2.0 mm apart, as in Figure P16.68. If the voltage difference between the plates is V  100 V, determine (a) how close, d, the electron will get to the bottom plate and (b) where the electron will strike the top plate.

62. A spherical capacitor consists of a spherical conducting shell of radius b and charge
Q concentric with a smaller conducting sphere of radius a and charge Q. (a) Find the capacitance of this device. (b) Show that as the radius b of the outer sphere approaches infi nity, the capacitance approaches the value a/ke  4pP0a. 63.

The immediate cause of many deaths is ventricular fibrillation, an uncoordinated quivering of the heart, as opposed to proper beating. An electric shock to the chest can cause momentary paralysis of the heart muscle, after which the heart will sometimes start organized beating again. A defibrillator is a device that applies a strong elec-

569

y Path of the electron D

0

x

θ0 d

FIGURE P16.68

V

17 Lester Lefkowitz/Getty Images

These power lines transfer energy from the power company to homes and businesses. The energy is transferred at a very high voltage, possibly hundreds of thousands of volts in some cases. The high voltage results in less loss of power due to resistance in the wires, so it is used even though it makes power lines very dangerous.

17.1 Electric Current 17.2

A Microscopic View: Current and Drift Speed

17.3

Current and Voltage Measurements in Circuits

17.4

Resistance, Resistivity, and Ohm’s Law

17.5

Temperature Variation of Resistance

17.6

Electrical Energy and Power

17.7 Superconductors 17.8 Electrical Activity in the Heart

CURRENT AND RESISTANCE Many practical applications and devices are based on the principles of static electricity, but electricity was destined to become an inseparable part of our daily lives when scientists learned how to produce a continuous flow of charge for relatively long periods of time using batteries. The battery or voltaic cell was invented in 1800 by Italian physicist Alessandro Volta. Batteries supplied a continuous flow of charge at low potential, in contrast to earlier electrostatic devices that produced a tiny flow of charge at high potential for brief periods. This steady source of electric current allowed scientists to perform experiments to learn how to control the flow of electric charges in circuits. Today, electric currents power our lights, radios, television sets, air conditioners, computers, and refrigerators. They ignite the gasoline in automobile engines, travel through miniature components making up the chips of microcomputers, and provide the power for countless other invaluable tasks. In this chapter we define current and discuss some of the factors that contribute to the resistance to the flow of charge in conductors. We also discuss energy transformations in electric circuits. These topics will be the foundation for additional work with circuits in later chapters.

17.1

ELECTRIC CURRENT

In Figure 17.1 charges move in a direction perpendicular to a surface of area A. (The area could be the cross-sectional area of a wire, for example.) The current is the rate at which charge flows through this surface. Suppose Q is the amount of charge that flows through an area A in a time interval t and that the direction of flow is perpendicular to the area. Then the average current Iav is equal to the amount of charge divided by the time interval: DQ I av ; [17.1a] Dt SI unit: coulomb/second (C/s), or the ampere (A)

570

17.1

Current is composed of individual moving charges, so for an extremely low current, it is conceivable that a single charge could pass through area A in one instant and no charge in the next instant. All currents, then, are essentially averages over time. Given the very large number of charges usually involved, however, it makes sense to define an instantaneous current. The instantaneous current I is the limit of the average current as the time interval goes to zero: DQ I 5 lim I av 5 lim [17.1b] Dt S 0 Dt S 0 Dt

Electric Current

571

TIP 17.1 Current Flow Is Redundant The phrases flow of current and current flow are commonly used, but here the word flow is redundant because current is already defined as a flow (of charge). Avoid this construction!

SI unit: coulomb/second (C/s), or the ampere (A) When the current is steady, the average and instantaneous currents are the same. Note that one ampere of current is equivalent to one coulomb of charge passing through the cross-sectional area in a time interval of 1 s. When charges flow through a surface as in Figure 17.1, they can be positive, negative, or both. The direction of conventional current used in this book is the direction positive charges flow. (This historical convention originated about 200 years ago, when the ideas of positive and negative charges were introduced.) In a common conductor such as copper, the current is due to the motion of negatively charged electrons, so the direction of the current is opposite the direction of motion of the electrons. On the other hand, for a beam of positively charged protons in an accelerator, the current is in the same direction as the motion of the protons. In some cases—gases and electrolytes, for example—the current is the result of the flows of both positive and negative charges. Moving charges, whether positive or negative, are referred to as charge carriers. In a metal, for example, the charge carriers are electrons. In electrostatics, where charges are stationary, the electric potential is the same everywhere in a conductor. That is no longer true for conductors carrying current: as charges move along a wire, the electric potential is continually decreasing (except in the special case of superconductors).

O Direction of current

+ + + + +

A I

FIGURE 17.1 Charges in motion through an area A. The time rate of flow of charge through the area is defined as the current I. The direction of the current is the direction of flow of positive charges.

EXAMPLE 17.1 Turn On the Light Goal

Apply the concept of current.

Problem The amount of charge that passes through the filament of a certain lightbulb in 2.00 s is 1.67 C. Find (a) the average current in the lightbulb and (b) the number of electrons that pass through the filament in 5.00 s. Strategy Substitute into Equation 17.1a for part (a), then multiply the answer by the time given in part (b) to get the total charge that passes in that time. The total charge equals the number N of electrons going through the circuit times the charge per electron. Solution (a) Compute the average current in the lightbulb. Substitute the charge and time into Equation 17.1a:

I av 5

DQ Dt

5

1.67 C 5 0.835 A 2.00 s

(b) Find the number of electrons passing through the filament in 5.00 s. The total number N of electrons times the charge per electron equals the total charge, Iav t:

(1) Nq  Iav t

Substitute and solve forN:

N(1.60  10
19 C/electron)  (0.835 A)(5.00 s) N  2.61 3 1019 electrons

572

Chapter 17

Current and Resistance

Remarks In developing the solution, it was important to use units to ensure the correctness of equations such as Equation (1). Notice the enormous number of electrons passing through a given point in a typical circuit. QUESTION 17.1 Is it possible to have an instantaneous current of e/2 per second? Explain. Can the average current take this value? EXERCISE 17.1 Suppose 6.40  1021 electrons pass through a wire in 2.00 min. Find the average current. Answer 8.53 A

QUICK QUIZ 17.1 Consider positive and negative charges moving horizontally through the four regions in Figure 17.2. Rank the magnitudes of the currents in these four regions from lowest to highest. (Ia is the current in Figure 17.2a, Ib the current in Figure 17.2b, etc.) (a) Id , Ia , Ic , Ib (b) Ia , Ic , Ib, Id (c) Ic , Ia , Id , Ib (d) Id , Ib, Ic , Ia (e) Ia , Ib, Ic , Id (f) None of these FIGURE 17.2 (Quick Quiz 17.1)

+

–

+

+ –

+

(a)

(b)

+

– + + –

+

∆x



17.2 vd A

q 

vd ∆t FIGURE 17.3 A section of a uniform conductor of cross-sectional area A. The charge carriers move with a speed vd , and the distance they travel in time t is given by x  vd t. The number of mobile charge carriers in the section of length x is given by nAvd t, where n is the number of mobile carriers per unit volume.

vd

–

(c)

– (d)

A MICROSCOPIC VIEW: CURRENT AND DRIFT SPEED

Macroscopic currents can be related to the motion of the microscopic charge carriers making up the current. It turns out that current depends on the average speed of the charge carriers in the direction of the current, the number of charge carriers per unit volume, and the size of the charge carried by each. Consider identically charged particles moving in a conductor of cross-sectional area A (Fig. 17.3). The volume of an element of length x of the conductor is A x. If n represents the number of mobile charge carriers per unit volume, the number of carriers in the volume element is nA x. The mobile charge Q in this element is therefore Q  number of carriers  charge per carrier  (nA x)q where q is the charge on each carrier. If the carriers move with a constant average speed called the drift speed vd , the distance they move in the time interval t is x  vd t. We can therefore write Q  (nAvd t)q If we divide both sides of this equation by t and take the limit as t goes to zero, we see that the current in the conductor is

–

E ACTIVE FIGURE 17.4 A schematic representation of the zigzag motion of a charge carrier in a conductor. The sharp changes in direction are due to collisions with atoms in the conductor. Notice that the net motion of electrons is opposite the direction of the electric field.

I 5 lim

Dt S 0

DQ Dt

5 nqv dA

[17.2]

To understand the meaning of drift speed, consider a conductor in which the charge carriers are free electrons. If the conductor is isolated, these electrons undergo random motion similar to the motion of the molecules of a gas. The drift speed is normally much smaller than the free electrons’ average speed between collisions with the fixed atoms of the conductor. When a potential difference is

17.2

A Microscopic View: Current and Drift Speed

applied between the ends of the conductor (say, with a battery), an electric field is set up in the conductor, creating an electric force on the electrons and hence a current. In reality, the electrons don’t simply move in straight lines along the conductor. Instead, they undergo repeated collisions with the atoms of the metal, and the result is a complicated zigzag motion with only a small average drift speed along the wire (Active Fig. 17.4). The energy transferred from the electrons to the metal atoms during a collision increases the vibrational energy of the atoms and causes a corresponding increase in the temperature of the conductor. Despite the collisions,Showever, the electrons move slowly along the conductor in a direction S opposite E with the drift velocity v d .

573

TIP 17.2 Electrons Are Everywhere in the Circuit Electrons don’t have to travel from the light switch to the lightbulb for the lightbulb to operate. Electrons already in the filament of the lightbulb move in response to the electric field set up by the battery. Also, the battery does not provide electrons to the circuit; it provides energy to the existing electrons.

EXAMPLE 17.2 Drift Speed of Electrons Goal

Calculate a drift speed and compare it with the rms speed of an electron gas.

Problem A copper wire of cross-sectional area 3.00  10
6 m2 carries a current of 10.0 A. (a) Assuming each copper atom contributes one free electron to the metal, find the drift speed of the electrons in this wire. (b) Use the ideal gas model to compare the drift speed with the random rms speed an electron would have at 20.0°C. The density of copper is 8.92 g/cm3, and its atomic mass is 63.5 u. Strategy All the variables in Equation 17.2 are known except for n, the number of free charge carriers per unit volume. We can find n by recalling that one mole of copper contains an Avogadro’s number (6.02  1023) of atoms and each atom contributes one charge carrier to the metal. The volume of one mole can be found from copper’s known density and atomic mass. The atomic mass is the same, numerically, as the number of grams in a mole of the substance. Solution (a) Find the drift speed of the electrons. Calculate the volume of one mole of copper from its density and its atomic mass: Convert the volume from cm3 to m3:

63.5 g m 5 5 7.12 cm3 r 8.92 g/cm3

V5

7.12 cm3 a

1m 3 b 5 7.12 3 1026 m3 102 cm

6.02 3 1023 electrons/mole 7.12 3 1026 m3 /mole

Divide Avogadro’s number (the number of electrons in one mole) by the volume per mole to obtain the number density:

n5

Solve Equation 17.2 for the drift speed and substitute:

vd 5

5 8.46 3 1028 electrons/m3

5

I nqA 10.0 C/s 1 8.46 3 10 electrons/m 2 1 1.60 3 10219 C 2 1 3.00 3 1026 m2 2 28

3

vd  2.46 3 1024 m/s (b) Find the rms speed of a gas of electrons at 20.0°C. Apply Equation 10.18:

v rms 5

Convert the temperature to the Kelvin scale and substitute values:

v rms 5

3k BT Å me Å

3 1 1.38 3 10223 J/K 2 1 293 K 2 9.11 3 10231 kg

 1.15 3 105 m/s

574

Chapter 17

Current and Resistance

Remark The drift speed of an electron in a wire is very small, only about one-billionth of its random thermal speed. QUESTION 17.2 True or False: The drift velocity in a wire of a given composition is inversely proportional to the number density of charge carriers. EXERCISE 17.2 What current in a copper wire with a cross-sectional area of 7.50  10
7 m2 would result in a drift speed equal to 5.00  10
4 m/s? Answer 5.08 A

Example 17.2 shows that drift speeds are typically very small. In fact, the drift speed is much smaller than the average speed between collisions. Electrons traveling at 2.46  10
4 m/s, as in the example, would take about 68 min to travel 1 m! In view of this low speed, why does a lightbulb turn on almost instantaneously when a switch is thrown? Think of the flow of water through a pipe. If a drop of water is forced into one end of a pipe that is already filled with water, a drop must be pushed out the other end of the pipe. Although it may take an individual drop a long time to make it through the pipe, a flow initiated at one end produces a similar flow at the other end very quickly. Another familiar analogy is the motion of a bicycle chain. When the sprocket moves one link, the other links all move more or less immediately, even though it takes a given link some time to make a complete rotation. In a conductor, the electric field driving the free electrons travels at a speed close to that of light, so when you flip a light switch, the message for the electrons to start moving through the wire (the electric field) reaches them at a speed on the order of 108 m/s! QUICK QUIZ 17.2 Suppose a current-carrying wire has a cross-sectional area that gradually becomes smaller along the wire so that the wire has the shape of a very long, truncated cone. How does the drift speed vary along the wire? (a) It slows down as the cross section becomes smaller. (b) It speeds up as the cross section becomes smaller. (c) It doesn’t change. (d) More information is needed.

17.3 CURRENT AND VOLTAGE MEASUREMENTS IN CIRCUITS To study electric current in circuits, we need to understand how to measure currents and voltages. The circuit shown in Figure 17.5a is a drawing of the actual circuit necessary for measuring the current in Example 17.1. Figure 17.5b shows a stylized figure called a circuit diagram that represents the actual circuit of Figure 17.5a. This circuit consists of only a battery and a lightbulb. The word circuit means “a closed loop of some sort around which current circulates.” The battery pumps charge through the bulb and around the loop. No charge would flow without a complete conducting path from the positive terminal of the battery into one side of the bulb, out the other side, and through the copper conducting wires back to the negative terminal of the battery. The most important quantities that characterize how the bulb works in different situations are the current I in the bulb and the potential difference V across the bulb. To measure the current in the bulb, we place an ammeter, the device for measuring current, in line with the bulb so there is no path for the current to bypass the meter; all the charge passing through the bulb must also pass through the ammeter. The voltmeter measures the potential difference, or volt-

17.4

Resistance, Resistivity, and Ohm’s Law

Battery – + c, Michael Dalton, Fundamental Photographs

–

+

Bulb

I – 2.91 V

=+0.835 A

A

+

Ammeter

I I

+

–

I

+

V – + Voltmeter

–

(a)

(c)

(b)

FIGURE 17.5 (a) A sketch of an actual circuit used to measure the current in a flashlight bulb and the potential difference across it. (b) A schematic diagram of the circuit shown in (a). (c) A digital multimeter can be used to measure both current and potential difference. Here, the meter is measuring the potential difference across a 9-V battery.

age, between the two ends of the bulb’s filament. If we use two meters simultaneously as in Figure 17.5a, we can remove the voltmeter and see if its presence affects the current reading. Figure 17.5c shows a digital multimeter, a convenient device, with a digital readout, that can be used to measure voltage, current, or resistance. An advantage of using a digital multimeter as a voltmeter is that it will usually not affect the current because a digital meter has enormous resistance to the flow of charge in the voltmeter mode. At this point, you can measure the current as a function of voltage (an I–V curve) of various devices in the lab. All you need is a variable voltage supply (an adjustable battery) capable of supplying potential differences from about
5 V to 5 V, a bulb, a resistor, some wires and alligator clips, and a couple of multimeters. Be sure to always start your measurements using the highest multimeter scales (say, 10 A and 1 000 V), and increase the sensitivity one scale at a time to obtain the highest accuracy without overloading the meters. (Increasing the sensitivity means lowering the maximum current or voltage that the scale reads.) Note that the meters must be connected with the proper polarity with respect to the voltage supply, as shown in Figure 17.5b. Finally, follow your instructor’s directions carefully to avoid damaging the meters and incurring a soaring lab fee. QUICK QUIZ 17.3 Look at the four “circuits” shown in Figure 17.6 and select those that will light the bulb. –

+

–

+

–

+

+

–

AMPS

+

(a)

(b)

(c)

FIGURE 17.6 (Quick Quiz 17.3)

17.4

RESISTANCE, RESISTIVITY, AND OHM’S LAW

Resistance and Ohm’s Law When a voltage (potential difference) V is applied across the ends of a metallic conductor as in Figure 17.7 (page 576), the current in the conductor is found to be

(d)

–

575

576

Chapter 17

Current and Resistance

proportional to the applied voltage; I V. If the proportionality holds, we can write V  IR, where the proportionality constant R is called the resistance of the conductor. In fact, we define the resistance as the ratio of the voltage across the conductor to the current it carries: Resistance R

R ;

/

A Va E

FIGURE 17.7 A uniform conductor of length l and cross-sectional area A. The current I in the conductor is proportional to the applied voltage V S  V b
Va . The electric field E set up in the conductor is also proportional to the current.

© Bettmann/CORBIS

DV 5 IR

GEORG SIMON OHM (1787–1854)

Courtesy of Henry Leap and Jim Lehman

A high school teacher in Cologne and later a professor at Munich, Ohm formulated the concept of resistance and discovered the proportionalities expressed in Equation 17.5.

An assortment of resistors used for a variety of applications in electronic circuits.

[17.3]

Resistance has SI units of volts per ampere, called ohms (). If a potential difference of 1 V across a conductor produces a current of 1 A, the resistance of the conductor is 1 . For example, if an electrical appliance connected to a 120-V source carries a current of 6 A, its resistance is 20 . The concepts of electric current, voltage, and resistance can be compared to the flow of water in a river. As water flows downhill in a river of constant width and depth, the flow rate (water current) depends on the steepness of descent of the river and the effects of rocks, the riverbank, and other obstructions. The voltage difference is analogous to the steepness, and the resistance to the obstructions. Based on this analogy, it seems reasonable that increasing the voltage applied to a circuit should increase the current in the circuit, just as increasing the steepness of descent increases the water current. Also, increasing the obstructions in the river’s path will reduce the water current, just as increasing the resistance in a circuit will lower the electric current. Resistance in a circuit arises due to collisions between the electrons carrying the current with fixed atoms inside the conductor. These collisions inhibit the movement of charges in much the same way as would a force of friction. For many materials, including most metals, experiments show that the resistance remains constant over a wide range of applied voltages or currents. This statement is known as Ohm’s law, after Georg Simon Ohm (1789–1854), who was the first to conduct a systematic study of electrical resistance. Ohm’s law is given by

I

Vb

DV I

[17.4]

where R is understood to be independent of V, the potential drop across the resistor, and I, the current in the resistor. We will continue to use this traditional form of Ohm’s law when discussing electrical circuits. A resistor is a conductor that provides a specified resistance in an electric circuit. The symbol for a resistor in circuit diagrams is a zigzag line: . Ohm’s law is an empirical relationship valid only for certain materials. Materials that obey Ohm’s law, and hence have a constant resistance over a wide range of voltages, are said to be ohmic. Materials having resistance that changes with voltage or current are nonohmic. Ohmic materials have a linear current–voltage relationship over a large range of applied voltages (Fig. 17.8a). Nonohmic materials have a nonlinear current–voltage relationship (Fig. 17.8b). One common semiconducting device that is nonohmic is the diode, a circuit element that acts like a one-way valve for current. Its resistance is small for currents in one direction (positive V ) and large for currents in the reverse direction (negative V ). Most modern electronic devices, such as transistors, have nonlinear current–voltage relationships; their operation depends on the particular ways in which they violate Ohm’s law.

QUICK QUIZ 17.4 In Figure 17.8b does the resistance of the diode (a) increase or (b) decrease as the positive voltage V increases? QUICK QUIZ 17.5 All electric devices are required to have identifying plates that specify their electrical characteristics. The plate on a certain steam iron states that the iron carries a current of 6.00 A when con-

17.4

Resistance, Resistivity, and Ohm’s Law

nected to a source of 1.20  102 V. What is the resistance of the steam iron? (a) 0.050 0  (b) 20.0  (c) 36.0 

I Slope = 1 R V

Resistivity Electrons don’t move in straight-line paths through a conductor. Instead, they undergo repeated collisions with the metal atoms. Consider a conductor with a voltage applied across its ends. An electron gains speed as the electric force associated with the internal electric field accelerates it, giving it a velocity in the direction opposite that of the electric field. A collision with an atom randomizes the electron’s velocity, reducing it in the direction opposite the field. The process then repeats itself. Together, these collisions affect the electron somewhat as a force of internal friction would. This step is the origin of a material’s resistance. The resistance of an ohmic conductor increases with length, which makes sense because the electrons going through it must undergo more collisions in a longer conductor. A smaller cross-sectional area also increases the resistance of a conductor, just as a smaller pipe slows the fluid moving through it. The resistance, then, is proportional to the conductor’s length  and inversely proportional to its crosssectional area A, R5r

, A

[17.5]

where the constant of proportionality, r, is called the resistivity of the material. Every material has a characteristic resistivity that depends on its electronic structure and on temperature. Good electric conductors have very low resistivities, and good insulators have very high resistivities. Table 17.1 lists the resistivities of various materials at 20°C. Because resistance values are in ohms, resistivity values must be in ohm-meters (  m).

TABLE 17.1 Resistivities and Temperature Coefficients of Resistivity for Various Materials (at 20°C)

Material

Resistivity (  m)

Silver Copper Gold Aluminum Tungsten Iron Platinum Lead Nichromea Carbon Germanium Silicon Glass Hard rubber Sulfur Quartz (fused)

1.59  10
8 1.7  10
8 2.44  10
8 2.82  10
8 5.6  10
8 10.0  10
8 11  10
8 22  10
8 150  10
8 3.5  105 0.46 640 10 10 –1014 1013 1015 75  1016

aA

577

nickel-chromium alloy commonly used in heating elements.

Temperature Coefficient of Resistivity [(°C) 1] 3.8  10
3 3.9  10
3 3.4  10
3 3.9  10
3 4.5  10
3 5.0  10
3 3.92  10
3 3.9  10
3 0.4  10
3
0.5  10
3
48  10
3
75  10
3

(a) I

V (b) FIGURE 17.8 (a) The current– voltage curve for an ohmic material. The curve is linear, and the slope gives the resistance of the conductor. (b) A nonlinear current–voltage curve for a semiconducting diode. This device doesn’t obey Ohm’s law.

578

Chapter 17

Current and Resistance

APPLYING PHYSICS 17.1

DIMMING OF AGING LIGHTBULBS

As a lightbulb ages, why does it gives off less light than when new? Explanation There are two reasons for the lightbulb’s behavior, one electrical and one optical, but both are related to the same phenomenon occurring within the bulb. The filament of an old lightbulb is made of a tungsten wire that has been kept at a high temperature for many hours. High temperatures evaporate tungsten from the filament, decreasing its radius. From R  r/A, we see that a decreased crosssectional area leads to an increase in the resistance of the filament. This increasing resistance with age

means that the filament will carry less current for the same applied voltage. With less current in the filament, there is less light output, and the filament glows more dimly. At the high operating temperature of the filament, tungsten atoms leave its surface, much as water molecules evaporate from a puddle of water. The atoms are carried away by convection currents in the gas in the bulb and are deposited on the inner surface of the glass. In time, the glass becomes less transparent because of the tungsten coating, which decreases the amount of light that passes through the glass.

EXAMPLE 17.3 The Resistance of Nichrome Wire Goal

Combine the concept of resistivity with Ohm’s law.

Problem (a) Calculate the resistance per unit length of a 22-gauge Nichrome wire of radius 0.321 mm. (b) If a potential difference of 10.0 V is maintained across a 1.00-m length of the Nichrome wire, what is the current in the wire? (c) The wire is melted down and recast with twice its original length. Find the new resistance R N as a multiple of the old resistance RO . Strategy Part (a) requires substitution into Equation 17.5, after calculating the cross-sectional area, whereas part (b) is a matter of substitution into Ohm’s law. Part (c) requires some algebra. The idea is to take the expression for the new resistance and substitute expressions for N and AN, the new length and cross-sectional area, in terms of the old length and cross-section. For the area substitution, remember that the volumes of the old and new wires are the same.

Solution (a) Calculate the resistance per unit length. Find the cross-sectional area of the wire:

A  pr 2  (0.321  10
3 m)2  3.24  10
7 m2

Obtain the resistivity of Nichrome from Table 17.1, solve Equation 17.5 for R/, and substitute:

r R 1.5 3 1026 V # m 5 5 5 4.6 V/m , A 3.24 3 1027 m2

(b) Find the current in a 1.00-m segment of the wire if the potential difference across it is 10.0 V. Substitute given values into Ohm’s law:

I5

DV 10.0 V 5 5 2.2 A R 4.6 V

(c) If the wire is melted down and recast with twice its original length, find the new resistance as a multiple of the old. Find the new area A N in terms of the old area AO , using the fact the volume doesn’t change and N  2O :

V N  VO :

Substitute into Equation 17.5:

RN 5

A N N  AO O

:

A N  AO(O/N)

A N  AO(O/2O)  AO/2 r,O r,N r 1 2,O 2 54 5 5 4R O 1 AO/2 2 AN AO

17.5

Temperature Variation of Resistance

579

Remarks From Table 17.1, the resistivity of Nichrome is about 100 times that of copper, a typical good conductor. Therefore, a copper wire of the same radius would have a resistance per unit length of only 0.052 /m, and a 1.00-m length of copper wire of the same radius would carry the same current (2.2 A) with an applied voltage of only 0.115 V. Because of its resistance to oxidation, Nichrome is often used for heating elements in toasters, irons, and electric heaters. QUESTION 17.3 Would replacing the Nichrome with copper result in a higher current or lower current? EXERCISE 17.3 What is the resistance of a 6.0-m length of Nichrome wire that has a radius 0.321 mm? How much current does it carry when connected to a 120-V source? Answers 28 ; 4.3 A

QUICK QUIZ 17.6 Suppose an electrical wire is replaced with one having every linear dimension doubled (i.e., the length and radius have twice their original values). Does the wire now have (a) more resistance than before, (b) less resistance, or (c) the same resistance?

17.5

TEMPERATURE VARIATION OF RESISTANCE

The resistivity r, and hence the resistance, of a conductor depends on a number of factors. One of the most important is the temperature of the metal. For most metals, resistivity increases with increasing temperature. This correlation can be understood as follows: as the temperature of the material increases, its constituent atoms vibrate with greater amplitudes. As a result, the electrons find it more difficult to get by those atoms, just as it is more difficult to weave through a crowded room when the people are in motion than when they are standing still. The increased electron scattering with increasing temperature results in increased resistivity. Technically, thermal expansion also affects resistance; however, this is a very small effect. Over a limited temperature range, the resistivity of most metals increases linearly with increasing temperature according to the expression [17.6]

where r is the resistivity at some temperature T (in Celsius degrees), r0 is the resistivity at some reference temperature T0 (usually taken to be 20°C), and  is a parameter called the temperature coefficient of resistivity. Temperature coefficients for various materials are provided in Table 17.1. The interesting negative values of  for semiconductors arise because these materials possess weakly bound charge carriers that become free to move and contribute to the current as the temperature rises. Because the resistance of a conductor with a uniform cross section is proportional to the resistivity according to Equation 17.5 (R  rl/A), the temperature variation of resistance can be written R  R 0[1  (T
T0)]

[17.7]

Precise temperature measurements are often made using this property, as shown by the following example.

© Royalty-free/Corbis

r  r0[1  (T
T0)]

In an old-fashioned carbon fi lament incandescent lamp, the electrical resistance is typically 10 , but changes with temperature.

580

Chapter 17

EXAMPLE 17.4 Goal

Current and Resistance

A Platinum Resistance Thermometer

Apply the temperature dependence of resistance.

Problem A resistance thermometer, which measures temperature by measuring the change in resistance of a conductor, is made of platinum and has a resistance of 50.0  at 20.0°C. (a) When the device is immersed in a vessel containing melting indium, its resistance increases to 76.8 . From this information, find the melting point of indium. (b) The indium is heated further until it reaches a temperature of 235°C. What

is the ratio of the new current in the platinum to the current Imp at the melting point? Strategy For part (a), solve Equation 17.7 for T
T0 and get  for platinum from Table 17.1, substituting known quantities. For part (b), use Ohm’s law in Equation 17.7.

Solution (a) Find the melting point of indium. Solve Equation 17.7 for T
T0:

T 2 T0 5

R 2 R0 76.8 V 2 50.0 V 5 3 3.92 3 1023 1 °C 2 21 4 3 50.0 V 4 aR 0

5 137°C Substitute T0  20.0°C and obtain the melting point of indium:

T  157°C

(b) Find the ratio of the new current to the old when the temperature rises from 157°C to 235°C. Write Equation 17.7, with R 0 and T0 replaced by R mp and Tmp, the resistance and temperature at the melting point.

R  R mp[1  (T
Tmp)]

According to Ohm’s law, R  V/I and R mp  V/Imp. Substitute these expressions into Equation 17.7:

DV DV 3 1 1 a 1 T 2 Tmp 2 4 5 I I mp

Cancel the voltage differences, invert the two expressions, and then divide both sides by Imp:

I 1 5 I mp 1 1 a 1 T 2 Tmp 2

Substitute T  235°C, Tmp  157°C, and the value for , obtaining the desired ratio:

I I mp

5 0.766

Remark As the temperature rises, both the rms speed of the electrons in the metal and the resistance increase. QUESTION 17.4 What happens to the drift speed of the electrons as the temperature rises? (a) It becomes larger. (b) It becomes smaller. (c) It remains unchanged. EXERCISE 17.4 Suppose a wire made of an unknown alloy and having a temperature of 20.0°C carries a current of 0.450 A. At 52.0°C the current is 0.370 A for the same potential difference. Find the temperature coefficient of resistivity of the alloy. Answer 6.76  10
3 (°C)
1

17.6 ELECTRICAL ENERGY AND POWER If a battery is used to establish an electric current in a conductor, chemical energy stored in the battery is continuously transformed into kinetic energy of the charge carriers. This kinetic energy is quickly lost as a result of collisions between the charge carriers and fi xed atoms in the conductor, causing an increase in the tem-

17.6

perature of the conductor. In this way the chemical energy stored in the battery is continuously transformed into thermal energy. To understand the process of energy transfer in a simple circuit, consider a battery with terminals connected to a resistor (Active Fig. 17.9; remember that the positive terminal of the battery is always at the higher potential). Now imagine following a quantity of positive charge Q around the circuit from point A, through the battery and resistor, and back to A. Point A is a reference point that is ), and its potential is taken to be zero. As the grounded (the ground symbol is charge Q moves from A to B through the battery, the electrical potential energy of the system increases by the amount Q V and the chemical potential energy in the battery decreases by the same amount. (Recall from Chapter 16 that PE  q V.) As the charge moves from C to D through the resistor, however, it loses this electrical potential energy during collisions with atoms in the resistor. In the process the energy is transformed to internal energy corresponding to increased vibrational motion of those atoms. Because we can ignore the very small resistance of the interconnecting wires, no energy transformation occurs for paths BC and DA. When the charge returns to point A, the net result is that some of the chemical energy in the battery has been delivered to the resistor and has caused its temperature to rise. The charge Q loses energy Q V as it passes through the resistor. If t is the time it takes the charge to pass through the resistor, the instantaneous rate at which it loses electric potential energy is lim

Dt S 0

DQ Dt

Electrical Energy and Power

581

I

B +

C R

– A

D

ACTIVE FIGURE 17.9 A circuit consisting of a battery and a resistance R. Positive charge flows clockwise from the positive to the negative terminal of the battery. Point A is grounded.

DV 5 I DV

where I is the current in the resistor and V is the potential difference across it. Of course, the charge regains this energy when it passes through the battery, at the expense of chemical energy in the battery. The rate at which the system loses potential energy as the charge passes through the resistor is equal to the rate at which the system gains internal energy in the resistor. Therefore, the power
, representing the rate at which energy is delivered to the resistor, is
5 I DV

[17.8]

O Power

Although this result was developed by considering a battery delivering energy to a resistor, Equation 17.8 can be used to determine the power transferred from a voltage source to any device carrying a current I and having a potential difference V between its terminals. Using Equation 17.8 and the fact that V  IR for a resistor, we can express the power delivered to the resistor in the alternate forms
5 I 2R 5

DV 2 R

[17.9]

When I is in amperes, V in volts, and R in ohms, the SI unit of power is the watt (introduced in Chapter 5). The power delivered to a conductor of resistance R is often referred to as an I 2R loss. Note that Equation 17.9 applies only to resistors and not to nonohmic devices such as lightbulbs and diodes. Regardless of the ways in which you use electrical energy in your home, you ultimately must pay for it or risk having your power turned off. The unit of energy used by electric companies to calculate consumption, the kilowatt-hour, is defined in terms of the unit of power and the amount of time it’s supplied. One kilowatthour (kWh) is the energy converted or consumed in 1 h at the constant rate of 1 kW. It has the numerical value 1 kWh  (103 W)(3 600 s)  3.60  106 J

[17.10]

On an electric bill, the amount of electricity used in a given period is usually stated in multiples of kilowatt-hours.

O Power delivered to a resistor

TIP 17.3 Misconception About Current Current is not “used up” in a resistor. Rather, some of the energy the charges have received from the voltage source is delivered to the resistor, making it hot and causing it to radiate. Also, the current doesn’t slow down when going through the resistor: it’s the same throughout the circuit.

582

Chapter 17

Current and Resistance

APPLYING PHYSICS 17.2

LIGHTBULB FAILURES

Why do lightbulbs fail so often immediately after they’re turned on? Explanation Once the switch is closed, the line voltage is applied across the bulb. As the voltage is applied across the cold filament when the bulb is first turned on, the resistance of the filament is low, the current is high, and a relatively large amount of

QUICK QUIZ 17.7 A voltage V is applied across the ends of a Nichrome heater wire having a cross-sectional area A and length L. The same voltage is applied across the ends of a second Nichrome heater wire having a crosssectional area A and length 2L. Which wire gets hotter? (a) The shorter wire does. (b) The longer wire does. (c) More information is needed.

30 W e

A

f

QUICK QUIZ 17.8 For the two resistors shown in Figure 17.10, rank the currents at points a through f from largest to smallest. (a) Ia  Ib Ie  If Ic  Id (b) Ia  Ib Ic  Id Ie  If (c) Ie  If Ic  Id Ia  Ib

60 W c

a

B

V

power is delivered to the bulb. This current spike at the beginning of operation is the reason lightbulbs often fail immediately after they are turned on. As the filament warms, its resistance rises and the current decreases. As a result, the power delivered to the bulb decreases and the bulb is less likely to burn out.

d

b

FIGURE 17.10 (Quick Quiz 17.8)

QUICK QUIZ 17.9 Two resistors, A and B, are connected in a series circuit with a battery. The resistance of A is twice that of B. Which resistor dissipates more power? (a) Resistor A does. (b) Resistor B does. (c) More information is needed. QUICK QUIZ 17.10 The diameter of wire A is greater than the diameter of wire B, but their lengths and resistivities are identical. For a given voltage difference across the ends, what is the relationship between
A and
B, the dissipated power for wires A and B, respectively? (a)
A 
B (b)
A <
B (c)
A >
B

EXAMPLE 17.5 Goal

The Cost of Lighting Up Your Life

Apply the electric power concept and calculate the cost of power usage using kilowatt-hours.

Problem A circuit provides a maximum current of 20.0 A at an operating voltage of 1.20  102 V. (a) How many 75 W bulbs can operate with this voltage source? (b) At $0.120 per kilowatt-hour, how much does it cost to operate these bulbs for 8.00 h? Strategy Find the necessary power with
 I V then divide by 75.0 W per bulb to get the total number of bulbs. To find the cost, convert power to kilowatts and multiply by the number of hours, then multiply by the cost per kilowatt-hour. Solution (a) Find the number of bulbs that can be lighted. Substitute into Equation 17.8 to get the total power:


total  I V  (20.0 A)(1.20  102 V)  2.40  103 W

Divide the total power by the power per bulb to get the number of bulbs:

Number of bulbs 5


total 2.40 3 103 W 5 5 32.0
bulb 75.0 W

17.6

(b) Calculate the cost of this electricity for an 8.00-h day.

Electrical Energy and Power

Energy 5
t 5 1 2.40 3 103 W 2 a

Find the energy in kilowatt-hours:

583

1.00 kW b 1 8.00 h 2 1.00 3 103 W

5 19.2 kWh Cost  (19.2 kWh)($0.12/kWh)  $2.30

Multiply the energy by the cost per kilowatt-hour:

Remarks This amount of energy might correspond to what a small office uses in a working day, taking into account all power requirements (not just lighting). In general, resistive devices can have variable power output, depending on how the circuit is wired. Here, power outputs were specified, so such considerations were unnecessary. QUESTION 17.5 Considering how hot the parts of an incandescent light bulb get during operation, guess what fraction of the energy emitted by an incandescent lightbulb is in the form of visible light. (a) 10% (b) 50% (c) 80% EXERCISE 17.5 (a) How many Christmas tree lights drawing 5.00 W of power each could be run on a circuit operating at 1.20  102 V and providing 15.0 A of current? (b) Find the cost to operate one such string 24.0 h per day for the Christmas season (two weeks), using the rate $0.12/kWh. Answers (a) 3.60  102 bulbs (b) $72.60

EXAMPLE 17.6

The Power Converted by an Electric Heater

Goal Calculate an electrical power output and link to its effect on the environment through the first law of thermodynamics. Problem An electric heater is operated by applying a potential difference of 50.0 V to a Nichrome wire of total resistance 8.00 . (a) Find the current carried by the wire and the power rating of the heater. (b) Using this heater, how long would it take to heat 2.50  103 moles of diatomic gas (e.g., a mixture of oxygen and nitrogen, or air) from a chilly 10.0°C to 25.0°C? Take the molar specific heat at constant volume of air to be 52R. Strategy For part (a), find the current with Ohm’s law and substitute into the expression for power. Part (b) is an isovolumetric process, so the thermal energy provided by the heater all goes into the change in internal energy, U. Calculate this quantity using the first law of thermodynamics and divide by the power to get the time. Solution (a) Compute the current and power output. DV 50.0 V 5 5 6.25 A R 8.00 V

Apply Ohm’s law to get the current:

I5

Substitute into Equation 17.9 to find the power:


 I 2R  (6.25 A)2(8.00 )  313 W

(b) How long does it take to heat the gas? Calculate the thermal energy transfer from the first law. Note that W  0 because the volume doesn’t change.

Q 5 DU 5 nC v DT

5 1 2.50 3 103 mol 2 1 52 # 8.31 J/mol # K 2 1 298 K 2 283 K 2

5 7.79 3 105 J Divide the thermal energy by the power to get the time:

t5

Q 7.79 3 105 J 5 2.49 3 103 s 5
313 W

584

Chapter 17

Current and Resistance

Remarks The number of moles of gas given here is approximately what would be found in a bedroom. Warming the air with this space heater requires only about 40 minutes. The calculation, however, doesn’t take into account conduction losses. Recall that a 20-cm-thick concrete wall, as calculated in Chapter 11, permitted the loss of more than 2 megajoules an hour by conduction! QUESTION 17.6 If the heater wire is replaced by a wire with lower resistance, is the time required to heat the gas (a) unchanged, (b) increased, or (c) decreased? EXERCISE 17.6 A hot-water heater is rated at 4.50  103 W and operates at 2.40  102 V. (a) Find the resistance in the heating element and the current. (b) How long does it take to heat 125 L of water from 20.0°C to 50.0°C, neglecting conduction and other losses? Answers (a) 12.8 , 18.8 A

R () 0.15

17.7

0.10

0.05 Tc 0.00

4.0

(b) 3.49  103 s

4.2 T (K)

4.4

FIGURE 17.11 Resistance versus temperature for a sample of mercury (Hg). The graph follows that of a normal metal above the critical temperature Tc . The resistance drops to zero at the critical temperature, which is 4.2 K for mercury, and remains at zero for lower temperatures.

TABLE 17.2 Critical Temperatures for Various Superconductors Material

Tc (K)

Zn Al Sn Hg Pb Nb Nb3Sn Nb3Ge Y Ba2Cu3O7 Bi–Sr–Ca–Cu–O Tl–Ba–Ca–Cu–O HgBa2Ca2Cu3O8

0.88 1.19 3.72 4.15 7.18 9.46 18.05 23.2 90 105 125 134

SUPERCONDUCTORS

There is a class of metals and compounds with resistances that fall virtually to zero below a certain temperature Tc called the critical temperature. These materials are known as superconductors. The resistance vs. temperature graph for a superconductor follows that of a normal metal at temperatures above Tc (Fig. 17.11). When the temperature is at or below Tc , however, the resistance suddenly drops to zero. This phenomenon was discovered in 1911 by Dutch physicist H. Kamerlingh Onnes as he and a graduate student worked with mercury, which is a superconductor below 4.1 K. Recent measurements have shown that the resistivities of superconductors below Tc are less than 4  10
25   m, around 1017 times smaller than the resistivity of copper and in practice considered to be zero. Today thousands of superconductors are known, including such common metals as aluminum, tin, lead, zinc, and indium. Table 17.2 lists the critical temperatures of several superconductors. The value of Tc is sensitive to chemical composition, pressure, and crystalline structure. Interestingly, copper, silver, and gold, which are excellent conductors, don’t exhibit superconductivity. A truly remarkable feature of superconductors is that once a current is set up in them, it persists without any applied voltage (because R  0). In fact, steady currents in superconducting loops have been observed to persist for years with no apparent decay! An important development in physics that created much excitement in the scientific community was the discovery of high-temperature copper-oxide-based superconductors. The excitement began with a 1986 publication by J. Georg Bednorz and K. Alex Müller, scientists at the IBM Zurich Research Laboratory in Switzerland, in which they reported evidence for superconductivity at a temperature near 30 K in an oxide of barium, lanthanum, and copper. Bednorz and Müller were awarded the Nobel Prize in Physics in 1987 for their important discovery. The discovery was remarkable because the critical temperature was significantly higher than that of any previously known superconductor. Shortly thereafter a new family of compounds was investigated, and research activity in the field of superconductivity proceeded vigorously. In early 1987 groups at the University of Alabama at Huntsville and the University of Houston announced the discovery of superconductivity at about 92 K in an oxide of yttrium, barium, and copper (YBa2Cu3O7), shown as the gray disk in Figure 17.12. Late in 1987, teams of scientists from Japan and the United States reported superconductivity at 105 K in an oxide of bismuth, strontium, calcium, and copper. More recently, scientists have reported superconductivity at temperatures as high as 150 K in an oxide containing mercury. The search for novel superconducting materials continues, with the hope of someday obtaining a room-temperature superconducting material. This research is important both for scientific reasons and for practical applications.

Electrical Activity in the Heart

An important and useful application is the construction of superconducting magnets in which the magnetic field intensities are about ten times greater than those of the best normal electromagnets. Such magnets are being considered as a means of storing energy. The idea of using superconducting power lines to transmit power efficiently is also receiving serious consideration. Modern superconducting electronic devices consisting of two thin-film superconductors separated by a thin insulator have been constructed. Among these devices are magnetometers (magnetic-field measuring devices) and various microwave devices.

17.8 ELECTRICAL ACTIVITY IN THE HEART Electrocardiograms Every action involving the body’s muscles is initiated by electrical activity. The voltages produced by muscular action in the heart are particularly important to physicians. Voltage pulses cause the heart to beat, and the waves of electrical excitation that sweep across the heart associated with the heartbeat are conducted through the body via the body fluids. These voltage pulses are large enough to be detected by suitable monitoring equipment attached to the skin. A sensitive voltmeter making good electrical contact with the skin by means of contacts attached with conducting paste can be used to measure heart pulses, which are typically of the order of 1 mV at the surface of the body. The voltage pulses can be recorded on an instrument called an electrocardiograph, and the pattern recorded by this instrument is called an electrocardiogram (EKG). To understand the information contained in an EKG pattern, it is necessary first to describe the underlying principles concerning electrical activity in the heart. The right atrium of the heart contains a specialized set of muscle fibers called the SA (sinoatrial) node that initiates the heartbeat (Fig. 17.13). Electric impulses that originate in these fibers gradually spread from cell to cell throughout the right and left atrial muscles, causing them to contract. The pulse that passes through the muscle cells is often called a depolarization wave because of its effect on individual cells. If an individual muscle cell were examined in its resting state, a double-layer electric charge distribution would be found on its surface, as shown in Figure 17.14a. The impulse generated by the SA node momentarily and locally allows positive charge on the outside of the cell to flow in and neutralize the negative charge on the inside layer. This effect changes the cell’s charge distribution to that shown in Figure 17.14b. Once the depolarization wave has passed through an individual heart muscle cell, the cell recovers the resting-state charge distribution (positive out, negative in) shown in Figure 17.14a in about 250 ms. When the impulse reaches the atrioventricular (AV) node (Fig. 17.13), the muscles of the atria begin to relax, and the pulse is directed to the ventricular muscles by the AV node. The muscles of the ventricles contract as the depolarization wave spreads through the ventricles along a group of fibers called the Purkinje fibers. The ventricles then relax after the pulse has passed through. At this point, the SA node is again triggered and the cycle is repeated.

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Courtesy of IBM Research Laboratory

17.8

FIGURE 17.12 A small permanent magnet floats freely above a ceramic disk made of the superconductor Y Ba 2Cu3O7, cooled by liquid nitrogen at 77 K. The superconductor has zero electric resistance at temperatures below 92 K and expels any applied magnetic field.

APPLICATION Electrocardiograms

Sinoatrial (SA) node

Purkinje fibers

LA RA LV RV

Atrioventricular (AV) node FIGURE 17.13 The electrical conduction system of the human heart. (RA: right atrium; LA: left atrium; RV: right ventricle; LV: left ventricle.)

FIGURE 17.14 (a) Charge distribution of a muscle cell in the atrium before a depolarization wave has passed through the cell. (b) Charge distribution as the wave passes.

Depolarization wave front (a)

(b)

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Voltage (mV)

1.0

Current and Resistance

R

0.5 T

P 0 Q

– 0.5

S 0

0.2

0.4

0.6

Time (s) FIGURE 17.15 An EKG response for a normal heart.

APPLICATION Cardiac Pacemakers

A sketch of the electrical activity registered on an EKG for one beat of a normal heart is shown in Figure 17.15. The pulse indicated by P occurs just before the atria begin to contract. The QRS pulse occurs in the ventricles just before they contract, and the T pulse occurs when the cells in the ventricles begin to recover. EKGs for an abnormal heart are shown in Figure 17.16. The QRS portion of the pattern shown in Figure 17.16a is wider than normal, indicating that the patient may have an enlarged heart. (Why?) Figure 17.16b indicates that there is no constant relationship between the P pulse and the QRS pulse. This suggests a blockage in the electrical conduction path between the SA and AV nodes which results in the atria and ventricles beating independently and inefficient heart pumping. Finally, Figure 17.16c shows a situation in which there is no P pulse and an irregular spacing between the QRS pulses. This is symptomatic of irregular atrial contraction, which is called fibrillation. In this condition, the atrial and ventricular contractions are irregular. As noted previously, the sinoatrial node directs the heart to beat at the appropriate rate, usually about 72 beats per minute. Disease or the aging process, however, can damage the heart and slow its beating, and a medical assist may be necessary in the form of a cardiac pacemaker attached to the heart. This matchbox-sized electrical device implanted under the skin has a lead that is connected to the wall of the right ventricle. Pulses from this lead stimulate the heart to maintain its proper rhythm. In general, a pacemaker is designed to produce pulses at a rate of about 60 per minute, slightly slower than the normal number of beats per minute, but sufficient to maintain life. The circuitry consists of a capacitor charging up to a certain voltage from a lithium battery and then discharging. The design of the circuit is such that if the heart is beating normally, the capacitor is not allowed to charge completely and send pulses to the heart.

An Emergency Room in Your Chest APPLICATION Implanted Cardioverter Defibrillators

In June 2001 an operation on Vice President Dick Cheney focused attention on the progress in treating heart problems with tiny implanted electrical devices. Aptly called “an emergency room in your chest” by Cheney’s attending physician, devices called Implanted Cardioverter Defibrillators (ICDs) can monitor, record, and logically process heart signals and then supply different corrective signals to hearts beating too slowly, too rapidly, or irregularly. ICDs can even monitor and send signals to the atria and ventricles independently! Figure 17.17a shows a sketch of an ICD with conducting leads that are implanted in the heart. Figure 17.17b shows an actual titanium-encapsulated dual-chamber ICD. The latest ICDs are sophisticated devices capable of a number of functions: 1. monitoring both atrial and ventricular chambers to differentiate between atrial and potentially fatal ventricular arrhythmias, which require prompt regulation

FIGURE 17.16

Abnormal EKGs.

R

R T

P Q

R TP

P

P

S

(b) R

R

(c)

R

P Q

Q

(a) R

P

R

T

17.8

Electrical Activity in the Heart

FIGURE 17.17 (a) A dual-chamber ICD with leads in the heart. One lead monitors and stimulates the right atrium, and the other monitors and stimulates the right ventricle. (b) Medtronic Dual Chamber ICD.

(a)

Courtesy of Medtronic, Inc.

Dual-chamber ICD

Blowup of defibrillator/ monitor lead

(b)

2. storing about a half hour of heart signals that can easily be read out by a physician 3. being easily reprogrammed with an external magnetic wand 4. performing complicated signal analysis and comparison 5. supplying either 0.25- to 10-V repetitive pacing signals to speed up or slow down a malfunctioning heart, or a high-voltage pulse of about 800 V to halt the potentially fatal condition of ventricular fibrillation, in which the heart quivers rapidly rather than beats (people who have experienced such a high-voltage jolt say that it feels like a kick or a bomb going off in the chest) 6. automatically adjusting the number of pacing pulses per minute to match the patient’s activity ICDs are powered by lithium batteries and have implanted lifetimes of 4 to 6 years. Some basic properties of these adjustable ICDs are given in Table 17.3. In the table tachycardia means “rapid heartbeat” and bradycardia means “slow heartbeat.” A key factor in developing tiny electrical implants that serve as defibrillators is the development of capacitors with relatively large capacitance (125 mf) and small physical size. TABLE 17.3 Properties of Implanted Cardioverter Defibrillators Physical Specifications Mass (g) Size (cm)

85 7.3  6.2  1.3 (about five stacked silver dollars)

Antitachycardia Pacing Number of bursts Burst cycle length (ms) Number of pulses per burst Pulse amplitude (V) Pulse width (ms)

ICD delivers a burst of critically timed low-energy pulses 1–15 200–552 2–20 7.5 or 10 1.0 or 1.9

High-Voltage Defibrillation Pulse energy (J) Pulse amplitude (V)

37 stored/33 delivered 801

Bradycardia Pacing Base frequency (beats/minute) Pulse amplitude (V) Pulse width (ms)

A dual-chamber ICD can steadily deliver repetitive pulses to both the atrium and the ventricle 40–100 0.25–7.5 0.05, 0.1–1.5, 1.9

Note: For more information, go to www.photonicd.com/specs.html.

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SUMMARY 17.1 Electric Current The average electric current I in a conductor is defined as I av ;

DQ

where Q is the charge that passes through a cross section of the conductor in time t. The SI unit of current is the ampere (A); 1 A  1 C/s. By convention, the direction of current is the direction of flow of positive charge. The instantaneous current I is the limit of the average current as the time interval goes to zero: Dt S 0

17.2

Dt S 0

DQ Dt

A Microscopic View: Current and Drift Speed

I 5 nqv dA

[17.2]

where n is the number of mobile charge carriers per unit volume, q is the charge on each carrier, vd is the drift speed of the charges, and A is the cross-sectional area of the conductor.

Resistance, Resistivity, and Ohm’s Law

The resistance R of a conductor is defined as the ratio of the potential difference across the conductor to the current in it: DV [17.3] R ; I The SI units of resistance are volts per ampere, or ohms (); 1   1 V/A. Ohm’s law describes many conductors for which the applied voltage is directly proportional to the current it causes. The proportionality constant is the resistance: V  IR

[17.4]

, A

[17.5]

where r is an intrinsic property of the conductor called the electrical resistivity. The SI unit of resistivity is the ohmmeter (  m).

17.5

Temperature Variation of Resistance

Over a limited temperature range, the resistivity of a conductor varies with temperature according to the expression

[17.1b]

The current in a conductor is related to the motion of the charge carriers by

17.4

R5r

[17.1a]

Dt

I 5 lim I av 5 lim

If a conductor has length  and cross-sectional area A, its resistance is

r  r0[1  (T
T0)]

[17.6]

where  is the temperature coefficient of resistivity and r0 is the resistivity at some reference temperature T0 (usually taken to be 20°C). The resistance of a conductor varies with temperature according to the expression R  R 0[1  (T
T0)]

17.6

[17.7]

Electrical Energy and Power

If a potential difference V is maintained across an electrical device, the power, or rate at which energy is supplied to the device, is
 I V

[17.8]

Because the potential difference across a resistor is V  IR, the power delivered to a resistor can be expressed as
5 I 2R 5

DV 2 R

[17.9]

A kilowatt-hour is the amount of energy converted or consumed in one hour by a device supplied with power at the rate of 1 kW. It is equivalent to 1 kWh  3.60  106 J

[17.10]

FOR ADDITIONAL STUDENT RESOURCES, GO TO W W W.SERWAYPHYSICS.COM

MULTIPLE-CHOICE QUESTIONS 1. A wire carries a current of 1.6 A. How many electrons per second pass a given point in the wire? Choose the best estimate. (a) 1017 (b) 1018 (c) 1019 (d) 1020 (e) 1021

R, what is the resistance of wire B? (a) 4R (b) 2R (c) R (d) R/2 (e) R/4

2. Wire A has the same length and twice the radius of wire B. Both wires are made of the same material and carry the same current. Which of the following equations is true concerning the drift velocities vA and vB of electrons in the wires? (a) vA  2vB (b) vA  vB (c) vA  vB /2 (d) vA  4vB (e) vA  vB /4

4. Three wires are made of copper having circular cross sections. Wire 1 has a length L and radius r. Wire 2 has a length L and radius 2r. Wire 3 has a length 2L and radius 3r. Which wire has the smallest resistance? (a) wire 1 (b) wire 2 (c) wire 3 (d) All three wires have the same resistance. (e) Not enough information is given to answer the question.

3. Wire B has the same resistivity, twice the length, and twice the radius of wire A. If wire A has a resistance

5. Which of the following combinations of units has units of energy? (a) C/s (b) kWh (c) kW (d) A  h (e) W/s

Conceptual Questions

6. The current versus voltage behavior of a certain electrical device is shown in Figure MCQ17.6. When the potential difference across the device is 2 V, what is the resistance? (a) 1  (b) 34 V (c) 43 V (d) undefined (e) none of these

Current (A)

9. A color television set draws about 2.5 A when connected to a 120 V source. What is the cost (with electrical energy at 8 cents/kWh) of running the set for 8.0 hours? (a) 2.0 cents (b) 4.0 cents (c) 19 cents (d) 40 cents (e) 62 cents 10. A potential difference of 1.0 V is maintained across a 10.0- resistor for a period of 20 s. What total charge passes through the wire in this time interval? (a) 200 C (b) 20 C (c) 2 C (d) 0.005 C (e) 0.05 C

3 2 1 0

589

0

1

2

3

4

Voltage (V) FIGURE MCQ17.6 Multiple-Choice Questions 6 and 7

7. The current vs. voltage behavior of a certain electrical device is shown in Figure MCQ17.6. When the potential difference across the device is 3 V, what is the resistance? (a) 0.83  (b) 1.2  (c) 3.0  (d) 1.33  (e) 2.3  8. A metal wire has a resistance of 10.00  at a temperature of 20.0°C. If the same wire has a resistance of 10.55  at 90.0°C, what is the resistance of this same wire when its temperature is
20°C? (a) 0.700  (b) 9.69  (c) 10.3  (d) 13.8  (e) 6.59 

11. Three resistors, A, B, and C, are connected in parallel and attached to a battery, with the resistance of A being the smallest and the resistance of C the greatest. Which resistor carries the highest current? (a) A (b) B (c) C (d) All wires carry the same current. (e) More information is needed to answer the question. 12. Three resistors, A, B, and C, are connected in series in a closed loop with a battery, with the resistance of A being the smallest and the resistance of C the greatest. Across which resistor is the voltage drop the greatest? (a) A (b) B (c) C (d) The voltage drops are the same for each. (e) More information is needed to answer the question. 13. The power delivered to a resistor is 4.0 W when a certain voltage is applied across it. How much power is delivered to the resistor when the voltage is doubled? (a) 2.0 W (b) 4.0 W (c) 8.0 W (d) 16 W (e) 32 W

CONCEPTUAL QUESTIONS 1. Car batteries are often rated in ampere-hours. Does this unit designate the amount of current, power, energy, or charge that can be drawn from the battery?

7. When the voltage across a certain conductor is doubled, the current is observed to triple. What can you conclude about the conductor?

2. We have seen that an electric field must exist inside a conductor that carries a current. How is that possible in view of the fact that in electrostatics we concluded that the electric field must be zero inside a conductor?

8. There is an old admonition given to experimenters to “keep one hand in the pocket” when working around high voltages. Why is this warning a good idea?

3. Why don’t the free electrons in a metal fall to the bottom of the metal due to gravity? And charges in a conductor are supposed to reside on the surface; why don’t the free electrons all go to the surface? 4. In an analogy between traffic flow and electrical current, what would correspond to the charge Q? What would correspond to the current I ? 5. Newspaper articles often have statements such as “10 000 volts of electricity surged through the victim’s body.” What is wrong with this statement? 6. Two lightbulbs are each connected to a voltage of 120 V. One has a power of 25 W, the other 100 W. Which lightbulb has the higher resistance? Which lightbulb carries more current?

9. When is more power delivered to a lightbulb, immediately after it is turned on and the glow of the filament is increasing or after it has been on for a few seconds and the glow is steady? 10. Some homes have light dimmers that are operated by rotating a knob. What is being changed in the electric circuit when the knob is rotated? 11. What could happen to the drift velocity of the electrons in a wire and to the current in the wire if the electrons could move through it freely without resistance? 12. Use the atomic theory of matter to explain why the resistance of a material should increase as its temperature increases.

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PROBLEMS (b) Starting with the density of iron and the result of part (a), compute the molar density of iron (the number of moles of iron per cubic meter). (c) Calculate the number density of iron atoms using Avogadro’s number. (d) Obtain the number density of conduction electrons given that there are two conduction electrons per iron atom. (e) If the wire carries a current of 30.0 A, calculate the drift speed of conduction electrons.

The Problems for this chapter may be assigned online at WebAssign. 1, 2, 3  straightforward, intermediate, challenging GP  denotes guided problem ecp  denotes enhanced content problem  biomedical application 䡺  denotes full solution available in Student Solutions Manual/ Study Guide

SECTION 17.1 ELECTRIC CURRENT SECTION 17.2 A MICROSCOPIC VIEW: CURRENT AND DRIFT SPEED 1. If a current of 80.0 mA exists in a metal wire, how many electrons flow past a given cross section of the wire in 10.0 min? In what direction do the electrons travel with respect to the current?

SECTION 17.4 RESISTANCE, RESISTIVITY, AND OHM’S LAW 10. An electric heater carries a current of 13.5 A when operating at a voltage of 1.20  102 V. What is the resistance of the heater? 11.

A person notices a mild shock if the current along a path through the thumb and index finger exceeds 80 mA. Compare the maximum possible voltage without shock across the thumb and index finger with a dry-skin resistance of 4.0  105  and a wet-skin resistance of 2 000 .

2. ecp A copper wire has a circular cross section with a radius of 1.25 mm. (a) If the wire carries a current of 3.70 A, find the drift speed of the electrons in the wire. (See Example 17.2 for relevant data on copper.) (b) All other things being equal, what happens to the drift speed in wires made of metal having a larger number of conduction electrons per atom than copper? Explain.

12. Suppose you wish to fabricate a uniform wire out of 1.00 g of copper. If the wire is to have a resistance R  0.500 , and if all the copper is to be used, what will be (a) the length and (b) the diameter of the wire?

3. In the Bohr model of the hydrogen atom, an electron in the lowest energy state moves at a speed of 2.19  106 m/s in a circular path having a radius of 5.29  10
11 m. What is the effective current associated with this orbiting electron?

13. Nichrome wire of cross-sectional radius 0.791 mm is to be used in winding a heating coil. If the coil must carry a current of 9.25 A when a voltage of 1.20  102 V is applied across its ends, find (a) the required resistance of the coil and (b) the length of wire you must use to wind the coil.

4. A proton beam in an accelerator carries a current of 125 mA. If the beam is incident on a target, how many protons strike the target in a period of 23.0 s?

14. Eighteen-gauge wire has a diameter of 1.024 mm. Calculate the resistance of 15 m of 18-gauge copper wire at 20°C.

5. If the current carried by a conductor is doubled, what happens to (a) the charge carrier density? (b) The electron drift velocity?

15. A potential difference of 12 V is found to produce a current of 0.40 A in a 3.2-m length of wire with a uniform radius of 0.40 cm. What is (a) the resistance of the wire? (b) The resistivity of the wire?

6. If 3.25  10
3 kg of gold is deposited on the negative electrode of an electrolytic cell in a period of 2.78 h, what is the current in the cell during that period? Assume the gold ions carry one elementary unit of positive charge.

16. ecp Aluminum and copper wires of equal length are found to have the same resistance. What is the ratio of their radii?

7. A 200-km-long high-voltage transmission line 2.0 cm in diameter carries a steady current of 1 000 A. If the conductor is copper with a free charge density of 8.5  1028 electrons per cubic meter, how many years does it take one electron to travel the full length of the cable?

17. A wire 50.0 m long and 2.00 mm in diameter is connected to a source with a potential difference of 9.11 V, and the current is found to be 36.0 A. Assume a temperature of 20°C and, using Table 17.1, identify the metal out of which the wire is made.

8. An aluminum wire carrying a current of 5.0 A has a crosssectional area of 4.0  10
6 m2. Find the drift speed of the electrons in the wire. The density of aluminum is 2.7 g/cm3. (Assume three electrons are supplied by each atom.)

18. A rectangular block of copper has sides of length 10 cm, 20 cm, and 40 cm. If the block is connected to a 6.0-V source across two of its opposite faces, what are (a) the maximum current and (b) the minimum current the block can carry?

GP An iron wire has a cross-sectional area of 5.00  10
6 m2. Carry out steps (a) through (e) to compute the drift speed of the conduction electrons in the wire. (a) How many kilograms are there in 1 mole of iron?

19. A wire of initial length L 0 and radius r 0 has a measured resistance of 1.0 . The wire is drawn under tensile stress to a new uniform radius of r  0.25r 0. What is the new resistance of the wire?

9.

Problems

20. ecp The human body can exhibit a wide range of resistances to current depending on the path of the current, contact area, and sweatiness of the skin. Suppose the resistance across the chest from the left hand to the right hand is 1.0  10 6 . (a) How much voltage is required to cause possible heart fibrillation in a man, which corresponds to 500 mA of direct current? (b) Why should rubber-soled shoes and rubber gloves be worn when working around electricity? 21. ecp Starting from Ohm’s law, show that E  Jr, where E is the magnitude of the electric field (assumed constant) and J  I/A is called the current density. The result is in fact true in general.

SECTION 17.5 TEMPERATURE VARIATION OF RESISTANCE 22. If a certain silver wire has a resistance of 6.00  at 20.0°C, what resistance will it have at 34.0°C? 23. While taking photographs in Death Valley on a day when the temperature is 58.0°C, Bill Hiker finds that a certain voltage applied to a copper wire produces a current of 1.000 A. Bill then travels to Antarctica and applies the same voltage to the same wire. What current does he register there if the temperature is
88.0°C? Assume no change occurs in the wire’s shape and size. 24. ecp A length of aluminum wire has a resistance of 30.0  at 20.0°C. When the wire is warmed in an oven and reaches thermal equilibrium, the resistance of the wire increases to 46.2  (a) Neglecting thermal expansion, find the temperature of the oven. (b) Qualitatively, how would thermal expansion be expected to affect the answer? 25. A certain lightbulb has a tungsten filament having a resistance of 15  at 20°C and 160  when hot. Assume Equation 17.7 can be used over the large temperature range here. Find the temperature of the filament when it is hot. 26. At what temperature will aluminum have a resistivity that is three times the resistivity of copper at room temperature? 27. At 20°C, the carbon resistor in an electric circuit connected to a 5.0-V battery has a resistance of 2.0  102 . What is the current in the circuit when the temperature of the carbon rises to 80°C? 28. A wire 3.00 m long and 0.450 mm2 in cross-sectional area has a resistance of 41.0  at 20°C. If its resistance increases to 41.4  at 29.0°C, what is the temperature coefficient of resistivity? 29. (a) A 34.5-m length of copper wire at 20.0°C has a radius of 0.25 mm. If a potential difference of 9.0 V is applied across the length of the wire, determine the current in the wire. (b) If the wire is heated to 30.0°C while the 9.0-V potential difference is maintained, what is the resulting current in the wire? 30. A toaster rated at 1 050 W operates on a 120-V household circuit and a 4.00-m length of Nichrome wire as its heat-

591

ing element. The operating temperature of this element is 320°C. What is the cross-sectional area of the wire? 31.

In one form of plethysmograph (a device for measuring volume), a rubber capillary tube with an inside diameter of 1.00 mm is filled with mercury at 20°C. The resistance of the mercury is measured with the aid of electrodes sealed into the ends of the tube. If 100.00 cm of the tube is wound in a spiral around a patient’s upper arm, the blood flow during a heartbeat causes the arm to expand, stretching the tube to a length of 100.04 cm. From this observation, and assuming cylindrical symmetry, you can find the change in volume of the arm, which gives an indication of blood flow. (a) Calculate the resistance of the mercury. (b) Calculate the fractional change in resistance during the heartbeat. (Hint: The fraction by which the cross-sectional area of the mercury thread decreases is the fraction by which the length increases, since the volume of mercury is constant.) Take rHg  9.4  10
7   m.

32. A platinum resistance thermometer has resistances of 200.0  when placed in a 0°C ice bath and 253.8  when immersed in a crucible containing melting potassium. What is the melting point of potassium? (Hint: First determine the resistance of the platinum resistance thermometer at room temperature, 20°C.)

SECTION 17.6 ELECTRICAL ENERGY AND POWER 33. Suppose your waffle iron is rated at 1.00 kW when connected to a 1.20  102 V source. (a) What current does the waffle iron carry? (b) What is its resistance? 34. If electrical energy costs 12 cents, or $0.12, per kilowatthour, how much does it cost to (a) burn a 100-W lightbulb for 24 h? (b) Operate an electric oven for 5.0 h if it carries a current of 20.0 A at 220 V? 35. A certain compact disc player draws a current of 350 mA at 6.0 V. How much power is required to operate the player? 36. A high-voltage transmission line with a resistance of 0.31 /km carries a current of 1 000 A. The line is at a potential of 700 kV at the power station and carries the current to a city located 160 km from the station. (a) What is the power loss due to resistance in the line? (b) What fraction of the transmitted power does this loss represent? 37. The heating element of a coffeemaker operates at 120 V and carries a current of 2.00 A. Assuming the water absorbs all the energy converted by the resistor, calculate how long it takes to heat 0.500 kg of water from room temperature (23.0°C) to the boiling point. 38. The power supplied to a typical black-and-white television set is 90 W when the set is connected to 120 V. (a) How much electrical energy does this set consume in 1 hour? (b) A color television set draws about 2.5 A when connected to 120 V. How much time is required for it to consume the same energy as the black-and-white model consumes in 1 hour?

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Chapter 17

Current and Resistance

39. What is the required resistance of an immersion heater that will increase the temperature of 1.50 kg of water from 10.0°C to 50.0°C in 10.0 min while operating at 120 V? 40. A certain toaster has a heating element made of Nichrome resistance wire. When the toaster is first connected to a 120-V source of potential difference (and the wire is at a temperature of 20.0°C), the initial current is 1.80 A but the current begins to decrease as the resistive element warms up. When the toaster reaches its final operating temperature, the current has dropped to 1.53 A. (a) Find the power the toaster converts when it is at its operating temperature. (b) What is the final temperature of the heating element? 41. A copper cable is designed to carry a current of 300 A with a power loss of 2.00 W/m. What is the required radius of this cable? 42. Batteries are rated in terms of ampere-hours (A  h). For example, a battery that can deliver a current of 3.0 A for 5.0 h is rated at 15 A  h. (a) What is the total energy, in kilowatt-hours, stored in a 12 V battery rated at 55 A  h? (b) At $0.12 per kilowatt-hour, what is the value of the electricity that can be produced by this battery? 43.

The potential difference across a resting neuron is about 75 mV and carries a current of about 0.20 mA. How much power does the neuron release?

44. The cost of electricity varies widely throughout the United States; $0.120/kWh is a typical value. At this unit price, calculate the cost of (a) leaving a 40.0-W porch light on for 2 weeks while you are on vacation, (b) making a piece of dark toast in 3.00 min with a 970-W toaster, and (c) drying a load of clothes in 40.0 min in a 5 200-W dryer. 45. An 11-W energy-efficient fluorescent lamp is designed to produce the same illumination as a conventional 40-W lamp. How much does the energy-efficient lamp save during 100 hours of use? Assume a cost of $0.080/kWh for electrical energy. 46. ecp An office worker uses an immersion heater to warm 250 g of water in a light, covered, insulated cup from 20°C to 100°C in 4.00 minutes. The heater is a Nichrome resistance wire connected to a 120-V power supply. Assume the wire is at 100°C throughout the 4.00-min time interval. Specify a diameter and a length the wire can have. Can it be made from less than 0.5 cm3 of Nichrome? 47. The heating coil of a hot-water heater has a resistance of 20  and operates at 210 V. If electrical energy costs $0.080/kWh, what does it cost to raise the 200 kg of water in the tank from 15°C to 80°C? (See Chapter 11.) 48. ecp A tungsten wire in a vacuum has length 15.0 cm and radius 1.00 mm. A potential difference is applied across it. (a) What is the resistance of the wire at 293 K? (b) Suppose the wire reaches an equilibrium temperature such that it emits 75.0 W in the form of radiation. Neglecting absorption of any radiation from its environment, what is the temperature of the wire? (Note: e  0.320 for tungsten.) (c) What is the resistance of the wire at the temperature found in part (b)? Assume the temperature changes

linearly over this temperature range. (d) What voltage drop is required across the wire? (e) Why are tungsten lightbulbs energetically inefficient as light sources? ADDITIONAL PROBLEMS 49. If a battery is rated at 60.0 A  h, how much total charge can it deliver before it goes “dead”? 50. A car owner forgets to turn off the headlights of his car while it is parked in his garage. If the 12-V battery in his car is rated at 90 A  h and each headlight requires 36 W of power, how long will it take the battery to completely discharge? 51. An electronic device requires a power of 15 W when connected to a 9.0-V battery. How much power is delivered to the device if it is connected to a 6.0-V battery? (Neglect the resistances of the batteries and assume the resistance of the device does not change.) 52. Determine the temperature at which the resistance of an aluminum wire will be twice its value at 20°C. (Assume its coefficient of resistivity remains constant.) 53. A particular wire has a resistivity of 3.0  10
8   m and a cross-sectional area of 4.0  10
6 m2. A length of this wire is to be used as a resistor that will develop 48 W of power when connected across a 20-V battery. What length of wire is required? 54. Birds resting on high-voltage power lines are a common sight. The copper wire on which a bird stands is 2.2 cm in diameter and carries a current of 50 A. If the bird’s feet are 4.0 cm apart, calculate the potential difference across its body. 55. An experiment is conducted to measure the electrical resistivity of Nichrome in the form of wires with different lengths and cross-sectional areas. For one set of measurements, a student uses 30-gauge wire, which has a crosssectional area of 7.30  10
8 m2. The student measures the potential difference across the wire and the current in the wire with a voltmeter and an ammeter, respectively. For each of the measurements given in the following table taken on wires of three different lengths, calculate the resistance of the wires and the corresponding value of the resistivity. L (m)

V (V)

I (A)

0.540 1.028 1.543

5.22 5.82 5.94

0.500 0.276 0.187

R ()

R (  m)

What is the average value of the resistivity, and how does this value compare with the value given in Table 17.1? 56. A carbon wire and a Nichrome wire are connected one after the other. If the combination has a total resistance of 10.0 k at 20°C, what is the resistance of each wire at 20°C so that the resistance of the combination does not change with temperature? 57. You are cooking breakfast for yourself and a friend using a 1 200-W waffle iron and a 500-W coffeepot. Usually, you

Problems

operate these appliances from a 110-V outlet for 0.500 h each day. (a) At 12 cents per kWh, how much do you spend to cook breakfast during a 30.0-day period? (b) You find yourself addicted to waffles and would like to upgrade to a 2 400-W waffle iron that will enable you to cook twice as many waffles during a half-hour period, but you know that the circuit breaker in your kitchen is a 20-A breaker. Can you do the upgrade? 58. The current in a conductor varies in time as shown in Figure P17.58. (a) How many coulombs of charge pass through a cross section of the conductor in the interval from t  0 to t  5.0 s? (b) What constant current would transport the same total charge during the 5.0-s interval as does the actual current?

Current (A)

4 2 0

current of 4.00 A. Is this system adequately protected against overload? 63. A resistor is constructed by forming a material of resistivity 3.5  105   m into the shape of a hollow cylinder of length 4.0 cm and inner and outer radii 0.50 cm and 1.2 cm, respectively. In use, a potential difference is applied between the ends of the cylinder, producing a current parallel to the length of the cylinder. Find the resistance of the cylinder. 64. A 50.0-g sample of a conducting material is all that is available. The resistivity of the material is measured to be 11  10
8   m, and the density is 7.86 g/cm3. The material is to be shaped into a solid cylindrical wire that has a total resistance of 1.5 . (a) What length of wire is required? (b) What must be the diameter of the wire? 65.

6

0

1

2

3

4

5

Time (s) FIGURE P17.58

59. An electric car is designed to run off a bank of 12.0-V batteries with a total energy storage of 2.00  107 J. (a) If the electric motor draws 8.00 kW, what is the current delivered to the motor? (b) If the electric motor draws 8.00 kW as the car moves at a steady speed of 20.0 m/s, how far will the car travel before it is “out of juice”? 60. (a) A 115-g mass of aluminum is formed into a right circular cylinder, shaped so that its diameter equals its height. Calculate the resistance between the top and bottom faces of the cylinder at 20°C. (b) Calculate the resistance between opposite faces if the same mass of aluminum is formed into a cube. 61. A length of metal wire has a radius of 5.00  10
3 m and a resistance of 0.100 . When the potential difference across the wire is 15.0 V, the electron drift speed is found to be 3.17  10
4 m/s. On the basis of these data, calculate the density of free electrons in the wire. 62. In a certain stereo system, each speaker has a resistance of 4.00 . The system is rated at 60.0 W in each channel. Each speaker circuit includes a fuse rated at a maximum

593

An x-ray tube used for cancer therapy operates at 4.0 MV, with a beam current of 25 mA striking the metal target. Nearly all the power in the beam is transferred to a stream of water flowing through holes drilled in the target. What rate of flow, in kilograms per second, is needed if the rise in temperature (T) of the water is not to exceed 50°C?

66. ecp When a straight wire is heated, its resistance changes according to the equation R  R 0[1  (T
T0)] (Eq. 17.7), where  is the temperature coefficient of resistivity. (a) Show that a more precise result, which includes the length and area of a wire change when it is heated, is R5

R 0 3 1 1 a 1 T 2 T0 2 4 3 1 1 ar 1 T 2 T0 2 4 3 1 1 2ar 1 T 2 T0 2 4

where  is the coefficient of linear expansion. (See Chapter 10.) (b) Compare the two results for a 2.00-m-long copper wire of radius 0.100 mm, starting at 20.0°C and heated to 100.0°C. 67. A man wishes to vacuum his car with a canister vacuum cleaner marked 535 W at 120 V. The car is parked far from the building, so he uses an extension cord 15.0 m long to plug the cleaner into a 120-V source. Assume the cleaner has constant resistance. (a) If the resistance of each of the two conductors of the extension cord is 0.900 , what is the actual power delivered to the cleaner? (b) If, instead, the power is to be at least 525 W, what must be the diameter of each of two identical copper conductors in the cord the young man buys? (c) Repeat part (b) if the power is to be at least 532 W. (Suggestion: A symbolic solution can simplify the calculations.)

18 This vision-imparied patient is wearing an artificial vision device. A camera on the lens sends video to a portable computer. The computer then converts the video to signals that stimulate implants in the patient’s visual cortex, giving him enough sight to move around on his own.

Sources of emf

© Najlah Feanny/Corbis

18.1

18.2 Resistors in Series 18.3

Resistors in Parallel

18.4 Kirchhoff’s Rules and Complex DC Circuits 18.5 RC Circuits 18.6 Household Circuits 18.7 Electrical Safety 18.8 Conduction of Electrical Signals by Neurons

DIRECT-CURRENT CIRCUITS Batteries, resistors, and capacitors can be used in various combinations to construct electric circuits, which direct and control the flow of electricity and the energy it conveys. Such circuits make possible all the modern conveniences in a home: electric lights, electric stove tops and ovens, washing machines, and a host of other appliances and tools. Electric circuits are also found in our cars, in tractors that increase farming productivity, and in all types of medical equipment that saves so many lives every day. In this chapter we study and analyze a number of simple direct-current circuits. The analysis is simplified by the use of two rules known as Kirchhoff’s rules, which follow from the principle of conservation of energy and the law of conservation of charge. Most of the circuits are assumed to be in steady state, which means that the currents are constant in magnitude and direction. We close the chapter with a discussion of circuits containing resistors and capacitors, in which current varies with time.

18.1

SOURCES OF EMF

A current is maintained in a closed circuit by a source of emf.1 Among such sources are any devices (for example, batteries and generators) that increase the potential energy of the circulating charges. A source of emf can be thought of as a “charge pump” that forces electrons to move in a direction opposite the electrostatic field inside the source. The emf of a source is the work done per unit charge; hence the SI unit of emf is the volt. Consider the circuit in Active Figure 18.1a consisting of a battery connected to a resistor. We assume the connecting wires have no resistance. If we neglect the internal resistance of the battery, the potential drop across the battery (the terminal voltage) equals the emf of the battery. Because a real battery always has some internal resistance r, however, the terminal voltage is not equal to the emf. The circuit of Active Figure 18.1a can be described schematically by the diagram

e

1The term was originally an abbreviation for electromotive force, but emf is not really a force, so the long form is discouraged.

594

18.2

Resistors in Series

in Active Figure 18.1b. The battery, represented by the dashed rectangle, consists of a source of emf in series with an internal resistance r. Now imagine a positive charge moving through the battery from a to b in the figure. As the charge passes from the negative to the positive terminal of the battery, the potential of the charge increases by . As the charge moves through the resistance r, however, its potential decreases by the amount Ir, where I is the current in the circuit. The terminal voltage of the battery, V  V b
Va , is therefore given by

595

Battery – +

e

e

V 

e
Ir

[18.1]

e

From this expression, we see that is equal to the terminal voltage when the current is zero, called the open-circuit voltage. By inspecting Figure 18.1b, we find that the terminal voltage V must also equal the potential difference across the external resistance R, often called the load resistance; that is, V  IR. Combining this relationship with Equation 18.1, we arrive at

e  IR  Ir

Resistor (a)

e+

r

a –

[18.2]

Solving for the current gives

b

I

I

I5

e

d

c

(b)

The preceding equation shows that the current in this simple circuit depends on both the resistance external to the battery and the internal resistance of the battery. If R is much greater than r, we can neglect r in our analysis (an option we usually select). If we multiply Equation 18.2 by the current I, we get I

R

R1r ACTIVE FIGURE 18.1 (a) A circuit consisting of a resistor connected to the terminals of a battery. (b) A circuit diagram of a source of emf e having internal resistance r connected to an external resistor R.

e  I 2R  I 2r

QUICK QUIZ 18.1 True or False: While discharging, the terminal voltage of a battery can never be greater than the emf of the battery. QUICK QUIZ 18.2 Why does a battery get warm while in use?

© Cengage Learning/George Semple

e

This equation tells us that the total power output I of the source of emf is converted at the rate I 2R at which energy is delivered to the load resistance, plus the rate I 2r at which energy is delivered to the internal resistance. Again, if r R, most of the power delivered by the battery is transferred to the load resistance. Unless otherwise stated, in our examples and end-of-chapter problems we will assume the internal resistance of a battery in a circuit is negligible.

An assortment of batteries.

18.2 RESISTORS IN SERIES When two or more resistors are connected end to end as in Active Figure 18.2, they are said to be in series. The resistors could be simple devices, such as lightbulbs or heating elements. When two resistors R 1 and R 2 are connected to a battery as in Active Figure 18.2 (page 596), the current is the same in the two resistors because any charge that flows through R1 must also flow through R 2 . This is analogous to water flowing through a pipe with two constrictions, corresponding to R 1 and R 2. Whatever volume of water flows in one end in a given time interval must exit the opposite end. Because the potential difference between a and b in Active Figure 18.2b equals IR 1 and the potential difference between b and c equals IR 2, the potential difference between a and c is V  IR 1  IR 2  I(R 1  R 2) Regardless of how many resistors we have in series, the sum of the potential differences across the resistors is equal to the total potential difference across the combination. As we will show later, this result is a consequence of the conservation

TIP 18.1 What’s Constant in a Battery? Equation 18.2 shows that the current in a circuit depends on the resistance of the battery, so a battery can’t be considered a source of constant current. Even the terminal voltage of a battery given by Equation 18.1 can’t be considered constant because the internal resistance can change (due to warming, for example, during the operation of the battery). A battery is, however, a source of constant emf.

596

Chapter 18

Direct-Current Circuits R1

R2

a

R1

R2

b

–

+

a

c

I

I Battery

Req

I

I V

V

+ (a)

c

+

–

– (c)

(b)

ACTIVE FIGURE 18.2 A series connection of two resistors, R 1 and R 2. The currents in the resistors are the same, and the equivalent resistance of the combination is given by R eq  R 1  R 2.

of energy. Active Figure 18.2c shows an equivalent resistor R eq that can replace the two resistors of the original circuit. The equivalent resistor has the same effect on the circuit because it results in the same current in the circuit as the two resistors. Applying Ohm’s law to this equivalent resistor, we have V  IR eq Equating the preceding two expressions, we have IR eq  I(R 1  R 2) or R eq  R 1  R 2 Equivalent resistance of a series combination of resistors R

(series combination)

[18.3]

An extension of the preceding analysis shows that the equivalent resistance of three or more resistors connected in series is R eq  R 1  R 2  R 3    

[18.4]

Therefore, the equivalent resistance of a series combination of resistors is the algebraic sum of the individual resistances and is always greater than any individual resistance. Note that if the filament of one lightbulb in Active Figure 18.2 were to fail, the circuit would no longer be complete (an open-circuit condition would exist) and the second bulb would also go out.

APPLYING PHYSICS 18.1

CHRISTMAS LIGHTS IN SERIES

A new design for Christmas lights allows them to be connected in series. A failed bulb in such a string would result in an open circuit, and all the bulbs would go out. How can the bulbs be redesigned to prevent that from happening?

I

I

Explanation If the string of lights contained the usual kind of bulbs, a failed bulb would be hard to locate. Each bulb would have to be replaced with a good bulb, one by one, until the failed bulb was found. If there happened to be two or more failed bulbs in

Filament I Jumper Glass insulator (a)

(b)

FIGURE 18.3 (Applying Physics 18.1) (a) Schematic diagram of a modern “miniature” holiday lightbulb, with a jumper connection to provide a current path if the filament breaks. When the fi lament is intact, charges flow in the filament. (b) A holiday lightbulb with a broken filament. In this case, charges flow in the jumper connection.

18.2

the string of lights, finding them would be a lengthy and annoying task. Christmas lights use special bulbs that have an insulated loop of wire (a jumper) across the conducting supports to the bulb filaments (Fig. 18.3). If the filament breaks and the bulb fails, the bulb’s resis-

Resistors in Series

597

tance increases dramatically. As a result, most of the applied voltage appears across the loop of wire. This voltage causes the insulation around the loop of wire to burn, causing the metal wire to make electrical contact with the supports. This produces a conducting path through the bulb, so the other bulbs remain lit.

R2

QUICK QUIZ 18.3 In Figure 18.4 the current is measured with the ammeter at the bottom of the circuit. When the switch is opened, does the reading on the ammeter (a) increase, (b) decrease, or (c) not change?

S R1

QUICK QUIZ 18.4 The circuit in Figure 18.4 consists of two resistors, a switch, an ammeter, and a battery. When the switch is closed, power
c is delivered to resistor R 1. When the switch is opened, which of the following statements is true about the power
o delivered to R 1? (a)
o
c (b)
o 
c (c)
o
c

A FIGURE 18.4 (Quick Quizzes 18.3 and 18.4)

EXAMPLE 18.1 Four Resistors in Series Goal

Analyze several resistors connected in series.

2.0  4.0  5.0  7.0 

Problem Four resistors are arranged as shown in Figure 18.5a. Find (a) the equivalent resistance of the circuit and (b) the current in the circuit if the emf of the battery is 6.0 V.

R1

Strategy Because the resistors are connected in series, summing their resistances gives the equivalent resistance. Ohm’s law can then be used to find the current.

(a)

R2

R3

6.0 V

18.0 

R4

(b)

6.0 V

FIGURE 18.5 (Example 18.1) (a) Four resistors connected in series. (b) The equivalent resistance of the circuit in (a).

Solution (a) Find the equivalent resistance of the circuit. Apply Equation 18.4, summing the resistances:

R eq  R 1  R 2  R 3  R 4  2.0   4.0   5.0   7.0   18.0 V

(b) Find the current in the circuit. Apply Ohm’s law to the equivalent resistor in Figure 18.5b, solving for the current:

I5

DV 6.0 V 5 5 R eq 18.0 V

1 3

A

Remarks A common misconception is that the current is “used up” and steadily declines as it progresses through a series of resistors. That would be a violation of conservation of charge. QUESTION 18.1 The internal resistance of the battery is neglected in this example. How would it affect the final current, if taken into account? EXERCISE 18.1 Because the current in the equivalent resistor is 13 A, the current in each resistor of the original circuit must also be 1 3 A. Find the voltage drop across each resistor. Answers

DV2V 5 23 V ; DV4V 5 43 V ; DV5V 5 53 V ; DV7V 5 73 V

598

Chapter 18

Direct-Current Circuits

18.3

RESISTORS IN PARALLEL

Now consider two resistors connected in parallel, as in Active Figure 18.6. In this case the potential differences across the resistors are the same because each is connected directly across the battery terminals. The currents are generally not the same. When charges reach point a (called a junction) in Active Figure 18.6b, the current splits into two parts: I1, flowing through R 1; and I2, flowing through R 2. If R 1 is greater than R 2, then I1 is less than I2. In general, more charge travels through the path with less resistance. Because charge is conserved, the current I that enters point a must equal the total current I1  I2 leaving that point. Mathematically, this is written I  I1  I 2 The potential drop must be the same for the two resistors and must also equal the potential drop across the battery. Ohm’s law applied to each resistor yields I1 5

DV R1

I2 5

DV R2

Ohm’s law applied to the equivalent resistor in Active Figure 18.6c gives I5

DV R eq

When these expressions for the currents are substituted into the equation I  I1  I2 and the V’s are cancelled, we obtain 1 1 1 5 1 R eq R1 R2

(parallel combination)

[18.5]

An extension of this analysis to three or more resistors in parallel produces the following general expression for the equivalent resistance: Equivalent resistance of a parallel combination of resistors R

1 1 1 1 5 1 1 1 ### R eq R1 R2 R3

[18.6]

From this expression, we see that the inverse of the equivalent resistance of two or more resistors connected in parallel is the sum of the inverses of the individual resistances and is always less than the smallest resistance in the group.

ACTIVE FIGURE 18.6 (a) A parallel connection of two lightbulbs with resistances R 1 and R 2. (b) Circuit diagram for the tworesistor circuit. The potential differences across R 1 and R 2 are the same. (c) The equivalent resistance of the combination is given by the reciprocal relationship 1/R eq  1/R 1  1/R 2.

R1

R2

I1

1

R1

R eq

= 1 + 1

R1

R2

a

–

+

∆V 1 = ∆V 2 = ∆V

b I2

I

I Battery + (a)

∆V –

(b)

+

∆V – (c)

R2

18.3

Resistors in Parallel S

QUICK QUIZ 18.5 In Figure 18.7 the current is measured with the ammeter on the right side of the circuit diagram. When the switch is closed, does the reading on the ammeter (a) increase, (b) decrease, or (c) remain the same?

599

R2

R1 A

QUICK QUIZ 18.6 When the switch is open in Figure 18.7, power
o is delivered to the resistor R 1. When the switch is closed, which of the following is true about the power
c delivered to R 1? (Neglect the internal resistance of the battery.) (a)
c
o (b)
c 
o (c)
c
o

FIGURE 18.7 and 18.6)

(Quick Quizzes 18.5

EXAMPLE 18.2 Three Resistors in Parallel Goal

Analyze a circuit having resistors connected in parallel.

I a

Problem Three resistors are connected in parallel as in Figure 18.8. A potential difference of 18 V is maintained between points a and b. (a) Find the current in each resistor. (b) Calculate the power delivered to each resistor and the total power. (c) Find the equivalent resistance of the circuit. (d) Find the total power delivered to the equivalent resistance. Strategy To get the current in each resistor we can use Ohm’s law and the fact that the voltage drops across parallel resistors are all the same. The rest of the problem just requires substitution into the equation for power delivered to a resistor,
 I 2R, and the reciprocal-sum law for parallel resistors.

I1

18 V

3.0 

I2

6.0 

FIGURE 18.8 (Example 18.2) Three resistors connected in parallel. The voltage across each resistor is 18 V.

I1 5

DV 18 V 5 5 6.0 A R1 3.0 V

I2 5

DV 18 V 5 5 3.0 A R2 6.0 V

I3 5

DV 18 V 5 5 2.0 A R3 9.0 V

(b) Calculate the power delivered to each resistor and the total power. Apply
 I 2R to each resistor, substituting the results from part (a):

3 :

Sum to get the total power:


1  I12R 1  (6.0 A)2(3.0 )  110 W

6 :
2  I22R 2  (3.0 A)2(6.0 )  54 W 9 :


3  I32R 3  (2.0 A)2(9.0 )  36 W


tot  110 W  54 W  36 W  2.0 3 102 W

(c) Find the equivalent resistance of the circuit. Apply the reciprocal-sum rule, Equation 18.6:

1 1 1 1 5 1 1 R eq R1 R2 R3 1 1 1 1 11 5 1 1 5 R eq 3.0 V 6.0 V 9.0 V 18 V R eq 5

18 V 5 1.6 V 11

9.0 

b

Solution (a) Find the current in each resistor. Apply Ohm’s law, solved for the current I delivered by the battery to find the current in each resistor:

I3

600

Chapter 18

Direct-Current Circuits

(d) Compute the power dissipated by the equivalent resistance. Use the alternate power equation:


5

1 18 V 2 2 1 DV 2 2 5 5 2.0 3 102 W R eq 1.6 V

Remarks There’s something important to notice in part (a): the smallest 3.0  resistor carries the largest current, whereas the other, larger resistors of 6.0  and 9.0  carry smaller currents. The largest current is always found in the path of least resistance. In part (b) the power could also be found with
 (V )2/R. Note that
1  108 W, but is rounded to 110 W because there are only two significant figures. Finally, notice that the total power dissipated in the equivalent resistor is the same as the sum of the power dissipated in the individual resistors, as it should be.

TIP 18.2 Don’t Forget to Flip It! The most common mistake in calculating the equivalent resistance for resistors in parallel is to forget to invert the answer after summing the reciprocals. Don’t forget to flip it!

QUESTION 18.2 If a fourth resistor were added in parallel to the other three, how would the equivalent resistance change? (a) It would be larger. (b) It would be smaller. (c) More information is required to determine the effect. EXERCISE 18.2 Suppose the resistances in the example are 1.0 , 2.0 , and 3.0 , respectively, and a new voltage source is provided. If the current measured in the 3.0- resistor is 2.0 A, find (a) the potential difference provided by the new battery and the currents in each of the remaining resistors, (b) the power delivered to each resistor and the total power, (c) the equivalent resistance, and (d) the total current and the power dissipated by the equivalent resistor. Answers (a)
eq  66 W

e  6.0 V, I1  6.0 A, I2  3.0 A (b)
1  36 W,
2  18 W,
3  12 W,
tot  66 W (c) 116 V (d) I  11

A,

QUICK QUIZ 18.7 Suppose you have three identical lightbulbs, some wire, and a battery. You connect one lightbulb to the battery and take note of its brightness. You add a second lightbulb, connecting it in parallel with the previous lightbulbs, and again take note of the brightness. Repeat the process with the third lightbulb, connecting it in parallel with the other two. As the lightbulbs are added, what happens to (a) the brightness of the lightbulbs? (b) The individual currents in the lightbulbs? (c) The power delivered by the battery? (d) The lifetime of the battery? (Neglect the battery’s internal resistance.) QUICK QUIZ 18.8 If the lightbulbs in Quick Quiz 18.7 are connected one by one in series instead of in parallel, what happens to (a) the brightness of the lightbulbs? (b) The individual currents in the lightbulbs? (c) The power delivered by the battery? (d) The lifetime of the battery? (Again, neglect the battery’s internal resistance.)

APPLICATION Circuit Breakers

Household circuits are always wired so that the electrical devices are connected in parallel, as in Active Figure 18.6a. In this way each device operates independently of the others so that if one is switched off, the others remain on. For example, if one of the lightbulbs in Active Figure 18.6 were removed from its socket, the other would continue to operate. Equally important is that each device operates at the same voltage. If the devices were connected in series, the voltage across any one device would depend on how many devices were in the combination and on their individual resistances. In many household circuits, circuit breakers are used in series with other circuit elements for safety purposes. A circuit breaker is designed to switch off and open the circuit at some maximum value of the current (typically 15 A or 20 A) that depends on the nature of the circuit. If a circuit breaker were not used, excessive currents caused by operating several devices simultaneously could result in excessive wire temperatures, perhaps causing a fire. In older home construction, fuses

18.3

Resistors in Parallel

601

were used in place of circuit breakers. When the current in a circuit exceeded some value, the conductor in a fuse melted and opened the circuit. The disadvantage of fuses is that they are destroyed in the process of opening the circuit, whereas circuit breakers can be reset.

APPLYING PHYSICS 18.2

LIGHTBULB COMBINATIONS

Compare the brightness of the four identical lightbulbs shown in Figure 18.9. What happens if bulb A fails and so cannot conduct current? What if C fails? What if D fails?

A

Explanation Bulbs A and B are connected in series across the emf of the battery, whereas bulb C is connected by itself across the battery. This means the voltage drop across C has the same magnitude as the battery emf, whereas this same emf is split between bulbs A and B. As a result, bulb C will glow more brightly than either of bulbs A and B, which will glow equally brightly. Bulb D has a wire connected across it—a short circuit—so the potential difference across bulb D is zero and it doesn’t glow. If bulb A fails, B goes out, but C stays lit. If C fails, there is no effect on the other bulbs. If D fails, the event is undetectable because D was not glowing initially.

C

APPLYING PHYSICS 18.3

FIGURE 18.9 18.2)

B

D

(Applying Physics

THREE-WAY LIGHTBULBS

Figure 18.10 illustrates how a three-way lightbulb is constructed to provide three levels of light intensity. The socket of the lamp is equipped with a three-way switch for selecting different light intensities. The bulb contains two filaments. Why are the filaments connected in parallel? Explain how the two filaments are used to provide the three different light intensities. Explanation If the filaments were connected in series and one of them were to fail, there would be no current in the bulb and the bulb would not glow, regardless of the position of the switch. When the filaments are connected in parallel and one of them (say, the 75-W filament) fails, however, the bulb will still operate in one of the switch positions because there is current in the other (100-W) filament. The three light intensities are made possible by selecting one of three values of filament resistance, using a single value of 120 V for the applied voltage. The 75-W filament offers one value of resistance, the 100-W filament offers a second value, and the third resistance is obtained by combining the two filaments in parallel. When switch S1 is closed and switch S2 is opened, only the 75-W filament carries current. When switch S1 is opened and switch S2 is closed, only the 100-W filament carries current. When both switches are closed, both filaments carry current and a total illumination corresponding to 175 W is obtained.

PROBLEM -SOLVING STRATEGY SIMPLIFYING CIRCUITS WITH RESISTORS

1. Combine all resistors in series by summing the individual resistances and draw the new, simplified circuit diagram. Useful facts: R eq  R 1  R 2  R 3     The current in each resistor is the same.

100-W filament 75-W filament

S1 S2 FIGURE 18.10 18.3)

120 V

(Applying Physics

602

Chapter 18

Direct-Current Circuits

2. Combine all resistors in parallel by summing the reciprocals of the resistances and then taking the reciprocal of the result. Draw the new, simplified circuit diagram. 1 1 1 1 5 1 1 1 ### R eq R1 R2 R3 The potential difference across each resistor is the same. 3. Repeat the first two steps as necessary, until no further combinations can be made. If there is only a single battery in the circuit, the result will usually be a single equivalent resistor in series with the battery. 4. Use Ohm’s law, V  IR, to determine the current in the equivalent resistor. Then work backwards through the diagrams, applying the useful facts listed in step 1 or step 2 to find the currents in the other resistors. (In more complex circuits, Kirchhoff’s rules will be needed, as described in the next section). Useful facts:

EXAMPLE 18.3 Equivalent Resistance Goal Solve a problem involving both series and parallel resistors. Problem Four resistors are connected as shown in Figure 18.11a. (a) Find the equivalent resistance between points a and c. (b) What is the current in each resistor if a 42-V battery is connected between a and c? Strategy Reduce the circuit in steps, as shown in Figures 18.11b and 18.11c, using the sum rule for resistors in series and the reciprocal-sum rule for resistors in parallel. Finding the currents is a matter of applying Ohm’s law while working backwards through the diagrams.

6.0 

(a)

8.0 

4.0 

I1 b

a

I

c

I2 3.0 

12.0 

2.0 

(b) a

b

c

14.0  (c) a

c

FIGURE 18.11 (Example 18.3) The four resistors shown in (a) can be reduced in steps to an equivalent 14- resistor.

Solution (a) Find the equivalent resistance of the circuit. The 8.0- and 4.0- resistors are in series, so use the sum rule to find the equivalent resistance between a and b : The 6.0- and 3.0- resistors are in parallel, so use the reciprocal-sum rule to find the equivalent resistance between b and c (don’t forget to invert!): In the new diagram, 18.11b, there are now two resistors in series. Combine them with the sum rule to find the equivalent resistance of the circuit:

R eq  R 1  R 2  8.0   4.0   12.0 

1 1 1 1 1 1 5 1 5 1 5 R eq R1 R2 6.0 V 3.0 V 2.0 V R eq  2.0  R eq  R 1  R 2  12.0   2.0   14.0 V

18.4

Kirchhoff’s Rules and Complex DC Circuits

603

(b) Find the current in each resistor if a 42-V battery is connected between points a and c. DVac 42 V 5 5 3.0 A R eq 14 V

Find the current in the equivalent resistor in Figure 18.11c, which is the total current. Resistors in series all carry the same current, so the value is the current in the 12- resistor in Figure 18.11b and also in the 8.0- and 4.0- resistors in Figure 18.11a.

I5

Apply the junction rule to point b:

(1) I  I1  I2

The 6.0- and 3.0- resistors are in parallel, so the voltage drops across them are the same:

V6  V3

Substitute this result into Equation (1), with I  3.0 A:

3.0 A  I1  2I1  3I1

→

(6.0 )I1  (3.0 )I2 → →

2.0I1  I2

I1  1.0 A

I2  2.0 A

Remarks As a final check, note that V bc  (6.0 )I1  (3.0 )I2  6.0 V and Vab  (12 )I1  36 V; therefore, Vac  Vab  V bc  42 V, as expected. QUESTION 18.3 Which of the original resistors dissipates energy at the greatest rate? EXERCISE 18.3 Suppose the series resistors in Example 18.3 are now 6.00  and 3.00  while the parallel resistors are 8.00  (top) and 4.00  (bottom), and the battery provides an emf of 27.0 V. Find (a) the equivalent resistance and (b) the currents I, I1, and I2. Answers

(a) 11.7  (b) I  2.31 A, I1  0.770 A, I2  1.54 A

18.4 KIRCHHOFF’S RULES AND COMPLEX DC CIRCUITS As demonstrated in the preceding section, we can analyze simple circuits using Ohm’s law and the rules for series and parallel combinations of resistors. There are, however, many ways in which resistors can be connected so that the circuits formed can’t be reduced to a single equivalent resistor. The procedure for analyzing more complex circuits can be facilitated by the use of two simple rules called Kirchhoff’s rules: 1. The sum of the currents entering any junction must equal the sum of the currents leaving that junction. (This rule is often referred to as the junction rule.) 2. The sum of the potential differences across all the elements around any closed circuit loop must be zero. (This rule is usually called the loop rule.) The junction rule is a statement of conservation of charge. Whatever current enters a given point in a circuit must leave that point because charge can’t build up or disappear at a point. If we apply this rule to the junction in Figure 18.12a, we get I1  I 2  I 3 Figure 18.12b represents a mechanical analog of the circuit shown in Figure 18.12a. In this analog water flows through a branched pipe with no leaks. The flow rate into the pipe equals the total flow rate out of the two branches. The loop rule is equivalent to the principle of conservation of energy. Any charge that moves around any closed loop in a circuit (starting and ending at the same

I2 I1 I3 (a)

Flow in Flow out

(b) FIGURE 18.12 (a) A schematic diagram illustrating Kirchhoff’s junction rule. Conservation of charge requires that whatever current enters a junction must leave that junction. In this case, therefore, I1  I 2  I 3. (b) A mechanical analog of the junction rule: the net flow in must equal the net flow out.

604

Chapter 18

Direct-Current Circuits

point) must gain as much energy as it loses. It gains energy as it is pumped through a source of emf. Its energy may decrease in the form of a potential drop
IR across a resistor or as a result of flowing backward through a source of emf, from the positive to the negative terminal inside the battery. In the latter case, electrical energy is converted to chemical energy as the battery is charged. When applying Kirchhoff’s rules, you must make two decisions at the beginning of the problem:

I (a)

a

V = Vb – Va = –IR

b

I (b)

a

V = Vb – Va = +IR

b

1. Assign symbols and directions to the currents in all branches of the circuit. Don’t worry about guessing the direction of a current incorrectly; the resulting answer will be negative, but its magnitude will be correct. (Because the equations are linear in the currents, all currents are to the first power.) 2. When applying the loop rule, you must choose a direction for traversing the loop and be consistent in going either clockwise or counterclockwise. As you traverse the loop, record voltage drops and rises according to the following rules (summarized in Fig. 18.13, where it is assumed that movement is from point a toward point b):

e (c)

– a

+

V = Vb – Va = +

e

b

e

b

e (d)

+ a

–

V = Vb – Va = –

(a) If a resistor is traversed in the direction of the current, the change in electric potential across the resistor is
IR (Fig. 18.13a). (b) If a resistor is traversed in the direction opposite the current, the change in electric potential across the resistor is IR (Fig. 18.13b). (c) If a source of emf is traversed in the direction of the emf (from
to  on the terminals), the change in electric potential is  (Fig. 18.13c). (d) If a source of emf is traversed in the direction opposite the emf (from  to
on the terminals), the change in electric potential is
(Fig. 18.13d).

FIGURE 18.13 Rules for determining the potential differences across a resistor and a battery, assuming the battery has no internal resistance.

e

e

There are limits to the number of times the junction rule and the loop rule can be used. You can use the junction rule as often as needed as long as, each time you write an equation, you include in it a current that has not been used in a previous junction-rule equation. (If this procedure isn’t followed, the new equation will just be a combination of two other equations that you already have.) In general, the number of times the junction rule can be used is one fewer than the number of junction points in the circuit. The loop rule can also be used as often as needed, so long as a new circuit element (resistor or battery) or a new current appears in each new equation. To solve a particular circuit problem, you need as many independent equations as you have unknowns.

PROBLEM -SOLVING STRATEGY

AIP ESVA/W.F. Meggers Collection

APPLYING KIRCHHOFF’S RULES TO A CIRCUIT

GUSTAV KIRCHHOFF German Physicist (1824–1887) Together with German chemist Robert Bunsen, Kirchhoff, a professor at Heidelberg, invented the spectroscopy that we study in Chapter 28. He also formulated another rule that states, “A cool substance will absorb light of the same wavelengths that it emits when hot.”

1. Assign labels and symbols to all the known and unknown quantities. 2. Assign directions to the currents in each part of the circuit. Although the assignment of current directions is arbitrary, you must stick with your original choices throughout the problem as you apply Kirchhoff’s rules. 3. Apply the junction rule to any junction in the circuit. The rule may be applied as many times as a new current (one not used in a previously found equation) appears in the resulting equation. 4. Apply Kirchhoff’s loop rule to as many loops in the circuit as are needed to solve for the unknowns. To apply this rule, you must correctly identify the change in electric potential as you cross each element in traversing the closed loop. Watch out for signs! 5. Solve the equations simultaneously for the unknown quantities, using substitution or any other method familiar to the student. 6. Check your answers by substituting them into the original equations.

18.4

EXAMPLE 18.4

Kirchhoff’s Rules and Complex DC Circuits

605

Applying Kirchhoff’s Rules 5.0 

Goal Use Kirchhoff’s rules to find currents in a circuit with three currents and one battery.

4.0 

Problem Find the currents in the circuit shown in Figure 18.14 by using Kirchhoff’s rules.

Select the bottom loop and traverse it clockwise starting at point a, generating an equation with the loop rule: Select the top loop and traverse it clockwise from point c. Notice the gain across the 9.0- resistor because it is traversed against the direction of the current! Rewrite the three equations, rearranging terms and dropping units for the moment, for convenience:

d

I3

Strategy There are three unknown currents in this circuit, so we must obtain three I1 independent equations, which then can be solved by substitution. We can find the equations with one application of the junction rule and two applications of the loop rule. We choose junction c. (Junction d gives the same equation.) For the loops, we b choose the bottom loop and the top loop, both shown by blue arrows, which indicate the direction we are going to traverse the circuit mathematically (not necessarily the FIGURE 18.14 direction of the current). The third loop gives an equation that can be obtained by a linear combination of the other two, so it provides no additional information and isn’t used. Solution Apply the junction rule to point c. I1 is directed into the junction, I2 and I3 are directed out of the junction.

I2 c

9.0 

a 6.0 V (Example 18.4)

I1  I 2  I 3

a DV 5 DVbat 1 DV4.0 V 1 DV9.0 V 5 0 6.0 V
(4.0 )I1
(9.0 )I3  0 a DV 5 DV5.0 V 1 DV9.0 V 5 0
(5.0 )I2  (9.0 )I3  0 (1) I1  I2  I3 (2)

4.0I1  9.0I3  6.0

(3)
5.0I2  9.0I3  0 Solve Equation 3 for I2 and substitute into Equation (1):

I2  1.8I3

Substitute the latter expression into Equation (2) and solve for I3:

4.0(2.8I3)  9.0I3  6.0

Substitute I3 back into Equation (3) to get I2:


5.0I2  9.0(0.30 A)  0

:

I2  0.54 A

Substitute I3 into Equation (2) to get I1:

4.0I1  9.0(0.30 A)  6.0

:

I1  0.83 A

I1  I2  I3  1.8I3  I3  2.8I3

Remarks Substituting these values back into the original equations verifies that they are correct, with any small discrepancies due to rounding. The problem can also be solved by first combining resistors. QUESTION 18.4 How would the answers change if the indicated directions of the currents in Figure 18.14 were all reversed? EXERCISE 18.4 Suppose the 6.0-V battery is replaced by a battery of unknown emf and an ammeter measures I1  1.5 A. Find the other two currents and the emf of the battery. Answers

I2  0.96 A, I3  0.54 A,

e  11 V

→

I3  0.30 A

TIP 18.3 More Current Goes in the Path of Less Resistance You may have heard the statement “Current takes the path of least resistance.” For a parallel combination of resistors, this statement is inaccurate because current actually follows all paths. The most current, however, travels in the path of least resistance.

606

Chapter 18

Direct-Current Circuits

EXAMPLE 18.5 Another Application of Kirchhoff’s Rules 2.0 

f I3

4.0 

10 V

6.0 

a 2.0  (a)

Solution Apply Kirchhoff’s junction rule to junction c. Because of the chosen current directions, I1 and I2 are directed into the junction and I3 is directed out of the junction. Apply Kirchhoff’s loop rule to the loops abcda and befcb. (Loop aefda gives no new information.) In loop befcb, a positive sign is obtained when the 6.0- resistor is traversed because the direction of the path is opposite the direction of the current I1. Using Equation (1), eliminate I3 from Equation (2) (ignore units for the moment):

1.0 

c I3

I2

I1

5.0 V

d

+

I1

+

1.0 

–

–

+

b

I2 –

Strategy Use Kirchhoff’s two rules, the junction rule once and the loop rule twice, to develop three equations for the three unknown currents. Solve the equations simultaneously.

+

Problem Find I1, I2, and I3 in Figure 18.15a.

14 V e

–

Goal Find the currents in a circuit with three currents and two batteries when some current directions are chosen wrongly.

5.0 V (b)

FIGURE 18.15 (a) (Example 18.5) (b) (Exercise 18.5)

(1) I3  I1  I2

(2)

Loop abcda:

(3)

Loop befcb:

10 V
(6.0 )I1
(2.0 )I3  0


(4.0 )I2
14 V  (6.0 )I1
10 V  0 10
6.0I1
2.0(I1  I2)  0 (4)

10  8.0I1  2.0I2

Divide each term in Equation (3) by 2 and rearrange the equation so that the currents are on the right side:

(5)
12 
3.0I1  2.0I2

Subtracting Equation (5) from Equation (4) eliminates I2 and gives I1:

22  11I1

Substituting this value of I1 into Equation (5) gives I2:

2.0I2  3.0I1
12  3.0(2.0)
12 
6.0 A

:

I1  2.0 A

I2  23.0 A Finally, substitute the values found for I1 and I2 into Equation (1) to obtain I3:

I3  I1  I2  2.0 A
3.0 A  21.0 A

Remarks The fact that I2 and I3 are both negative indicates that the wrong directions were chosen for these currents. Nonetheless, the magnitudes are correct. Choosing the right directions of the currents at the outset is unimportant because the equations are linear, and wrong choices result only in a minus sign in the answer. QUESTION 18.5 Is it possible for the current in a battery to be directed from the positive terminal toward the negative terminal? EXERCISE 18.5 Find the three currents in Figure 18.15b. (Note that the direction of one current was deliberately chosen wrongly!) Answers

I1 
1.0 A, I2  1.0 A, I3  2.0 A

18.5

e 0.632C e C

e

t =RC

R t

607

ACTIVE FIGURE 18.16 (a) A capacitor in series with a resistor, a battery, and a switch. (b) A plot of the charge on the capacitor versus time after the switch on the circuit is closed. After one time constant t, the charge is 63% of the maximum value, C e. The charge approaches its maximum value as t approaches infinity.

q

C

RC Circuits

t

S (a)

18.5

(b)

RC CIRCUITS

So far, we have been concerned with circuits with constant currents. We now consider direct-current circuits containing capacitors, in which the currents vary with time. Consider the series circuit in Active Figure 18.16. We assume the capacitor is initially uncharged with the switch opened. After the switch is closed, the battery begins to charge the plates of the capacitor and the charge passes through the resistor. As the capacitor is being charged, the circuit carries a changing current. The charging process continues until the capacitor is charged to its maximum equilibrium value, Q  C , where is the maximum voltage across the capacitor. Once the capacitor is fully charged, the current in the circuit is zero. If we assume the capacitor is uncharged before the switch is closed, and if the switch is closed at t  0, we find that the charge on the capacitor varies with time according to the equation

e

t !  E

 +    5 7 ; ? @ C %

Nu Xi Omicron Pi Rho Sigma Tau Upsilon Phi Chi Psi Omega

) # . 0 2 " 8 <  A  

$ , /  3  9 =  B D 

TABLE C.4 Physical Data Often Useda Average Earth–Moon distance Average Earth–Sun distance Equatorial radius of Earth Density of air (20°C and 1 atm) Density of water (20°C and 1 atm) Free-fall acceleration Mass of Earth Mass of Moon Mass of Sun Standard atmospheric pressure a

3.84  108 m 1.496  1011 m 6.38  106 m 1.20 kg/m3 1.00  103 kg/m3 9.80 m/s2 5.98  1024 kg 7.36  1022 kg 1.99  1030 kg 1.013  105 Pa

These are the values of the constants as used in the text.

A.19

A.20

Appendix C

Some Useful Tables

TABLE C.5 Some Fundamental Constants Quantity

Symbol

Valuea

Atomic mass unit

u

Avogadro’s number

NA

1.660 538 86 (28)  10
27 kg 931.494 043 (80) MeV/c 2 6.022 141 5 (10)  1023 particles/mol

Bohr magneton

mB 5

Deuteron mass

eU 2m e U2 a0 5 m ee 2k e R kB 5 NA h lC 5 m ec 1 ke 5 4pP0 md

Electron mass

me

Electron volt Elementary charge Gas constant Gravitational constant

eV e R G 2e h

Bohr radius Boltzmann’s constant Compton wavelength Coulomb constant

Josephson frequency–voltage ratio

F0 5

Neutron mass

mn

Nuclear magneton

mn 5

Permeability of free space

m0

Permittivity of free space

P0 5

Planck’s constant

h

Proton mass

mp

Rydberg constant Speed of light in vacuum

RH c

5.291 772 108 (18)  10
11 m 1.380 650 5 (24)  10
23 J/K 2.426 310 238 (16)  10
12 m 8.987 551 788 . . .  109 N  m2/C2 3.343 583 35 (57)  10
27 kg 2.013 553 212 70 (35) u 9.109 382 6 (16)  10
31 kg 5.485 799 094 5 (24)  10
4 u 0.510 998 918 (44) MeV/c 2 1.602 176 53 (14)  10
19 J 1.602 176 53 (14)  10
19 C 8.314 472 (15) J/mol  K 6.674 2 (10)  10
11 N  m2/kg2 4.835 978 79 (41)  1014 Hz/V

Magnetic flux quantum

U5

9.274 009 49 (80)  10
24 J/T

h 2e

2.067 833 72 (18)  10
15 T  m2 1.674 927 28 (29)  10
27 kg 1.008 664 915 60 (55) u 939.565 360 (81) MeV/c 2

eU 2m p

5.050 783 43 (43)  10
27 J/T 4p  10
7 T  m/A (exact)

1 m0c 2

8.854 187 817 . . .  10
12 C2/N  m2 6.626 069 3 (11)  10
34 J  s

h 2p

1.054 571 68 (18)  10
34 J  s 1.672 621 71 (29)  10
27 kg 1.007 276 466 88 (13) u 938.272 029 (80) MeV/c 2 1.097 373 156 852 5 (73)  107 m
1 2.997 924 58  108 m/s (exact)

Note: These constants are the values recommended in 2002 by CODATA, based on a least-squares adjustment of data from different measurements. For a more complete list, see P. J. Mohr and B. N. Taylor, “CODATA Recommended Values of the Fundamental Physical Constants: 2002.” Rev. Mod. Phys. 77:1, 2005. aThe

numbers in parentheses for the values represent the uncertainties of the last two digits.

APPENDIX D SI Units

TABLE D.1 SI Base Units Base Quantity

SI Base Unit Name Symbol

Length Mass Time Electric current Temperature Amount of substance Luminous intensity

meter kilogram second ampere kelvin mole candela

m kg s A K mol cd

TABLE D.2 Derived SI Units

Quantity

Name

Symbol

Expression in Terms of Base Units

Plane angle Frequency Force Pressure Energy: work Power Electric charge Electric potential (emf ) Capacitance Electric resistance Magnetic flux Magnetic field intensity Inductance

radian hertz newton pascal joule watt coulomb volt farad ohm weber tesla henry

rad Hz N Pa J W C V F  Wb T H

m/m s
1 kg  m/s2 kg/m  s2 kg  m2/s2 kg  m2/s3 As kg  m2/A  s3 A2  s4/kg  m2 kg  m2/A2  s3 kg  m2/A  s2 kg/A  s2 kg  m2/A2  s2

Expression in Terms of Other SI Units

J/m N/m2 Nm J/s W/A, J/C C/V V/A V  s, T  m2 Wb/m2 Wb/A

A.21

APPENDIX E

MCAT Skill Builder Study Guide

VECTORS EXAMPLE 1 A hiker walks due north at 4.00 km/h from his campsite. A grizzly bear, starting 3.00 km due east of the campsite, walks at 7.00 km/h in a direction 30° west of north. After one hour, how far apart are they? Solution Write the components of the position vector of the hiker after one hour, choosing the campsite as the origin:

Hx  0

Hy  4.00 m

The bear travels at an angle of 30°  90°  120° with respect to due east, where due east corresponds to the positive x-direction. Write the components of the position vector of the bear after one hour, again relative the campsite:

Bx  3.00 km  (7.00 km) cos 120° 
0.500 km

Subtracting the bear’s position vectorSfrom the hiker’s position vector will result in a vector R pointing from the bear to the hiker:

R x  Hx
Bx  0
(
0.500 km)  0.500 km

Calculate the magnitude of the resultant vector, obtaining the distance between the hiker and the bear:

By  (7.00 km) sin 120°  6.06 km

Ry  Hy
By  4.00 km
6.06 km 
2.06 km R 5 1 R x2 1 R y2 2 1/2 5 3 1 0.500 km 2 2 1 1 22.06 km 2 2 4 1/2  2.12 km

EXAMPLE 2 S

S

S

S

S

Two vectors A and B are not parallel to each other (nor “antiparallel”—pointing in opposite directions). If R 5 A 1 B, which of the following must be true of the magnitudes A, B, and R? (a) R A  B (b) R  A  B (c) R A  B Conceptual SolutionS S Because the vectors A and B are not parallel, they can be arranged to form two sides of a triangle. By the geometric S law of vector addition, the resultant vector R forms the third side of the same triangle. The shortest distance between two points is a straight line, so the distance along R is shorter than the sum of the lengths of A and B. Hence, the correct answer is (c).

MULTIPLE-CHOICE PROBLEMS 1. A car travels due east for a distance of 3 mi and then due north for an additional 4 mi before stopping. What is the shortest straight-line distance between the starting and ending points of this trip? (a) 3 mi (b) 4 mi (c) 5 mi (d) 7 mi 2. In the previous problem, what is the angle a of the shortest path relative to due north? (a) a  arc cos 3/5 (b) a  arc sin 5/3 (c) a  arc sin 4/3 (d) a  arc tan 3/4 S

3. A vector A makes an angle of 60° with the x-axis of a Cartesian coordinate system. Which of the following statements is true of the indicated magnitudes? (a) Ax Ay (b) Ay Ax (c) Ay A (d) Ax A 4. Force is a vector quantity measured in units of newtons, N. What must be the angle between two concurrently acting forces of 5 N and 3 N, respectively, if the resultant vector has a magnitude of 8 N? (a) 0° (b) 45° (c) 90° (d)180°

A.22

Motion

A.23

5. Two forces act concurrently on an object. Both vectors have the same magnitude of 10 N and act at right angles to each other. What is the closest estimate of the magnitude of their resultant? (a) 0 N (b) 14 N (c) 20 N (d) 100 N

Answers 1. (c). The two legs of the trip are perpendicular; therefore, the shortest distance is given by the hypotenuse of the corresponding right triangle. The value is given by the Pythagorean theorem: R  (32  42)1/2  5 mi. 2. (d). The angle a that gives the direction of the hypotenuse relative to the y-axis is given by tan a  opposite side/adjacent side  3/4 :

a  arc tan 3/4

3. (b). Choices (c) and (d) can be eliminated immediately because the projection of a vector can never be greater than the magnitude of the vector itself. As the angle between the vector and the axis increases towards 90°, the magnitude of the projection decreases toward zero. The angle between the vector and the x-axis is 60°, which means that the angle between the vector and the y-axis must be 30°. Because the angle between the vector and the y-axis is smaller than the angle between the vector and the x-axis, the projection onto the y-axis, Ay, must be greater than the projection onto the x-axis, Ax . 4. (a). The only way the vector sum of a 3-N and a 5-N force can equal 8 N is if both forces act in the same direction. The angle between them must therefore be zero degrees. 5. (b). The two vectors form the legs of a right triangle. The resultant vector is the hypotenuse. Choices (a) and (c) can be eliminated immediately because they require that the two forces be antiparallel or parallel respectively. Choice (d) can also be eliminated because it is too large to be the hypotenuse. The magnitude of the resultant can be calculated with the Pythagorean theorem: R  (102  102)1/2  (200)1/2  10(2)1/2  10(1.41)  14 N

MOTION EXAMPLE 1 A motorist traveling 24.0 m/s slams on his brakes and comes to rest in 6.00 s. What is the car’s acceleration assuming it’s constant, and how far does the car travel during this time? Solution Apply the kinematics equation for velocity:

(1) v  at  v0

Solve Equation (1) for the acceleration a and substitute v  0, t  6.00 s, and v 0  24.0 m/s:

a5

v 2 v0 0 2 24.0 m/s 5 5 24.00 m/s 2 t 6.00 s

The displacement can then be found with the timeindependent equation:

(2)

v 2
v 02  2a x

Solve Equation (2) for x and substitute values:

Dx 5

v 2 2 v 02 0 2 1 24.0 m/s 2 2 5 72.0 m 5 2a 2 1 24.00 m/s 2 2

EXAMPLE 2 A student on top of a building releases a rock from rest. When the rock is halfway to the ground, he releases a golf ball from rest. As both objects fall, does the distance between them increase, decrease, or stay the same? Conceptual Solution At the instant the golf ball is released the rock is traveling faster, hence will always travel faster before it reaches the ground because the velocities of both objects are changing at the same rate. Because the rock is always falling faster, the distance between it and the golf ball will increase.

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EXAMPLE 3 A baseball player throws a ball at an angle u above the horizontal and at speed v 0. Neglecting air drag, what is the ball’s speed when it returns to the same height at which it was thrown? (a) The ball’s speed will be the same. (b) The ball’s speed will be larger. (c) The ball’s speed will be smaller. Conceptual Solution The ball’s speed will be the same, so the correct answer is (a). Because there is no acceleration in the horizontal direction, the velocity in the x-direction is constant, and only the y-direction need be considered. By symmetry, the time taken to reach maximum height is the same time taken to fall back to the original height. If the ball’s velocity in the y-direction changes from v 0y to 0 on the way to maximum height, then it will change from 0 to
v 0y on returning to its original height. Hence the ball’s velocity components will be the same as before, except for a sign change in the y-component, which doesn’t affect the speed. Quantitative Solution Let the point of release correspond to x  0, y  0. Set the equation of displacement in the y-direction equal to zero:

(1) Dy 5 12at 2 1 v 0yt 5 0

22v 0y

Solve Equation (1), obtaining the times when y  0 (i.e., at the start and end of the trajectory):

t 5 0; t 5

Substitute these two solutions into the equation for the velocity in the y-direction:

v y 5 at 1 v 0y

Case 1: t  0

v y 5 a # 0 1 v 0y 5 v 0y

Case 2: t 5

22v 0y

vy 5 a # a

a

The speed at any time is given by the following expression:

a

22v 0y a

b 1 v 0y 5 2v 0y

v  (v 0x2  v 0y2)1/2

Because the y-component is squared, the negative sign in Case 2 makes no difference and the speeds at the two times in question are the same, again giving answer (a).

EXAMPLE 4 A merry-go-round has a radius of 2.00 m and turns once every 5.00 s. What magnitude net force is required to hold a 35.0-kg boy in place at the rim? Solution First, calculate the speed of an object at the rim, which is the distance around the rim divided by the time for one rotation: A body traveling in a circle at uniform speed requires a centripetal acceleration of ac  v 2/r. The force producing this acceleration can be found by substituting this expression into Newton’s second law:

v5

2p 1 2.00 m 2 d 2pr 5 5 5 2.51 m/s t t 5.00 s

F 5 ma c 5 m

1 2.51 m/s 2 2 v2 5 1 35.0 kg 2 r 2.00 m

5 1.10 3 102 N

MULTIPLE-CHOICE PROBLEMS 1. A bird flies 4.0 m due north in 2.0 s and then flies 2.0 m due west in 1.0 s. What is the bird’s average speed? (a) 2.0 m/s (b) 4.0 m/s (c) 8.0 m/s (d) 2 !5/3 m/s

Motion

2. Applying the brakes to a car traveling at 45 km/h provides an acceleration of 5.0 m/s2 in the opposite direction. How long will it take the car to stop? (a) 0.40 s (b) 2.5 s (c) 5.0 s (d) 9.0 s 3. A ball rolls down a long inclined plane with a uniform acceleration of magnitude 1.0 m/s2. If its speed at some instant of time is 10 m/s, what will be its speed 5.0 seconds later? (a) 5 m/s (b) 10 m/s (c) 15 m/s (d) 16 m/s 4. A ball rolls down an inclined plane with a uniform acceleration of 1.00 m/s2. If its initial velocity is 1.00 m/s down the incline, how far will it travel along the incline in 10.0 s? (a) 10.0 m (b) 12.0 m (c) 60.0 m (d) 1.00  102 m 5. An object with an initial velocity of 25.0 m/s accelerates uniformly for 10.0 s to a final velocity of 75.0 m/s. What is the magnitude of its acceleration? (a) 3.00 m/s2 (b) 5.00 m/s2 (c) 25.0 m/s2 (d) 50.0 m/s2 6. A rock is dropped from a height of 19.6 m above the ground. How long does it take the rock to hit the ground? (a) 2.0 s (b) 4.0 s (c) 4.9 s (d) 9.8 s 7. A spacecraft hovering above the surface of a distant planet releases a probe to explore the planet’s surface. The probe falls freely a distance of 40.0 m during the first 4.00 s after its release. What is the magnitude of the acceleration due to gravity on this planet? (a) 4.00 m/s2 (b) 5.00 m/s2 (c) 10.0 m/s2 (d) 16.0 m/s2 8. A ball is dropped from the roof of a very tall building. What is its speed after falling for 5.00 s? (a) 1.96 m/s (b) 9.80 m/s (c) 49.0 m/s (d) 98.0 m/s 9. A quarterback throws a football with a velocity of 7.00 m/s at an angle of 15.0° with the horizontal. How far away should the receiver be to catch the football successfully? (a) 1.25 m (b) 2.50 m (c) 5.00 m (d) 6.25 m 10. A bomber is flying parallel to the ground with a speed of 500 km/h at an altitude of 1 960 m when it drops a bomb. How long does it take the bomb to hit the ground? (a) 0.500 s (b) 20.0 s (c) 50.0 s (d) 2.00  102 s 11. A ball is thrown horizontally with a speed of 6.0 m/s. What is its speed after 3.0 seconds of flight? (a) 30 m/s (b) 15.8 m/s (c) 18 m/s (d) 4.9 m/s 12. A 70-kg woman standing at the equator rotates with the Earth around its axis at a tangential speed of about 5  102 m/s. If the radius of the Earth is approximately 6  106 m, which is the best estimate of the centripetal acceleration experienced by the woman? (a) 4  10
2 m/s2 (b) 4 m/s2 (c) 10
4 m/s2 (d) 24 m/s2 13. A ball rolls with uniform speed around a flat, horizontal, circular track. If the speed of the ball is doubled, its centripetal acceleration is (a) quadrupled (b) doubled (c) halved (d) unchanged

Answers 1. (a). The average speed is the total distance traveled divided by the elapsed time. v  d/t  6.0 m/3.0 s  2.0 m/s 2. (b). t  (vf
vi)/a. The final velocity is 0 km/h, and the acceleration is
5.0 m/s2. (The negative sign appears because the acceleration is antiparallel to the velocity.) Remember to convert from kilometers to meters and from hours to seconds. 0 2 1 45 km/h 2 1 103 m/1 km 2 1 1 h/3 600 s 2 Dv t5 5 5 2.5 s a 25.0 m/s 2 3. (c). vf = vi  at  10 m/s  (1.0 m/s2)(5.0 s)  15 m/s

4. (c). d 5 v it 1 12at 2 5 1 1.00 m/s 2 1 10.0 s 2 1 12 1 1.00 m/s 2 2 1 10.0 s 2 2 5 60.0 m 5. (b). a  v/t  (75.0 m/s
25.0 m/s)/10.0 s  5.00 m/s2 6. (a). The rock is in free fall with zero initial velocity and initial height of y 0  19.6 m. The displacement equation becomes y  y
y 0 
12gt 2. Solving for time t with y  0 gives t  (2y 0/g)1/2  (2  19.6 m/9.8 m/s2)1/2  (4.0 s2)1/2  2.0 s 7. (b). This question is a free-fall problem. The object starts from rest, v 0  0. Rearrange Dy 5 212gt 2 and solve for the acceleration of gravity, g : g 5 22 1 y 2 y0 2 /t 2 5 22 1 240.0 m 2 0 2 /16.0 s 2 5 5.00 m/s 2

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8. (c). The free-fall velocity starting with an initial velocity of zero is v 
gt 
(9.8 m/s2)(5.00 s) 
49 m/s 9. (b). The initial velocity in the y-direction is v 0y  v 0 sin u  (7.00 m/s) sin 15.0°  1.81 m/s. Maximum height is reached when the velocity in the y-direction is zero: v  0 
gt  v 0y : 0 
(9.80 m/s2)t  1.81 m/s : t  0.185 s. Doubling this result gives the total time of flight. The range is x  (v 0 cos u)t  (7.00 m/s)(cos 15.0°)(0.370 s)  2.50 m. 10. (b). The horizontal and the vertical components of velocity are independent. The vertical component is due only to the acceleration of gravity: y 5 12 gt 2

S

t 5 1 2y/g 2 1/2

t 5 3 2 1 1 960 m 2 / 1 9.80 m/s 2 2 4 1/2 5 1 4.00 3 102 s 2 2 1/2 5 20.0 s

11. (a). The horizontal component of velocity is vx  6.00 m/s and the vertical component is vy  gt  (9.80 m/s2)(3.00 s)  29.4 m/s, so v 5 1 v x 2 1 v y 2 2 1/2 5 3 1 6.00 m/s 2 2 1 1 29.4 m/s 2 2 4 1/2 5 3.00 m/s

12. (a). The woman is moving with constant speed equal to the rotation of the Earth on its axis in a circle equal to the radius of the Earth. The centripetal acceleration is ac  v 2/R  (500 m/s)2/6  106 m  (25  104 m2/s2)/6  106 m  4  10
2 m/s2 13. (a). ac  v 2/R. The acceleration and the square of the velocity are directly proportional. Because the velocity is squared, doubling the velocity quadruples the acceleration.

FORCE EXAMPLE 1 A net force F accelerates an object of mass m at 6 m/s2. The same net force is applied to a body with mass 3m. What is the acceleration of the more massive body? Conceptual Solution According to the Newton’s second law, the acceleration imparted to an object by a force is inversely proportional to the object’s mass. The acceleration of an object three times as massive is therefore 13 1 6 m/s 2 2 5 2 m/s2. Quantitative Solution Write Newton’s second law for each of the objects:

(1) mam  F (2)

MaM  F

Divide Equation (2) by Equation (1):

(3)

Ma M F 5 51 ma m F

Substitute M  3m into Equation (3) and solve for aM :

1 3m 2 a M 51 ma m

S

a M 5 13a m 5 13 # 1 6 m/s 2 2

aM  2 m/s 2

EXAMPLE 2 A block of mass m is attached by a horizontal string of negligible mass to the left side of a second block of mass M, with m M. A second such string is attached to the right side of mass M, and the system is pulled in the positive x-direction by a constant force F. If the surface is frictionless, what can be said about the tension T in the string connecting the two blocks? (a) T  F (b) T F (c) T F Conceptual Solution The answer is (b). The acceleration of the block m is the same as that of the system of both, which has total mass m  M. The tension T must accelerate only the mass m, whereas the force F must provide the same acceleration to a system of greater mass, m  M. It therefore follows that F T.

Force

Quantitative Solution In this problem the y-component of Newton’s second law gives only the normal forces acting on the blocks in terms of the gravity force. Because there is no friction, these forces don’t affect the acceleration. Write the x-component of Newton’s second law for the system:

(1)

1m 1 M2a 5 F

Write the x-component of Newton’s second law for the less massive block:

(2)

ma  T

Divide Equation (1) by Equation (2), canceling the acceleration, which is the same for both:

(3)

1m 1 M 2a F 5 ma T

S

A.27

1m 1 M 2 F 5 m T

Inspecting Equation (3), it’s clear that F T because m  M m, and again the answer is (b).

EXAMPLE 3 A block with mass m is started at the top of an incline with coefficient of kinetic friction mk and allowed to slide to the bottom. A second block with mass M  2m is also allowed to slide down the incline. If mk is the same for both blocks, how does the acceleration aM of the second block compare with the acceleration am of the first block? (a) aM  2am (b) aM  am (c) aM  12am Conceptual Solution The accelerations are the same because the force of gravity, the force of friction, and the ma side of Newton’s second law when applied in this physical context are all proportional to the mass. Hence, a different mass should not affect the acceleration of the body, and the answer is (b).

EXAMPLE 4 Planet A has twice the mass and twice the radius of Planet B. On which planet would an astronaut have a greater weight? (a) Planet A (b) Planet B (c) The astronaut’s weight would be unaffected. Conceptual Solution Weight as measured on a given planet is the magnitude of the gravitational force at the surface of that planet. In Newton’s law of gravitation, the gravitational force is directly proportional to the mass of each of two bodies but inversely proportional to the square of the distance between them. Hence, in considering only the masses, planet A exerts twice the gravitational force of planet B. Planet A also has twice the radius of planet B, however, and the inverse square of 2 is 14. Overall, therefore, Planet A has a weaker gravitational acceleration at its surface than Planet B by a factor of one-half, and the weight of an astronaut will be greater on planet B [answer (b)]. Quantitative Solution Write Newton’s law of gravitation for the weight of an astronaut of mass m on planet A: Write the same law for the weight of the astronaut on Planet B:

Divide Equation (2) by Equation (1):

(1) wA 5

mMAG rA2

(2)

mMBG rB2

wB 5

mMBG MBr A 2 wB rB2 5 5 wA mMAG MAr B 2 2 rA

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Now substitute rA  2rB and MA  2MB :

wB MB 1 2r B 2 2 5 52 wA 1 2MB 2 r B 2

Solve for wB :

wB  2wA [answer (b)]

MULTIPLE-CHOICE PROBLEMS 1. Which of the following will result from the application of a nonzero net force on an object? (a) The velocity of the object will remain constant. (b) The velocity of the object will remain constant, but the direction in which the object moves will change. (c) The velocity of the object will change. (d) None of the above. 2. Body A has a mass that is twice as great as that of body B. If a force acting on body A is half the value of a force acting on body B, which statement is true? (a) The acceleration of A will be twice that of B. (b) The acceleration of A will be half that of B. (c) The acceleration of A will be equal to that of B. (d) The acceleration of A will be one-fourth that of B. 3. Which of the following is a statement of Newton’s second law of motion? (a) For every action there is an equal and opposite reaction. (b) Force and the acceleration it produces are directly proportional, with mass as the constant of proportionality. (c) A body at rest tends to remain at rest unless acted upon by a force. (d) None of the these. 4. If a body has an acceleration of zero, which statement is most true? (a) The body must be at rest. (b) The body may be at rest. (c) The body must slow down. (d) The body may speed up. 5. What is the weight of a 2.00-kg body on or near the surface of the Earth? (a) 4.90 N (b) 16.0 lb (c) 19.6 N (d) 64.0 kg  m/s2 6. Two objects of equal mass are separated by a distance of 2 m. If the mass of one object is doubled, the force of gravity between the two objects will be (a) half as great (b) twice as great (c) one-fourth as great (d) four times as great 7. The distance between a spaceship and the center of the Earth increases from one Earth radius to three Earth radii. What happens to the force of gravity acting on the spaceship? (a) It becomes 1/9 as great. (b) It becomes 9 times as great. (c) It becomes 1/3 as great. (d) It becomes 3 times as great. 8. A 100-kg astronaut lands on a planet with a radius three times that of the Earth and a mass nine times that of the Earth. The acceleration due to gravity, g, experienced by the astronaut will be (a) Nine times the value of g on the Earth. (b) Three times the value of g on the Earth. (c) The same value of g as on the Earth. (d) One-third the value of g on the Earth. 9. The measured weight of a person standing on a scale and riding in an elevator will be the greatest when (a) the elevator rises at a constant velocity. (b) the elevator accelerates upward. (c) the elevator falls at a constant velocity. (d) none of the these, because weight is constant. 10. A block with mass 25.0 kg rests on a rough level surface. A horizontal force of 1.50  102 N is applied to it, and an acceleration of 4.00 m/s2 is subsequently observed. What is the magnitude of the force of friction acting on the block? (a) 25.0 N (b) 1.00  102 N (c) 75.0 N (d) 50.0 N

Answers 1. (c). Force produces acceleration, which results in a change in velocity. 2. (d). Obtain the ratio of the accelerations from Newton’s second law, F  ma: FA m Aa A 5 m Ba B FB

1 12FB 2 1 2m B 2 a A 5 m Ba B FB

aA 1 5 aB 4 3. (b). Choice (a) is a statement of the third law, and choice (c) is a statement of the first law. S

S

4. (b). Choices (c) and (d) can be eliminated immediately because changing velocity (either its magnitude or direction) requires the application of a force. Choice (a) can

Equilibrium

A.29

be eliminated because the first law applies to both bodies at rest and those moving with uniform velocity. 5. (c). w  F  mg  (2.00 kg)(9.80 m/s2)  19.6 N 6. (b). In the law of universal gravitation, the gravitational force is directly proportional to the product of the masses. Doubling one of the masses doubles their product and therefore doubles the force. 7. (a). Choices (b) and (d) can be eliminated because the force of gravitational attraction must decrease with increasing distance. Because Fgrav  Gm1m 2/r 2, the force decreases by a factor of 9 when r is tripled. 8. (c). See Example 4. Gravitational force is proportional to mass and inversely proportional to the radius squared, so nine times the mass and three times the radius yields a factor of 9/32  1. There is no discernible difference in weight because the two effects exactly cancel each other. 9. (b). The weight measured on the scale is essentially the magnitude of the normal force. Apply Newton’s second law to the passenger and solve for the normal force: ma  n
mg

:

n  ma  mg

Inspecting the equation for the normal force, it’s clear that a positive acceleration results in a larger normal force, hence a larger reading on the scale. 10. (d). Use Newton’s second law: ma  Fapp
F fric (25.0 kg)(4.00 m/s2)  1.50  102 N
F fric F fric  50.0 N

EQUILIBRIUM EXAMPLE 1 A uniform steel beam with length L and mass M  60.0 kg rests on two pivots. The first pivot is located at the left end of the beam and exerts a normal force n1 at that point. A second pivot is two-thirds of the distance from the left end to the right end. What is the normal force n2 exerted on the beam by the second pivot set? (a) 441 N (b) 264 N (c) 372 N (d) 188 N Strategy In equilibrium, the sum of the torques is zero. Let tn1 be the torque exerted by the normal force n1 associated with the first pivot, tn2 the torque exerted by the normal force n2 associated with the second pivot, and tM the torque exerted by the force of gravity on the beam. Choose the pivot at the left end as the point around which to calculate S S torques. Use t  rF sin u for each torque, where u is the angle between the position vector of r and the force F . The sign of a given torque is positive if the force tends to rotate the beam counterclockwise and negative if the force tends to rotate the beam clockwise. Solution In equilibrium, the sum of the torques is zero: The torque due to the normal force n1 is zero because its lever arm is zero. The weight of the uniform beam may be considered to act at the middle of the beam, its center of gravity. Equation (1) becomes: Notice that for n2 the torque is positive, whereas the gravitational torque is negative (see the Strategy). Cancel the factors of L in Equation (2) and solve for n2:

(1)

a ti 5 tn1 1 tn 2 1 tM 5 0

(2)

n2 1 2L/3 2 sin 90° 2 Mg 1 L/2 2 sin 90° 5 0

n2 5 34Mg 5 34 1 60.0 kg 2 1 9.80 m/s 2 2 5 441 N  441 N [answer (a)]

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EXAMPLE 2 Ice skater A has mass M and is at rest on a frozen pond, holding a backpack of mass m. She tosses the backpack to ice skater B, who also has mass M and is initially at rest. If the surface of the pond is considered frictionless, which skater has the larger subsequent speed? (Changes of momentum in the vertical direction can be ignored, here.) (a) skater A (b) skater B (c) Both skaters have the same subsequent speed. Conceptual Solution By conservation of momentum, the final momentum of skater A is equal and opposite the momentum of the backpack. Skater B acquires this same momentum when catching the backpack, but also increases her mass from m to M  m. Because momentum is proportional to both mass and velocity, skater B, with greater mass, has a smaller subsequent speed. Thus, the answer is (a). Quantitative Solution Apply conservation of momentum to skater A. The initial momentum equals the final momentum:

pi  pf

The initial total momentum is zero. Skater A’s final momentum is MVA , whereas the backpack has momentum mv :

0  MVA  mv

Solve for the velocity VA of skater A:

(1) VA 5 2

Apply conservation of momentum to skater B. The initial momentum of the backpack, mv, equals the final momentum of the system of skater and backpack:

mv  (m  M)V B

Solve for the velocity V B of the backpack–skater B system:

(2)

VB 5

(3)

0 VA 0 5 0 VB 0

Divide the magnitude of Equation (1) by Equation (2):

mv M

mv m1M `2

mv ` m1M M 5 .1 mv M ` ` m1M

By Equation (3),
VA
/
VB
1, so
VA

VB
[answer (a)].

MULTIPLE-CHOICE PROBLEMS 1. A nonuniform bar 8.00 m long is placed on a pivot 2.00 m from the right end of the bar, which is also the lighter end. The center of gravity of the bar is located 2.00 m from the heavier left end. If a weight of W  5.00  102 N on the right end balances the bar, what must be the weight w of the bar? (a) 1.25  102 N (b) 2.50  102 N (c) 5.00  102 N (d) 1.00  103 N 2. A rod of negligible mass is 10 m in length. If a 30-kg object is suspended from the left end of the rod and a 20-kg object from the right end, where must the pivot point be placed to ensure equilibrium? (a) 4 m from the 30-kg object (b) 4 m from the 20-kg object (c) 8 m from the 30-kg object (d) 5 m from the 20-kg object 3. A car with a mass of 8.00  102 kg is stalled on a road. A truck with a mass of 1 200 kg comes around the curve at 20.0 m/s and hits the car. The two vehicles remain locked together after the collision. What is their combined speed after impact? (a) 3.0 m/s (b) 6.0 m/s (c) 12 m/s (d) 24 m/s 4. A car of mass 1.00  103 kg traveling at 5.00 m/s overtakes and collides with a truck of mass 3.00  103 kg traveling in the same direction at 1.00 m/s. During the colli-

Equilibrium

sion, the two vehicles remain in contact and continue to move as one unit. What is the speed of the coupled vehicles right after impact? (a) 2.00 m/s (b) 4.00 m/s (c) 5.00 m/s (d) 6.00 m/s 5. A pistol with a mass of 0.20 kg fires a 0.50-gram bullet with an initial speed of 1.0  102 m/s. What is the magnitude of the recoil velocity of the pistol? (a) 0.25 m/s (b) 0.50 m/s (c) 1.0 m/s (d) 10 m/s 6. A 0.20-kg ball is bounced against a wall. It hits the wall with a speed of 20.0 m/s and rebounds in an elastic collision. What is the magnitude of the change in momentum of the ball? (a) 0 kg  m/s (b) 4.0 kg  m/s (c) 8.0 kg  m/s (d) 10 kg  m/s 7. A tennis ball is hit with a tennis racket and the change in the momentum of the ball is 4.0 kg  m/s. If the collision time of the ball and racket is 0.01 s, what is the magnitude of the force exerted by the ball on the racket? (a) 2.5  10
3 N (b) 4  10
2 N (c) 4.0 N (d) 400 N 8. A 30-kg cart traveling due north at 5 m/s collides with a 50-kg cart that had been traveling due south. Both carts immediately come to rest after the collision. What must have been the speed of the southbound cart? (a) 3 m/s (b) 5 m/s (c) 6 m/s (d) 10 m/s

Answers 1. (b). At equilibrium, (1)

S

S

S

S

a ti 5 tp 1 tB 1 tW 5 0

S

The torque tp exerted by the pivot is zero because the torques are computed around that point, meaning a zero lever arm for the force exerted by the pivot. The center of gravity is 4.00 m from the pivot point, and the gravitational force exerts a counS terclockwise torque tB on the bar as if all the mass were concentrated at that point. The torque due to the weight of the bar is therefore equal to rF  (4.00 m)wB . The S weight W on the right side gives a clockwise torque tW  2(2.00 m)W. Hence, from Equation (1), 0 1 wB 1 4.00 m 2 2 1 5.00 3 102 N 2 1 2.00 m 2 5 0

S

wB 5 2.50 3 102 N

2. (a). The pivot point must be closer to the heavier weight; otherwise, the torques due to the weights could not be equal in magnitude and opposite in direction. Given this information, choices (b) and (c) can be eliminated immediately. Choice (d) can also be eliminated because it puts the pivot at the center of the rod, which would only give equilibrium if the two weights were identical. The choice must be (a), which can be confirmed using the second condition of equilibrium. If x is the distance of the pivot from the 30-kg mass, then 10
x is the distance from the pivot to the 20-kg mass. Apply the second condition of equilibrium, " ti  0: (30 kg)g(x)
(20 kg)g(10
x)  0

:

x4m

3. (c). Conservation of momentum ensures that the total momentum after the collision must equal the total momentum before the collision. If A is the car of mass 8.00  102 kg and B is the 1 200-kg truck, then m AvA initial  m Bv B initial  m AvA final  m Bv B final Because the car is stalled, its initial velocity is zero and m AvA initial is zero. After the collision, the vehicles are joined and vA final  v B final  v final. The equation becomes m Bv B initial  (m A m B)v final Solve for v final:

v final 5 m Bv B initial / 1 m A 1 m B 2  (1 200 kg)(20.0 m/s)/(1 200 kg  8.00  102 kg)  12 m/s.

4. (a). Conservation of momentum gives p before  p after. The total momentum before the collision is the sum of momenta for the two vehicles. After the collision, p is the momentum associated with their coupled masses: (1.00  103 kg)(5.00 m/s)  (3.00  103 kg)(1.00 m/s)  (4.00  103 kg)v final v final  (5.00  103  3.00  103)(kg  m /s)/(4.00  103 kg)  2.00 m/s 5. (a). The momentum is conserved because there are no external forces acting on the pistol–bullet system during the firing. The initial momentum, however, is zero. The

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MCAT Skill Builder Study Guide

final momentum is the sum of the momentum of the bullet and the recoil momentum of the pistol: p before  p after

:

0  m pistolv pistol  m bulletv bullet

v pistol 
(0.50 g)(1.0  102 m/s)/0.20  103 g 
0.25 m/s 6. (c). The change in momentum is given by p  mvf
mvi , where vi is the initial velocity and vf is the rebound velocity. Because the collision is elastic, the ball rebounds in the opposite direction but with the same speed with which it hit the wall. Therefore, vf  2vi , and the change in momentum is p  m(
vi)
mvi 
2mvi 
2(0.20 kg)(20.0 m/s) 
8.0 kg  m/s The magnitude of a number is its absolute value:

8.0 kg  m/s
 8.0 kg  m/s 7. (d). Force is related to momentum by the impulse-momentum equation, F t p. Therefore, F  p/t  (4 kg  m/s)/0.01 s  400 kg  m/s2  400 N 8. (a). Momentum is conserved. Because both carts come to rest, the total momentum after the collision must be zero, which means that the total momentum prior to the collision was also zero. If the two carts are labeled A and B, then m AvA  m Bv B : v B  (30 kg)(5 m/s)/50 kg  3 m/s

WORK EXAMPLE 1 A car traveling at speed v can stop in distance d due to kinetic friction. At twice the speed, what stopping distance is required? (a) d (b) 2d (c) 4d (d) 8d Conceptual Solution The answer is (c). Answer (a) can be eliminated immediately because at double the velocity, the car’s energy must be greater than before and hence the stopping distance must be greater. Because kinetic energy is proportional to v 2, doubling the velocity quadruples the energy. Work done by friction is linear in distance d, so four times the distance is required. This result can more readily be seen through a proportion. Quantitative Solution Apply the work–energy theorem:

(1) W 5 DKE 5 KE f 2 KE i

Assume the car is traveling in the positive x-direction. Substitute expressions for work done by a kinetic friction force with magnitude Fk performed over a displacement x and for kinetic energy, noting that KEf  0:

(2)

2Fk Dx 5 0 2 12 mv 2

Multiply both sides by
1 and drop the zero term:

(3)

Fk Dx 5 12 mv 2

Denote the two possible different displacements by d1 and d2, the two velocities by v1 and v2. Substitute into Equation (3), creating one equation for the first case and a similar equation for the second case:

(4)

Fkd1 5 12 mv 12

(5)

Fkd2 5 12 mv 22

Now divide Equation (5) by Equation (4), canceling terms Fk, m, and 12: It follows, then, that answer (c) is correct:

1 2 Fkd2 2 mv 2 51 2 Fkd1 2 mv 1

d2  4d1

S

1 2v 2 2 d2 v 22 5 25 54 d1 v1 v2

Work

A.33

EXAMPLE 2 A truck of mass M and a car of mass m are traveling at the same speed. They both slam on their brakes, locking them and skidding along the road, which has friction coefficient mk . The car stops after going a distance d. Assuming mk is the same for both vehicles, which of the following is true of the distance D that the truck travels before stopping? (a) D  d (b) D d (c) D d Conceptual Solution Kinetic energy is directly proportional to mass, and because the truck is more massive than the car, its kinetic energy is greater and hence it might be thought that a greater braking distance would be required. The friction force, however, is also directly proportional to mass, so the friction force is proportionately greater for the truck than the car. All other things being equal, the truck and car stop in the same distance, D  d, which is answer (a). Quantitative Solution Write the expression for the force of friction, where n is the normal force:

(1)

Now use Equation (4) from Example 1:

(2)

Substitute the expressions for the kinetic friction force from Equation (1): The answer is (a), as expected:

Fk 5 mk n 5 mkmg Fk,truckD Fk,card

1 mkMg 2 D 1 mkmg 2 d

5

5

1 2 2 Mv 1 2 2 mv

1 2 2 Mv 1 2 2 mv

S

MD M 5 m md

S

D 51 d

D5d

EXAMPLE 3 A car undergoes uniform acceleration from rest. What can be said about the instantaneous power delivered by the engine? (a) The instantaneous power increases with increasing speed. (b) The instantaneous power remains the same. (c) The instantaneous power decreases with increasing speed. Conceptual Solution The correct answer is (a). Because the acceleration is uniform, the force is constant. The instantaneous power is given by
 Fv, so with increasing velocity the delivered power must also increase.

EXAMPLE 4 A circus stuntman is blasted straight up out of a cannon. If air resistance is negligible, which of the following is true about his total mechanical energy as he rises? (a) It increases. (b) It remains the same. (c) It decreases. Conceptual Solution In the absence of nonconservative forces, mechanical energy is conserved, so it will remain the same, and answer (b) is correct. Application If the stuntman has mass 65.0 kg and his initial velocity is 8.00 m/s, find the maximum height reached. Solution Apply conservation of mechanical energy:

DKE 1 DPE 5 0 1 2 2 mv f

2 12 mv i 2 1 mghf 2 mghi 5 0

The final velocity is zero, and initial height may be taken as zero:

0 2 12 mv i 2 1 mghf 2 0 5 0

Solve for hf and substitute values:

hf 5

1 8.00 m/s 2 2 vi2 5 3.27 m 5 2g 2 1 9.80 m/s 2 2

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Appendix E

MCAT Skill Builder Study Guide

MULTIPLE-CHOICE PROBLEMS 1. If the speed at which a car is traveling is tripled, by what factor does its kinetic energy increase? (a) 31/2 (b) 3 (c) 6 (d) 9 2. What is the magnitude of the force exerted by air on a plane if 500 kilowatts of power are needed to maintain a constant speed of 100 meters per second? (a) 5 N (b) 50 N (c) 500 N (d) 5 000 N 3. What happens to the speed of a body if its kinetic energy is doubled? (a) It increases by a factor of 21/2. (b) It is doubled. (c) It is halved. (d) It increases by a factor of 4. 4. A ball with a mass of 1.0 kg sits at the top of a 30° incline plane that is 20.0 meters long. If the potential energy of the ball relative the bottom of the incline is 98 J at the top of the incline, what is its potential energy once it rolls halfway down the incline? (a) 0 J (b) 49 J (c) 98 J (d) 196 J 5. How much work is done by a horizontal force of 20.0 N when applied to a mass of 0.500 kg over a distance of 10.0 m? (a) 5.00 J (b) 10.0 J (c) 49.0 J (d) 2.00  102 J 6. A body located 10.0 meters above the surface of the Earth has a gravitational potential energy of 4.90  102 J relative to the Earth’s surface. What is the new gravitational potential energy if the body drops to a height of 7.00 m above the Earth? (a) 70.0 J (b) 147 J (c) 281 J (d) 343 J 7. Cart A has a mass of 1.0 kg and a constant velocity of 3.0 m/s. Cart B has a mass of 1.5 kg and a constant velocity of 2.0 m/s. Which of the following statements is true? (a) Cart A has the greater kinetic energy. (b) Cart B has the greater kinetic energy. (c) Cart A has the greater acceleration. (d) Cart B has the greater acceleration. 8. The work done in raising a body must (a) increase the kinetic energy of the body. (b) decrease the total mechanical energy of the body. (c) decrease the internal energy of the body. (d) increase the gravitational potential energy. 9. What is the average power output of a 50.0-kg woman who climbs a 2.00-m step ladder at constant speed in 10.0 seconds? (a) 10.0 W (b) 49.0 W (c) 98.0 W (d) 2.50  102 W

Answers 1. (d). The kinetic energy is directly proportional to the square of the speed. If the speed is tripled, its square becomes (3v)2  9v 2 and KE must increase by a factor of nine. 2. (d). The instantaneous power is the product of the force acting on the plane and the speed,
 Fv. Solving for the required force gives F 
/v  500  103 W/(100 m/s)  5  103 N 3. (a). Let KEi be the initial kinetic energy, and KEf the final kinetic energy. Make a proportion: KE f KE i

5

1 2 2 mv f 1 2 2 mv i

S

1 2KE i 2 KE i

5

1 2 2 mv f 1 2 2 mv i

S

25

vf 2 vi 2

Solve for vf , getting v f 5 !2v i. 4. (b). Because the height is decreasing, the gravitational potential energy is decreasing and must have a value less than its initial reading of 98 J. Choice (d) can be eliminated because its value is higher than the initial value of the potential energy, as can choice (c) because its value is the same as the initial value of the potential energy. Choice (a) can be eliminated because, by the problem statement, the gravitational potential energy is zero at the bottom of the ramp. That leaves (b) as the only option. Quantitatively: PE  mgh  (1.0 kg)(9.8 m/s2)[(10.0 m)  sin (30°)]  49 J 5. (d). W  F  s  (20.0 N)(10.0 m)  2.00  102 J 6. (d). The gravitational potential energy and the height of the body above the ground are directly proportional: PE  mgh. Reducing the height by a factor of 7/10 must reduce the gravitational potential energy by the same factor: (4.90  102 J)(0.700)  343 J. 7. (a). Because the velocity is constant, the acceleration must be zero and we can eliminate choices (c) and (d). Solving KE  12 mv 2 for the two carts yields KE A 5 12 1 1.0 kg 2 1 3.0 m/s 2 2 5 4.5 J

KE B 5 12 1 1.5 kg 2 1 4.0 m/s 2 2 5 3.0 J

Matter

A.35

8. (d). By definition, gravitational potential energy changes with the height of a body above the ground. 9. (c). The average power is the work done on an object divided by the length of the time interval,
 W/t. The work done here is against gravity. In the period under consideration, the woman must increase her mechanical energy by an amount equal to her change in gravitational potential energy, so
 mgh/t  (50.0 kg)(9.80 m/s2) (2.00 m)/10.0 s  98.0 W.

MATTER EXAMPLE 1 Tank A is filled with water, and, in an identical tank B, the bottom half is filled with water and the top half with oil. For which tank is the pressure at the bottom of the tank the greatest? (a) Tank A (b) Tank B (c) The pressure in both tanks is the same. Conceptual Solution The oil in tank B floats on top of the water because it’s less dense than water. The total weight of the fluid in tank B must therefore be less than an equal volume of water alone. Consequently, the fluid weight per unit area is greater at the bottom of tank A, and the answer is (a).

EXAMPLE 2 A steel cable with cross-sectional area A and length L stretches by length L1 when suspending a given weight. A second steel cable with half the cross-sectional area and three times the length is used to support the same weight, and stretches by an amount L2. What is the ratio L2/L1? (a) 2 (b) 4 (c) 6 (d) 8 Conceptual Solution Tensile stress, a force per unit area, is proportional to tensile strain, which is the fractional change in length of an object or the change in length divided by the original length. The amount the cable stretches is thus inversely proportional to the cross-sectional area and directly proportional to the length. Hence, the second cable, being three times as long, should stretch three times as far under a given load. With only half the cross-sectional area, it should stretch farther by a factor of 1 12 2 21, or 2. Taken together, the second cable should stretch six times as far as the first shorter and thicker cable, so L2/L1  6, which is answer (c). Quantitative Solution Write the equation relating tensile stress and strain for the first cable: Rewrite Equation (1) for the second cable, noting that Y, Young’s modulus, is constant for a given material:

Divide Equation (2) by Equation (1), canceling Y:

Cancel F 2 and F 1 in Equation (3) because they are equal in this case, rearrange the equation, and substitute L2  3L1 and A 2  12A1: The answer is (c), as expected.

(1)

DL 1 F1 5Y A1 L1

(2)

DL 2 F2 5Y A2 L2

(3)

F2 DL 2 A2 L 5 2 F1 DL 1 A1 L1

S

DL 2 L 1 F2 A 1 5 F1 A2 DL 1 L 2

DL 2 A1 L 2 A1 3L 1 3 5 51 5 1 5 6 DL 1 A2 L 1 2 A1 L 1 2

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Appendix E

MCAT Skill Builder Study Guide

EXAMPLE 3

An object floats with three-quarters of its volume submerged in water. What is its density? 1 r water 5 1.00 3 103 kg/m3 2 (a) 1.00  103 kg/m3 (b) 8.70  102 kg/m3 (c) 7.50  102 kg/m3 (d) 1.25  103 kg/m3 Conceptual Solution Choice (d) can be eliminated immediately because the object would sink if it were more dense than water. If choice (a) were true, the object would be floating but would be completely submerged. These considerations leave only choices (b) and (c). The buoyant force depends on the amount of fluid displaced, and this force must exactly balance the gravitational force, or weight. Because the object is only three-quarters submerged, its weight must be only three-quarters that of a similar volume of water. It follows that the density of the object must be three-quarters that of water, which is answer (c). Quantitative Solution Neglecting the buoyancy of air, two forces act on the object: the buoyant force FB , which is equal to the weight of displaced fluid, and the gravitational force:

S

S

S

a F 5 F B 1 F grav 5 0

(1)

FB
mg  0

By definition, the mass of the object is given by m  robjV, where robj is the object’s density and V is its volume. The buoyant force is the weight of the displaced water: FB  rwaterVsubg, where Vsub is the volume of water displaced. Substitute these expressions into Equation (1):

(2)

rwaterVsub g
robjV g  0

Solve Equation (2) for robj, obtaining answer (c):

robj 5

Substitute the expressions for the buoyant and gravitational forces:

Vsub r 5 1 0.750 2 1 1.00 3 103 kg/m3 2 V water

 7.50 3 102 kg/m3

EXAMPLE 4 At point 1, water flows smoothly at speed v1 through a horizontal pipe with radius r 1. The pipe then narrows to half that radius at point 2. What can be said of the speed v1 of the water at point 1 compared with its speed v 2 at point 2? (a) v1  v 2 (b) v1 v 2 (c) v1 v 2 Conceptual Solution The volume flow rate is proportional to the velocity and cross-sectional area of the pipe. A larger radius at point 1 means a larger cross-sectional area. Because water is essentially incompressible, the flow rate must be the same in both sections of pipe, so the larger cross-section at point 1 results in a smaller fluid velocity and the answer is (c). Quantitative Solution Apply the equation of continuity for an incompressible fluid:

(1) A1v1  A 2v 2

Substitute an expression for the area on each side of Equation (1):

(2)

Solve Equation (2) for v1, and substitute r 2 5 12 r 1, obtaining answer (c):

v1 5

pr 12v1  pr 22v 2 r 22 r 12

v2 5

r 22

1 2r 2 2 2

v2 5

1 4v2

Matter

MULTIPLE-CHOICE PROBLEMS 1. A diver is swimming 10.0 m below the surface of the water in a reservoir. There is no current, the air has a pressure of 1.00 atmosphere, and the density of the water is 1.00  103 kilograms per cubic meter. What is the pressure as measured by the diver? (a) 1.10 atm (b) 1.99  105 Pa (c) 11.0 atm (d) 1.01  105 Pa 2. The aorta of a 70.0-kg man has a cross-sectional area of 3.00 cm2 and carries blood with a speed of 30.0 cm/s. What is the average volume flow rate? (a) 10.0 cm/s (b) 33.0 cm3/s (c) 10.0 cm2/s (d) 90.0 cm3/s 3. At 20.0°C the density of water is 1.00 g/cm3. What is the density of a body that has a weight of 0.980 N in air but registers an apparent weight of only 0.245 N on a spring scale when fully immersed in water? (a) 0.245 g/cm3 (b) 0.735 g/cm3 (c) 1.33 g/cm3 (d) 4.00 g/cm3 4. Two insoluble bodies, A and B, appear to lose the same amount of weight when submerged in alcohol. Which statement is most applicable? (a) Both bodies have the same mass in air. (b) Both bodies have the same volume. (c) Both bodies have the same density. (d) Both bodies have the same weight in air. 5. The bottom of each foot of an 80-kg man has an area of about 400 cm2. What is the effect of his wearing snowshoes with an area of about 0.400 m2? (a) The pressure exerted on the snow becomes 10 times as great. (b) The pressure exerted on the snow becomes 1/10 as great. (c) The pressure exerted on the snow remains the same. (d) The force exerted on the snow is 1/10 as great. 6. In a hydraulic lift, the surface of the input piston is 10 cm2 and that of the output piston is 3 000 cm2. What is the work done if a 100 N force applied to the input piston raises the output piston by 2.0 m? (a) 20 kJ (b) 30 kJ (c) 40 kJ (d) 60 kJ 7. Young’s modulus for steel is 2.0  1011 N/m2. What is the stress on a steel rod that is 100 cm long and 20 mm in diameter when it is stretched by a force of 6.3  103 N? (a) 2.01  107 N/m2 (b) 12.6  1012 N/m2 (c) 3.15  108 N/m2 (d) 4.0  1011 N/m2

Answers 1. (b). The fluid is at rest (no currents), so this problem is a hydrostatic pressure calculation. In SI units 1 atm  1.01  105 Pa. Choice (d) can be eliminated because the pressure below the water must be greater than the pressure at the surface. Pdiver  Patm  rgh  (1.01  105 Pa) (1.00  103 kg/m3)(9.80 m/s2)(10.0 m)  1.99  105 Pa 2. (d). The answer follows from the definition: Volume flow rate  vA  (30 cm/s)(3 cm2)  90 cm3/s 3. (c). This problem is an application of Archimedes’ principle. The apparent loss of weight of a submerged body equals the weight of the fluid displaced. The weight w of the displaced water is w  0.980 N
0.245 N  0.735 N. Using the definition of density, rwaterg  w/V, so the volume occupied by this weight of water is V  0.735 N/rwaterg. The volume of the body must equal the volume of the water displaced. The density of the body is r  m/V  (0.980 N/0.735 N)rwater  1.33 g/cm3. 4. (b). According to Archimedes’ principle, the apparent weight lost is equal to the weight of the displaced fluid; therefore, both must have the same volume because the volume of the fluid equals the volume of the body. 5. (b). The force exerted by the man is his weight and it is assumed to be constant, which eliminates choice (d). For a constant force, pressure and area are inversely proportional. The area of the snowshoes is ten times the area of the foot, so the pressure associated with the snowshoes is the inverse of 10, or 1/10 the pressure exerted by the foot. 6. (d). By Pascal’s principle, the pressure at the input and output pistons is the same, so Fin Fout 5 Ain Aout

S

Fout 5

Aout F Ain in

3 000 cm2 b 100 N 5 3 3 104 N 10 cm2 Work is a force times a displacement: 5a

W 5 F Ds 5 1 3 3 104 N 2 1 2 m 2 5 6 3 104 J 5 60 kJ

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Appendix E

MCAT Skill Builder Study Guide

7. (a). Stress is force per unit area, so neither Young’s modulus nor the length of the rod are needed to solve the problem. Stress  F/A  (6.30  103 N)/p(1  10
2 m)2  2.01  107 N/m2

WAVES EXAMPLE 1 A transverse wave travels at speed v1 and has twice the frequency and one-quarter the wavelength of a second transverse wave. (They could be waves on two different strings, for example.) How does the speed v1 of the first wave compare with the speed v 2 of the second wave? (a) v1  v 2 (b) v1  2v 2 (c) v1  12v 2 Solution The speed of a wave is proportional to both the frequency and wavelength: Make a ratio of v 2 to v1 using Equation (1) and substitute f 1  2f 2 and l1  l2/4: Solve for v1, obtaining answer (c):

(1) v  f l 1 2f2 2 1 l2 /4 2 f 1l 1 v1 2 1 5 5 5 5 v2 f 2l 2 4 2 f2l 2 v1 

1 2

v2

EXAMPLE 2 A block of mass m oscillates at the end of a horizontal spring on a frictionless surface. At maximum extension, an identical block drops onto the top of the first block and sticks to it. How does the new period Tnew compare to the original period, T0? (a) Tnew 5 !2T0 (b) Tnew 5 T0 (c) Tnew 5 2T0 (d) Tnew 5 12 T0 Conceptual Solution An increased mass would increase the inertia of the system without augmenting the mechanical energy, so a mass of 2m should move more slowly than a mass of m. That in turn would lengthen the period, eliminating choices (b) and (d). The period is proportional to the square root of the mass, so doubling the mass increases the period by a factor of the square root of 2, which is answer (a). Quantitative Solution Apply the equation for the period of a mass–spring system:

Using Equation (1), make a ratio of the new period Tnew to the old period, T0, canceling common terms and substituting m new  2m 0: Solving Equation (2) for the new period Tnew yields answer (a):

m (1) T 5 2p Åk

(2)

Tnew 5 T0

2p

m new Å k

2p

m0 Å k

5

!m new !m 0

5

2m 0 5 !2 Å m0

Tnew 5 !2 T0 [answer (a)]

EXAMPLE 3 A simple pendulum swings back and forth 36 times in 68 s. What is its length if the local acceleration of gravity is 9.80 m/s2? (a) 0.067 7 m (b) 0.356 m (c) 1.25 m (d) 0.887 m Solution First, find the period T of the motion:

T5

68 s 5 1.89 s 36

Waves

Write the equation for the period of a pendulum and solve it for the length L, obtaining answer (d):

T 5 2p

L Åg

S

L5

4p2

 0.887 m [answer (d)]

MULTIPLE-CHOICE PROBLEMS 1. A simple pendulum has a period of 4.63 s at a place on the Earth where the acceleration of gravity is 9.82 m/s2. At a different location, the period increases to 4.64 s. What is the value of g at this second point? (a) 9.78 m/s2 (b) 9.82 m/s2 (c) 9.86 m/s2 (d) Cannot be determined without knowing the length of the pendulum. 2. What is the wavelength of a transverse wave having a speed of 15 m/s and a frequency of 5.0 Hz? (a) 3.0 m (b) 10 m (c) 20 m (d) 45 m 3. What is the optimum difference in phase for maximum destructive interference between two waves of the same frequency? (a) 360° (b) 270° (c) 180° (d) 90° 4. Standing waves can be formed if coincident waves have (a) the same direction of propagation. (b) the same frequency. (c) different amplitudes. (d) different wavelengths. 5. A simple pendulum with a length L has a period of 2 s. For the pendulum to have a period of 4 s, we must (a) halve the length. (b) quarter the length. (c) double the length. (d) quadruple the length. 6. If a simple pendulum 12 m long has a frequency of 0.25 Hz, what will be the period of a second pendulum at the same location if its length is 3.0 m? (a) 2.0 s (b) 3.0 s (c) 4.0 s (d) 6.0 s 7. A pendulum clock runs too slowly (i.e., is losing time). Which of the following adjustments could rectify the problem? (a) The weight of the bob should be decreased so it can move faster. (b) The length of the wire holding the bob should be shortened. (c) The amplitude of the swing should be reduced so the path covered is shorter. (d) None of the above. 8. A 20.0-kg object placed on a frictionless floor is attached to a wall by a spring. A 5.00-N force horizontally displaces the object 1.00 m from its equilibrium position. What is the period of oscillation of the object? (a) 2.00 s (b) 6.08 s (c) 12.6 s (d) 16.4 s

Answers 1. (a). The answer can be determined without doing a numerical solution. Rearrange T  2p(L/g)1/2 to Tg 1/2  constant. The period is inversely related to the square root of the acceleration due to gravity. Because T has increased, g 1/2 and hence g must decrease. Choice (a) is the only value of g that is less than the original 9.82 m/s2. Quantitatively, g 2  (T12g 1/T22)  (4.63 s)2(9.82 m/s2)/(4.64 s)2  9.78 m/s2 2. (a). Wavelength is velocity divided by frequency. The formula does not depend upon the type of wave involved. l  v/f  (15 m/s)/5.0 s
1  3.0 m 3. (c). Two waves are completely out of phase when their antinodes coincide so that each crest on one wave coincides with a trough on the other. This situation occurs when the waves differ in phase by 180°. 4. (b). In standing waves, the nodes are stationary, which can be accomplished when two waves with the same frequency travel in opposite directions. 5. (d). In a pendulum the period and the square root of the length are directly proportional: T  2p(L/g)1/2 so that T/L1/2  constant To double the period, you must double the square root of the length. To double the square root of the length, you must quadruple the length: (4L)1/2  41/2 L1/2  2L1/2

gT 2

5

A.39

1 9.80 m/s 2 2 1 1.89 s 2 2 4p2

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Appendix E

MCAT Skill Builder Study Guide

6. (a). The frequency is the reciprocal of the period, f  1/T, so the first pendulum has period T1  4.0 s. For pendulums, the period and the square root of the length are directly proportional, so the ratio of the two periods is T2/T1  (L 2/L1)1/2  (12/3)1/2  (4.0)1/2  2.0 It follows that T2  2.0 s. 7. (b). The period of a pendulum is directly related to the square root of the length of the cord holding the bob. It is independent of the mass and amplitude. 8. (c). The force constant is k  F/x  5.0 N/1.0 m  5.0 N  m
1. The period is T  2p(m/k)1/2  2p(20.0 kg/5.00 N  m
1)1/2  12.6 s

SOUND EXAMPLE 1 When a given sound intensity doubles, by how much does the sound intensity level (or decibel level) increase? Solution Write the expression for a difference in decibel level, b: Factor and apply the logarithm rule log a
log b  log(a/b)  log(a  b
1):

Substitute I2  2I1 and evaluate the expression:

Db 5 b2 2 b1 5 10 log a Db 5 10 c log a 5 10 log a Db 5 10 log a

I2 I1 b 210 log a b I0 I0

I2 I2 I0 I1 b 2 log a b d 5 10 log a # b I0 I0 I0 I1

I2 b I1 2I 1 b 5 10 log 1 2 2 5 3.01 dB I1

EXAMPLE 2 A sound wave passes from air into water. What can be said of its wavelength in water compared with air? (a) The wavelengths are the same. (b) The wavelength in air is greater than in water. (c) The wavelength in air is less than in water. Conceptual Solution The frequency of the sound doesn’t change in going from one type of media to the next because it’s caused by periodic variations of pressure in time. The wavelength must change, however, because the speed of sound changes and during a single period the sound wave will travel a different distance, which corresponds to a single wavelength. The speed of sound in water is greater than in air, so in a single period the sound wave will travel a greater distance. Hence, the wavelength of a given sound wave is greater in water than in air, which is (c). This result can be made quantitative by using v  fl, where v is the wave speed, f the frequency, and l the wavelength.

EXAMPLE 3 A sound wave emitted from a sonar device reflects off a submarine that is traveling away from the sonar source. How does the frequency of the reflected wave, f R , compare with the frequency of the source, f S ? (a) f R  f S (b) f R f S (c) f R f S Solution The reflected wave will have a lower frequency [answer (b)]. To see why, consider pressure maximums impinging sequentially on the submarine. The first pressure maximum hits the submarine and reflects, but the second reaches the submarine when it has moved farther away. The distance between consecutive maximums in the reflected waves is thereby increased; hence the wavelength also increases. Because v  fl and the speed isn’t affected, the frequency must decrease.

Sound

MULTIPLE-CHOICE PROBLEMS 1. The foghorn of a ship echoes off an iceberg in the distance. If the echo is heard 5.00 seconds after the horn is sounded and the air temperature is
15.0°C, how far away is the iceberg? (a) 224 m (b) 805 m (c) 827 m (d) 930 m 2. What is the sound level of a wave with an intensity of 10
3 W/m2 ? (a) 30 dB (b) 60 dB (c) 90 dB (d) 120 dB 3. At 0°C, approximately how long does it take sound to travel 5.00 km through air? (a) 15 s (b) 30 s (c) 45 s (d) 60 s 4. If the speed of a transverse wave of a violin string is 12.0 m/s and the frequency played is 4.00 Hz, what is the wavelength of the sound in air? (Use 343 m/s for the speed of sound.) (a) 48.0 m (b) 12.0 m (c) 3.00 m (d) 85.8 m 5. If two identical sound waves interact in phase, the resulting wave will have a (a) shorter period. (b) larger amplitude. (c) higher frequency. (d) greater velocity. 6. What is the speed of a longitudinal sound wave in a steel rod if Young’s modulus for steel is 2.0  1011 N/m2 and the density of steel is 8.0  103 kg/m3? (a) 4.0  10
8 m/s (b) 5.0  103 m/s (c) 25  10 6 m/s (d) 2.5  109 m/s 7. If two frequencies emitted from two sources are 48 Hz and 54 Hz, how many beats per second are heard? (a) 3 (b) 6 (c) 9 (d) 12 8. The frequency registered by a detector is higher than the frequency emitted by the source. Which of the statements below must be true? (a) The source must be moving away from the detector. (b) The source must be moving toward the detector. (c) The distance between the source and the detector must be decreasing. (d) The detector must be moving away from the source.

Answers 1. (b). The normal speed of sound at 0°C in air is 331 m/s. The speed of sound in air at various temperatures is given by v 5 1 331 m/s 2

273 K 2 15.0 K T 5 1 331 m/s 2 Å 273 K Å 273 K

5 322 m/s Calculate the distance traveled in half of 5 seconds: d  vt  (322 m/s)(2.50 s)  805 m 2. (c). Substitute: b  10 log I/I 0  10 log (1  10
3 W/m2/10
12 W/m2)  90 dB. 3. (a). At 0°C, the speed of sound in air is 331 m/s, so t  (5.00  103 m)/(331 m/s)  15 s. 4. (d). The frequency is the same for the string as for the sound wave the string produces in the surrounding air, but the wavelength differs. The speed of sound is equal to the product of the frequency and wavelength, so l  v/f  (343 m/s)/4.00 s
1  85.8 m. 5. (b). Two waves are in phase if their crests and troughs coincide. The amplitude of the resulting wave is the algebraic sum of the amplitudes of the two waves being superposed at that point, so the amplitude is doubled. 6. (b). The solution requires substituting into an expression for the velocity of sound through a rod of solid material having Young’s modulus Y: v  (Y/r)1/2 (2.0  1011 kg  m/s2)/(8.0  103 kg/m3)1/2  25  106 m2/s2)1/2  5.0  103 m/s 7. (b). f beat  f 1
f 2  54
48  6 beats 8. (c). Movement of a sound source relative to an observer gives rise to the Doppler effect. Because the frequency is shifted to a higher value, the source and the detector must be getting closer together, effectively shortening the wavelength from the observer’s point of view. This observation eliminates choices (a) and (d). Choice (b) can be eliminated because it isn’t necessarily true: the same kind of effect occurs when the source is held steady and the detector moves towards it.

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LIGHT EXAMPLE 1 Which substance will have smaller critical angle for total internal reflection, glass with an index of refraction of 1.50 or diamond with an index of refraction of 2.42? (a) glass (b) diamond (c) The critical angles are the same for both. Conceptual Solution A larger index of refraction indicates a slower speed of light inside the material, which in turn means a larger refraction angle, or a larger bending towards the normal on entering the material from air and a larger bending away from the normal on passing from the material into air. (Recall that a normal line is perpendicular to the surface of the material.) For total internal reflection to occur, a refraction angle of 90° must be possible, which occurs whenever the refracting medium has a lower index of refraction than the incident medium. Diamond bends light more than glass, so the incident ray can be closer to the normal and still be bent enough so the angle of refraction results in total internal reflection. Closer to the normal means a smaller critical angle, so the answer is (b). Quantitative Solution Write Snell’s law for the diamond–air interface:

nD sin u D 5 nA sin u A

Compute the critical angle for diamond, using n A  1.00 for air and uA  90°, together with nD  2.42:

2.42 sin u D 5 1.00

S

u D 5 sin21 a

1.00 b 5 24.4° 2.42

Repeat the calculation for glass:

1.50 sin u G 5 1.00

S

u G 5 sin21 a

1.00 b 5 41.8° 1.50

The calculation explicitly shows that diamond has a smaller critical angle than glass [answer (b)].

EXAMPLE 2 A patient’s near point is 85.0 cm. What focal length prescription lens will allow the patient to see objects clearly that are at a distance of 25.0 cm from the eye? Neglect the eye–lens distance. Solution Use the thin lens equation:

An object at distance p  25.0 cm must form an image at the patient’s near point of 85.0 cm. The image must be virtual, so q 
85.0 cm:

1 1 1 5 1 q f p 1 1 1 5 1 5 2.82 3 1022 cm21 f 25.0 cm 285.0 cm f 5 35.4 cm

EXAMPLE 3 Two photons traveling in vacuum have different wavelengths. Which of the following statements is true? (a) The photon with a smaller wavelength has greater energy. (b) The photon with greater wavelength has greater energy. (c) The energy of all photons is the same. (d) The photon with the greater wavelength travels at a lesser speed. Solution Answer (d) can be eliminated immediately because the speed of light in vacuum is the same for all wavelengths of light. The energy of a photon, or particle of light, is given by E  hf and consequently is proportional to the frequency f, which in turn is inversely proportional to the wavelength because f = c/l, where c is the speed of light. A smaller wavelength photon has a greater frequency and therefore a greater energy, so answer (a) is true.

Light

MULTIPLE-CHOICE PROBLEMS 1. Glass has an index of refraction of 1.50. What is the frequency of light that has a wavelength of 5.00  102 nm in glass? (a) 1.00 Hz (b) 2.25 Hz (c) 4.00  1014 Hz (d) 9.00  1016 Hz 2. Water has an index of refraction of 1.33. If a plane mirror is submerged in water, what can be said of the angle of reflection u if light strikes the mirror with an angle of incidence of 30°? (a) u 30°(b) u  30° (c) 30° u (d) No light is reflected because 30° is the critical angle for water. 3. The index of refraction for water is 1.33 and that for glass is 1.50. A light ray strikes the water–glass boundary with an incident angle of 30.0° on the water side. Which of the following is the refraction angle in the glass? (a) 26.3° (b) 34.7° (c) 30.0° (d) 60.0° 4. Light is incident on a prism at an angle of 90° relative to its surface. The index of refraction of the prism material is 1.50. Which of the following statements is most accurate about the angle of refraction u? (a) 0° u 45°(b) 45° u 90°(c) u  0° (d) 90° u 5. White light incident on an air–glass interface is split into a spectrum within the glass. Which color light has the greatest angle of refraction? (a) red light (b) violet light (c) yellow light (d) The angle is the same for all wavelengths. 6. A real object is placed 10.0 cm from a converging lens that has a focal length of 6.00 cm. Which statement is most accurate? (a) The image is real, upright, and enlarged. (b) The image is real, inverted, and enlarged. (c) The image is real, upright, and reduced. (d) The image is real, inverted, and reduced. 7. What is the focal length of a lens that forms a virtual image 30.0 cm from the lens when a real object is placed 15.0 cm from the lens? (a) 10.0 cm (b) 15.0 cm (c) 30.0 cm (d) 45.0 cm 8. What is the magnification of a lens that forms an image 20.0 cm to its right when a real object is placed 10.0 cm to its left? (a) 0.500 (b) 1.00 (c) 1.50 (d)
2.00 9. The human eye can respond to light with a total energy of as little as 10
18 J. If red light has a wavelength of 600 nm, what is the minimum number of red light photons the eye can perceive? (a) 1 (b) 2 (c) 3 (d) 5 10. Which phenomenon occurs for transverse waves but not for longitudinal waves? (a) reflection (b) refraction (c) diffraction (d) polarization

Answers 1. (c). The velocity of light in glass can be found from the definition of refractive index, n  c/v; wavelength and frequency are related to velocity by the general wave relation, v  f l. Therefore f  v/l  c/nl  (3.00  108 m/s)/(1.50)(5.00  10
7 m)  4.00  1014 Hz 2. (b). The law of reflection is independent of the medium involved. The angle of reflection is always equal to the angle of incidence. 3. (a). Snell’s law is given by n1 sin u1  n 2 sin u2; hence, 1.33 sin 30.0°  1.50 sin u2. From this expression, we see that sin u2 sin 30.0o, so u2 must be less than 30.0°. Therefore, (a) is the only reasonable choice. 4. (c). Snell’s law is n1 sin u1 n 2 sin u2. Because the incident rays are normal to the prism surface, u1  0° and sin 0°  0. Consequently, 0  n 2 sin u2 : u2  0°. 5. (b). The greater the frequency of light, the greater its energy and the faster its speed through any material medium. From Snell’s law, the velocity and sin u with respect to the normal are inversely proportional. Of the choices, violet light has the highest frequency and therefore the highest velocity and the greatest angle of refraction. 6. (b). By the thin lens equation, 1/f  1/p  1/q; hence, with p  10.0 cm and f  6.00 cm, substitution results in q  15.0 cm. The image distance is positive; hence, the image is real. The magnification is given by M 
q/p 
15.0 cm/10.0 
1.50. Because M is negative, the image is inverted, whereas
M
1 means that the image is enlarged.

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7. (c). The focal length is given by the thin lens equation, 1/f  1/p  1/q. Because the image formed is virtual, the sign of q is negative. Therefore, 1 1 1 1 1 1 5 1 5 1 5 q 30.0 cm f p 15.0 cm 230.0 cm

S

f 5 30.0 cm

8. (d). Magnification is given by M 
q/p 
20.0 cm/10.0 cm 
2.00. The sign of q is positive because the image is real. The negative value of M means that the image is inverted. 9. (c). E  hc/l  (6.63  10
34 J s)(3.00  108 m/s)/(6.00  10
7 m)  3.31  10
19 J. This result is the energy of each red photon. The number of such photons needed to produce a total of 10
18 J of energy is (10
18 J)/(3.31  10
19 J/photon)  3 photons 10. (d). Polarization can only occur with transverse waves because the motion must be perpendicular to the direction of propagation.

ELECTROSTATICS EXAMPLE 1 Two protons, each of charge qp  e, exert an electric force of magnitude Fp
p on each other when they are a distance rp apart. A pair of alpha particles, each of charge q a  2e, exert an electric force Fa
a  14 Fp
p on each other. What is the distance between the alpha particles, ra, in terms of the distance between the protons, rp ? (a) ra  4rp (b) ra  2rp (c) ra  rp (d) More information is needed. Solution Use Coulomb’s law to find an expression for the force between the protons:

(1)

Fp2p 5

Use Coulomb’s law to find an expression for the force between the alpha particles:

(2)

Fa2a 5

Fp2p

k e qpqp rp2 k e qa qa ra2

5

kee2 rp2

5

k e 1 2e 2 2 4k e e 2 5 ra2 ra2

kee2 rp2 ra2 5 2 5 4k e e 4r p 2 2 ra

Divide Equation (2) by Equation (1) and cancel common terms:

(3)

Now substitute Fa
a  14 Fp
p and solve for ra2:

Fp2p Fp2p ra2 51 54 2 5 Fa2a 4r p 4 Fp2p

Take square roots, obtaining ra in terms of rp:

r a 5 4r p

Fa2a

S

r a 2 5 16r p 2

The distance between the alpha particles is four times the distance between the protons, which is answer (a).

EXAMPLE 2 Sphere A has twice the radius of a second, very distant sphere B. Let the electric potential at infinity be taken as zero. If the electric potential at the surface of sphere A is the same as at the surface of sphere B, what can be said of the charge Q A on sphere A compared with charge Q B on B ? (a) Q A  2Q B (b) Q A  Q B (c) Q A  Q B/2 Conceptual Solution By Gauss’s law, a spherical distribution of charge creates an electric field outside the sphere as if all the charge were concentrated as a point charge at the center of the sphere. The electric potential due to a point charge is proportional to the charge Q and inversely proportional to the distance from that charge. Twice the radius reduces the electric potential at the surface of A by a factor of one-half. The charge of sphere A must be twice that of B so that the electric potentials will be the same for both spheres. Hence, the answer is (a).

Electrostatics

Quantitative Solution Write the equation for the electric potential of a point charge q:

A.45

keQ r

(1)

V5

Make a ratio of Equation (1) for charge A and charge B, respectively:

(2)

k eQ A Q Ar B rA VA 5 5 k eQ B VB Q Br A rB

Substitute VA  V B and rA  2rB into Equation (2) and solve for QA , again obtaining [answer (a)]:

15

Q Ar B Q B 1 2r B 2

S

Q A 5 2Q B [answer (a)]

EXAMPLE 3 How much work is required to bring a proton with charge 1.6  10
19 C and an alpha particle with charge 3.2  10
19 C from rest at a great distance (effectively infinity) to rest positions a distance of 1.00  10
15 m away from each other? Solution Use the work–energy theorem: The velocities are zero both initially and finally, so the kinetic energies are zero. Substitute values into the potential energy and find the necessary work to assemble the configuration:

W 5 DKE 1 DPE 5 KE f 2 KE i 1 PE f 2 PE i W50201 5

k e qp qa r

20

1 9.00 3 109 kg # m3 /C2 # s 2 2 1 1.60 3 10219 C 2 1 3.20 3 10219 C 2 1.00 3 10215 m

 4.61 3 10

213

J

EXAMPLE 4 A fixed, constant electric field E accelerates a proton from rest through a displacement s. A fully ionized lithium atom with three times the charge of the proton accelerates through the same constant electric field and displacement. Which of the following is true of the kinetic energies of the particles? (a) The kinetic energies of the two particles are the same. (b) The kinetic energy of the proton is larger. (c) The kinetic energy of the lithium ion is larger. Solution The work done by the electric field on a particle of charge q is given by W  F s  qE s. The electric field E and displacement s are the same for both particles, so the field does three times as much work on the lithium ion. The work–energy theorem for this physical context is W  KE, so the lithium ion’s kinetic energy is three times that of the proton and the answer is (c).

MULTIPLE-CHOICE PROBLEMS 1. What is the potential difference between point A and point B if 10.0 J of work is required to move a charge of 4.00 C from one point to the other? (a) 0.400 V (b) 2.50 V (c) 14.0 V (d) 40.0 V 2. How much work would have to be done by a nonconservative force in moving an electron through a positive potential difference of 2.0  106 V? Assume the electron is at rest both initially and at its final position. (a) 3.2  10
13 J (b)
8.0  10
26 J (c) 1.25  10
5 J (d)
3.2  10
13 J 3. Two electrically neutral materials are rubbed together. One acquires a net positive charge. The other must have (a) lost electrons. (b) gained electrons. (c) lost protons. (d) gained protons. 4. What is the magnitude of the charge on a body that has an excess of 20 electrons? (a) 3.2  10
18 C (b) 1.6  10
18 C (c) 3.2  10
19 C (d) 2.4  10
19 C

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5. Two point charges, A and B, with charges of 2.00  10
4 C and
4.00  10
4 C, respectively, are separated by a distance of 6.00 m. What is the magnitude of the electrostatic force exerted on charge A? (a) 2.20  10
9 N (b) 1.30 N (c) 20.0 N (d) 36.0 N 6. Two point charges, A and B, are separated by 10.0 m. If the distance between them is reduced to 5.00 m, the force exerted on each (a) decreases to one-half its original value. (b) increases to twice its original value. (c) decreases to one-quarter of its original value. (d) increases to four times its original value. 7. Sphere A with a net charge of 3.0  10
3 C is touched to a second sphere B, which has a net charge of
9.0  10
3 C. The two spheres, which are exactly the same size and composition, are then separated. The net charge on sphere A is now (a) 3.0  10
3 C (b)
3.0  10
3 C (c)
6.0  10
3 C (d)
9.0  10
3 C 8. If the charge on a particle in an electric field is reduced to half its original value, the force exerted on the particle by the field is (a) doubled. (b) halved. (c) quadrupled. (d) unchanged. 9. In the figure below, points A, B, and C are at various distances from a given point charge. A

B

+ C

Which statement is most accurate? The electric field strength is (a) greatest at point A. (b) greatest at point B. (c) greatest at point C. (d) the same at all three points. 10. The electrostatic force between two point charges is F. If the charge of one point charge is doubled and that of the other charge is quadrupled, the force becomes which of the following? (a) F/2 (b) 2F (c) 4F (d) 8F

Answers 1. (b). Potential difference between two points in an electric field is the work per unit charge required to move a charge between the two points: V  W/q  10.0 J/4.00 C  2.50 V 2. (d). If the only effect on the particle is a change of position, negative work must be done; otherwise, negative charges gain kinetic energy on moving through a positive potential difference. W  KE  q V  0  (
1.6  10
19 C)(2.0  106 V) 3.

4. 5.

6.

7.


3.2  10
13 J (b). Protons are fixed in the nucleus and cannot be transferred by friction. Electrons can be transferred by friction. Therefore, net charges are due to the transfer of electrons between two bodies. Conservation of charge means that if there is a net positive charge, one body must have lost electrons and the other body must have gained the electrons. (a). The elementary charge e  1.6  10
19 C, so the total charge of 20 electrons has a magnitude of 20(1.6  10
19 C)  3.2  10
18 C. (c). Apply Coulomb’s law, F  ke(qAq B)/r 2: F  (9.00  109 N m2/C2)(2.00  10
4 C)(
4.00  10
4 C)/(6.00 m)2 
20.0 N so the magnitude of the force is 20.0 N. (d). From Coulomb’s law, F  ke(qAq B)/r 2 and force is inversely proportional to the square of the distance separating the points. Decreasing the distance to half its original value means that the force quadruples. (b). This problem is an application of the law of conservation of charge. The initial net charge is (3.0  10
3 C)  (
9.0  10
3 C) 
6.0  10
3 C. The same net charge must exist after contact. The
6.0  10
3 C must be evenly distributed between the two spheres because physically they are identical.

Circuits

A.47

8. (b). The electric field strength is the ratio of the force exerted on a unit charge in the field: E  F/q. Therefore, F and q are directly proportional and linearly related. 9. (c). From Coulomb’s law, force varies inversely with the square of the distance from the charge. The strength of the electric field at a point is the ratio of this force to the charge: E  keq/r 2. Therefore, E and r 2 are inversely proportional. The smaller the value of r, the smaller the value of r 2 and the greater the value of E. 10. (d). From Coulomb’s law, F  ke(q 1q 2)/r 2, force is directly proportional to the product of the charges. If q 1 is doubled and q 2 is quadrupled, the product of the charges increases eightfold and so does the force.

CIRCUITS EXAMPLE 1 Three resistors are connected together. How should they be combined so as to minimize the resistance of the combination? (a) They should be connected in series. (b) The two larger resistors should be put in parallel, and the remaining resistor put in series with the first two. (c) All three resistors should be placed in parallel. Conceptual Solution Resistance is proportional to length and inversely proportional to the cross- sectional area of a resistor. Putting all three resistors in parallel effectively minimizes the overall length of the combined resistor and maximizes the effective cross-sectional area, so (c) is the correct answer. Quantitative Solution Let R 1 and R 2 be the two larger resistors. Calculate the resistance R S of the three resistors in series: Calculate the resistance R combo of a parallel pair in series with a third resistor: Divide the numerator and denominator of the parallel resistor term by R 1:

(1) R S  R 1  R 2  R 3 R combo 5 a

(2)

R 1R 2 1 1 21 1 1 R3 b 1 R3 5 R1 R2 R1 1 R2

R combo 5

Calculate the resistance RP of three resistors in parallel:

RP 5 a

Divide the numerator and denominator by R1R 2:

(3)

R2 1 R3 R2 11 R1

R 1R 2R 3 1 1 1 21 1 1 b 5 R1 R2 R3 R 1R 2 1 R 2R 3 1 R 1R 3

RP 5

R3 R3 R3 11 1 R1 R2

Notice in Equation (2) that R combo is less than R 2  R 3 because the denominator of the first term is greater than 1, which in turn means that R combo is less than R S  R 1  R 2  R 3. Finally, notice in Equation (3) that R P is less than R 3, again because of a denominator larger than 1, and so R P is less than R combo. The purely parallel combination therefore yields the least resistance, and the answer is (c).

EXAMPLE 2 Two resistors are to dissipate as much energy as possible when a fixed voltage difference is placed across them. Should they be installed in parallel or in series? Conceptual Solution The power dissipated by a resistor is proportional to the voltage difference squared and inversely proportional to the resistance. The smallest possible resistance will therefore result in the largest power output. Consequently, the two resistors should be placed in parallel, which results in the lowest combined resistance.

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EXAMPLE 3 A certain off-the-shelf resistor of resistivity r has resistance R. Suppose it is commercially desirable to design a new resistor that has one-third the length and one-fourth the cross-sectional area of the existing resistor, but the same overall resistance. What should the resistivity rn of the new resistor be, in terms of the resistivity r of the original device? Solution Write an expression for the resistance Rn of the new resistor in terms of its resistivity, cross-sectional area A, and its length L:

(1) R n 5

rnL n An

Divide Equation (1) by the same expression for the original resistor:

(2)

rnL n rnL nA Rn An 5 5 rL R rL An A

Substitute An  A/4, Rn  R, and Ln  L/3 into Equation (2):

(3)

15

Solve Equation (3) for the new resistivity, rn:

rn 

3 4

4rn rn 1 L/3 2 A 5 3r rL 1 A/4 2

r

MULTIPLE-CHOICE PROBLEMS 1. Three resistors of resistance 1.0 , 2.0 , and 3.0 , respectively, are in series. If a potential difference of 12 V is applied across the combination, what is the resulting current in the circuit? (a) 0.50 A (b) 2.0 A (c) 6.0 A (d) 12 A 2. If the length of a conducting wire with resistance R is doubled, what will the resistance of the longer wire be? (a) R/4 (b) R/2 (c) 2R (d) 4R 3. If a 2.0- resistor and a 6.0  resistor are connected in parallel, what is their combined resistance? (a) 1.5  (b) 4.0  (c) 8.0  (d) 12  4. If all the components of an electric circuit are connected in series, which of the following physical quantities must be the same at all points in the circuit? (a) voltage (b) current (c) resistance (d) power 5. The current in a conductor is 3.0 A when it is connected across a 6.0-V battery. How much power is delivered to the conductor? (a) 0.50 W (b) 2.0 W (c) 9.0 W (d) 18 W 6. A 12- resistor is connected across a 6.0-V dc source. How much energy is delivered to the resistor in half an hour? (a) 1.5  10
3 kWh (b) 2.0  10
3 kWh (c) 3.0  10
3 kWh (d) 12  10
3 kWh 7. A battery with an emf of 6.20 V carries a current of 20.0 A. If the internal resistance of the battery is 0.01 , what is the terminal voltage? (a) 1.24 V (b) 6.00 V (c) 6.40 V (d) 31.0 V 8. Devices A and B are connected in parallel to a battery. If the resistance R A of device A is four times as great as the resistance R B of device B, what is true of IA and IB , the currents in devices A and B? (a) IA  2IB (b) IA  IB/2 (c) IA  4IB (d) IA  IB/4 9. A wire has resistance R. What is the resistance of a wire of the same substance that has the same length but twice the cross-sectional area? (a) 2R (b) R/2 (c) 4R (d) R/4 10. What must be the reading in the ammeter A for the circuit section shown? 1 4.0 A

6.0 A

(a) 0 A (b) 6.0 A (c) 8.0 A (d) 12 A

2.0 A

2

A

Atoms

A.49

11. What is the current in a wire if 5.00 C of charge passes through the wire in 0.500 s? (a) 1.00 A (b) 2.50 A (c) 5.00 A (d) 10.0 A

Answers 1. (b). The three resistors are connected in series, so R tot   R  6.0 . From Ohm’s law, Vtot/R tot  Itot  12 V/6.0   2.0 A. 2. (c). Resistance is directly proportional to length and inversely proportional to the cross-sectional area of the conductor. If the area remains constant, doubling the length will double the resistance. 3. (a). The resistors are in parallel: R eq  R 1R 2/(R 1  R 2)  12 2/8.0   1.5  In fact, the equivalent resistance of parallel resistors is always less than the smallest resistance in the combination, which means that answers (b), (c), and (d) could have been eliminated immediately. 4. (b). A series circuit has only one path for current, so it must be the same at all points in the circuit. 5. (d).
 I  V  18 W. 6. (a). The energy used by a load is the product of the power it uses per unit time and the length of time it is operated: E  W 
t  (V )2 tFR  (6.0 V)2 (0.50 h)/12   1.5 Wh  1.5  10
3 kWh Note the conversion from watt-hours to kilowatt-hours. 7. (b). The product Ir is the potential drop occurring within the battery: Ir  (20.0 A)(0.01 )  0.20 V. Because the battery is producing current and not being recharged, the terminal voltage will be less than the emf by the internal potential drop, which eliminates choices (c) and (d) immediately. So, V  
Ir  6.20 V
0.20 V  6.00 V. 8. (d). More current will follow the circuit branch with less resistance. Use Kirchoff’s loop law around the loop consisting of the two parallel resistances and R A  4R B : a DVi 5 0

S

I AR A 2 I B R B 5 0

S

IA RB RB 1 5 5 5 IB RA 4R B 4 Hence, one-fourth as much current goes through resistor A. 9. (b). Resistance is inversely proportional to the area; doubling the area reduces the resistance by half. 10. (d). Kirchhoff’s current rule says that the current entering a junction must equal the current leaving the junction. The current entering junction 1 is 6.0 A, which subsequently enters junction 2 from above, so the total current entering junction 2 is 12 A and the current then leaving junction 2 and entering the ammeter is 12 A. 11. (d). By definition, the current is the amount of charge that passes a point in the circuit in a given time: I  Q /t  5.00 C/0.500 s  10.0 A.

ATOMS EXAMPLE 1 Tritium is an isotope of hydrogen with a half-life of t 1/2  12.33 yr. How long would it take 1.60  102 g of tritium to decay to 20.0 g? (a) 6.17 yr (b) 26.7 yr (c) 37.0 yr (d) 74.0 yr Solution Calculate the number n of half-lives required. The equation can be solved by inspection (logarithms are ordinarily required):

20.0 g 1 n 1 5 a b 5 2 8.00 1.60 3 102 g

S

n53

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Appendix E

MCAT Skill Builder Study Guide

t  nt 1/2  3 (12.33 yr)  37.0 yr

Multiply the number of half-lives by the length of the half-life to find the necessary time interval in question, verifying that the answer is (c):

EXAMPLE 2 What is the mass number of a carbon atom having 6 protons and 8 neutrons? (a) 6 (b) 8 (c) 14 Solution Don’t confuse mass number with the atomic number, Z, which is the number of protons in the nucleus. The mass number A is the number of nucleons in the nucleus. To calculate the mass number, just add the number of protons and neutrons together: 6  8  14, which is answer (c).

EXAMPLE 3 Which of the following particles is given by ZAX in the following reaction? (a) 73Li (b) 74Be (c) 147 N (d) 106C 1H 1

 ZAX

→

4 He 2

Solution Equate the sum of the mass numbers on both sides of the reaction: By charge conservation, the number of protons must also be the same on both sides: Based on these two results, the correct answer is (a),

7 3Li

 42He

1A448

→

A7

1Z224

→

Z3

.

MULTIPLE-CHOICE PROBLEMS 1. In the nuclear reaction below, what particle does X represent? 22Na 11

→

22Ne 10

 X  ne

(a) an a-particle (b) a b-particle (c) a positron (d) a g-photon 2. What is the atomic number of the daughter nuclide in the following reaction? 3 0P 15

→

A Z Si

 e  ne

(a) 14 (b) 16 (c) 30 (d) 31 3. If 137 N has a half-life of about 10.0 min, how long will it take for 20 g of the isotope to decay to 2.5 g? (a) 5 min (b) 10 min (c) 20 min (d) 30 min 4. A certain radionuclide decays by emitting an a-particle. What is the difference between the atomic numbers of the parent and the daughter nuclides? (a) 1 (b) 2 (c) 4 (d) 6 5. What is the difference in mass number between the parent and daughter nuclides after a b-decay process? (a)
1 (b) 0 (c) 1 (d) 2 6. A nitrogen atom has 7 protons and 6 neutrons. What is its mass number? (a) 1 (b) 6 (c) 7 (d) 13 7. Which one of the following is an isotope of 18632X? (a) 18622X (b) 18642X (c) 18630X (d) 18620X 8. In the following nuclear equation, what is X? 14 N 7

 42He →

17 O 8

X

(a) a proton (b) a positron (c) a b-particle (d) an a-particle 9. What is the number of neutrons in 14540Xe? (a) 54 (b) 86 (c) 140 (d) 194

Atoms

10. Radon-222 has a half-life of about 4 days. If a sample of 222Rn gas in a container is initially doubled, the half-life will be (a) halved. (b) doubled. (c) quartered. (d) unchanged. 11. A radionuclide decays completely. In the reaction flask, the only gas that is found is helium, which was not present when the flask was sealed. The decay process was probably (a) b-decay. (b) a-decay. (c) g-decay. (d) positron emission. 12. What is the half-life of a radionuclide if 1/16 of its initial mass is present after 2 h? (a) 15 min (b) 30 min (c) 45 min (d) 60 min 13. The half-life of 2112Na is 2.6 y. If X grams of this sodium isotope are initially present, how much is left after 13 yr? (a) X/32 (b) X/13 (c) X/8 (d) X/5

Answers 1. (c). Notice that the mass number is unchanged, whereas the atomic number has been reduced by 1, which implies that a proton has changed into a neutron. The emitting particle must have a charge equal to a proton but an atomic mass number of zero. The positron is the only choice that has both these attributes. 2. (a). The atomic number is Z. Conservation of charge means that 15  Z  1. Therefore, Z  14. 3. (d). First, find the number of half-lives and then multiply by the value of the half-life to get the elapsed time: 2.5 g 20 g

5

1 1 3 5a b 8 2

S

n53

t  nt 1/2  3(10 min)  30 min 4. (b). An a-particle is a 42He helium nucleus. In a-decay, two protons are effectively removed. 5. (b). b-decay emits a high-energy electron,
10e. In the process, a neutron decays into a proton plus the emitted electron and an antineutrino. The number of nucleons remains unchanged, with the proton replacing the neutron in the sum of nucleons. 6. (d). The mass number is the sum of neutrons and protons: 6  7  13 nucleons. 7. (c). Isotopes of an element have the same atomic number but different numbers of neutrons, so their mass numbers are different. The atomic number of X is 63. 8. (a). The mass numbers must be the same on both sides of the reaction. If A is the mass number of X, then 14  4  17  A, so A  1. As for atomic number, 7  2  8  Z. Therefore, Z  1, which describes a proton, 11H. 9. (b). The number of neutrons is the mass number minus the atomic number: A
Z  140
54  86. 10. (d). The half-life is a constant that depends on the identity of the nuclide, not on the amount of nuclide present. 11. (b). The a-particle is a helium nucleus. Each a-particle then acquires two electrons to form a neutral helium atom. 12. (b). In every half-life, the mass decreases to half its previous value: 1/16  1/24. It takes four half-lives to decay down to 1/16 the original mass. Each must be 30 min long because the entire process takes two hours. 13. (a). In 13 yr, there will be 5 half-lives of 2.6 yr each (5  2.6  13). The isotope decreases to 1/25  1/32 of its original amount.

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ANSWERS TO QUICK QUIZZES, EXAMPLE QUESTIONS, ODD-NUMBERED MULTIPLE CHOICE QUESTIONS, CONCEPTUAL QUESTIONS, AND PROBLEMS CHAPTER 15 QUICK QUIZZES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

(b) (b) (c) (a) (c) and (d) (a) (c) (b) (d) (b) and (d)

EX A MPLE QUESTIONS 1. 14 2. (a) 3. Fourth quadrant 4. The suspended droplet would accelerate downward at twice the acceleration of gravity. 5. The angle f would increase. 6. (c) 7. The charge on the inner surface would be negative. 8. Zero

MULTIPLE-CHOICE QUESTIONS 1. 3. 5. 7. 9. 11. 13.

(a) (b) (b) (d) (b) (e) (d)

CONCEPTUAL QUESTIONS 1. Electrons have been removed from the object. 3. No. The charge on the metallic sphere resides on its outer surface, so the person is able to touch the surface without causing any charge transfer. 5. Move an object A with a net positive charge so it is near, but not touching, a neutral metallic object B that is insulated from the ground. The presence of A will polarize B, causing an excess negative charge to exist on the side nearest A and an excess positive charge of equal magnitude to exist on the side farthest from A. While A is still near B, touch B with your hand. Additional electrons will then flow from ground, through your body and onto B. With A continuing to be near but not in contact with B, remove your hand from B, thus trapping the excess electrons on B. When A is now removed, B is left with excess electrons, or a net negative charge. By means of mutual repulsion, this negative charge will now spread uniformly over the entire surface of B. 7. An object’s mass decreases very slightly (immeasurably) when it is given a positive charge, because it loses electrons. When the object is given a negative charge, its mass increases slightly because it gains electrons. 9. Electric field lines start on positive charges and end on negative charges. Thus, if the fair-weather field is directed into the ground, the ground must have a negative charge.

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11. The electric shielding effect of conductors depends on the fact that there are two kinds of charge: positive and negative. As a result, charges can move within the conductor so that the combination of positive and negative charges establishes an electric field that exactly cancels the external field within the conductor and any cavities inside the conductor. There is only one type of gravitation charge, however, because there is no negative mass. As a result, gravitational shielding is not possible. A room cannot be gravitationally shielded because mass is always positive or zero, never negative. 13. When the comb is nearby, charges separate on the paper, and the paper is attracted to the comb. After contact, charges from the comb are transferred to the paper, so that it has the same type of charge as the comb. The paper is thus repelled. 15. You can only conclude that the net charge inside the Gaussian surface is positive.

PROBLEMS

1. 8.7  10
8 N; the force is repulsive. 3. (a) 2.36  10
5 C (b) The charges induce opposite charges in the bulkheads, but the induced charge in the bulkhead near ball B is greater because of ball B’s greater charge. The system therefore moves slowly toward the bulkhead closer to ball B. 5. (a) 36.8 N (b) 5.54  1027 m/s2 7. 5.12  105 N 9. (a) 2.2  10
5 N (attraction) (b) 9.0  10
7 N (repulsion) 11. 1.38  10
5 N at 77.5° below the negative x -axis 13. 0.437 N at
85.3° from the x-axis. 15. 7.2 nC 17. 2.07  103 N/C down 19. 7.20  105 N/C (downward) 21. 27.0 N/C; negative x-direction 23. (a) 6.12  1010 m/s2 (b) 19.6 ms (c) 11.8 m (d) 1.20  10
15 J 25. 0.849 m 27. 1.8 m to the left of the
2.5-mC charge 29. zero 35. (a) 0 (b) 5 mC inside,
5 mC outside (c) 0 inside,
5 mC outside (d) 0 inside,
5 mC outside 37. 1.3  10
3 C 39. (a) 2.54  10
15 N (b) 1.59  104 N/C radially outward 41. (a) 858 N  m2/C (b) 0 (c) 657 N  m2/C 43.
Q /P 0 for S1; 0 for S2;
2Q /P 0 for S3; 0 for S 4 45. (a) 0 (b) keq/r 2 outward 47. (a)
7.99 N/C (b) 0 (c) 1.44 N/C (d) 2.00 nC on the inner surface; 1.00 nC on the outer surface 49. 115 N 51. 24 N/C in the positive x-direction 53. (a) E  2ke qb (a 2  b 2)
3/2 in the positive x-direction (b) E  keQb(a 2  b 2)
3/2 in the positive x-direction 55. (a) 4.64  10
2 m (b) 2.54  10
2 m 57. 4.4  105 N/C 59. 10
7 C 61. (a) 0 (b) 7.99  107 N/C (outward) (c) 0 (d) 7.34  10 6 N/C (outward)

Answers to Quick Quizzes, Example Questions, Odd-Numbered Multiple-Choice Questions, Conceptual Questions, and Problems 63. (a) 1.00  103 N/C (b) 3.37  10
8 s (c) accelerate at 1.76  1014 m/s2 in the direction opposite that of the electric field

CHAPTER 16 QUICK QUIZZES (b) (a) (b) (d) (d) (c) (a) (c) (a) C decreases. (b) Q stays the same. (c) E stays the same. (d) V increases. (e) The energy stored increases. 10. (a) C increases. (b) Q increases. (c) E stays the same. (d) V remains the same. (e) The energy stored increases. 11. (a) 1. 2. 3. 4. 5. 6. 7. 8. 9.

EX A MPLE QUESTIONS True True False (a) Three Each answer would be reduced by a factor of one-half. One-quarter The voltage drop is the smallest across the 24-mF capacitor and largest across the 3.0-mF capacitor. 9. The 3.0-mF capacitor 10. (c) 11. (b) 1. 2. 3. 4. 5. 6. 7. 8.

tions of higher potential.) All of the charges would continue to move until the potential became equal everywhere in the conductor. 9. The capacitor often remains charged long after the voltage source is disconnected. This residual charge can be lethal. The capacitor can be safely handled after discharging the plates by short-circuiting the device with a conductor, such as a screwdriver with an insulating handle. 11. Field lines represent the direction of the electric force on a positive test charge. If electric field lines were to cross, then, at the point of crossing, there would be an ambiguity regarding the direction of the force on the test charge, because there would be two possible forces there. Thus, electric field lines cannot cross. It is possible for equipotential surfaces to cross. (However, equipotential surfaces at different potentials cannot intersect.) For example, suppose two identical positive charges are at diagonally opposite corners of a square and two negative charges of equal magnitude are at the other two corners. Then the planes perpendicular to the sides of the square at their midpoints are equipotential surfaces. These two planes cross each other at the line perpendicular to the square at its center. 13. You should use a dielectric-filled capacitor whose dielectric constant is very large. Further, you should make the dielectric as thin as possible, keeping in mind that dielectric breakdown must also be considered.

PROBLEMS

1. (a) 1.92  10
18 J (b)
1.92  10
18 J (c) 2.05  10 6 m/s in the negative x-direction 3. 1.4  10
20 J 5. 1.7  10 6 N/C 7. (a) 1.13  105 N/C (b) 1.80  10
14 N (c) 4.38  10
17 J 9. (a) The spring stretches by 4.36 cm.

(b) (b) (a) (b) (a) (c)

11.

13. 15.

CONCEPTUAL QUESTIONS 1. (a) The proton moves in a straight line with constant acceleration in the direction of the electric field. (b) As its velocity increases, its kinetic energy increases and the electric potential energy associated with the proton decreases. 3. The work done in pulling the capacitor plates farther apart is transferred into additional electric energy stored in the capacitor. The charge is constant and the capacitance decreases, but the potential difference between the plates increases, which results in an increase in the stored electric energy. 5. If the power line makes electrical contact with the metal of the car, it will raise the potential of the car to 20 kV. It will also raise the potential of your body to 20 kV, because you are in contact with the car. In itself, this is not a problem. If you step out of the car, however, your body at 20 kV will make contact with the ground, which is at zero volts. As a result, a current will pass through your body and you will likely be injured. Thus, it is best to stay in the car until help arrives. 7. If two points on a conducting object were at different potentials, then free charges in the object would move and we would not have static conditions, in contradiction to the initial assumption. (Free positive charges would migrate from locations of higher to locations of lower potential. Free electrons would rapidly move from locations of lower to loca-

2kA 2 Q (a)
5.75  10
7 V (b)
1.92  10
7 V, V  3.84  10
7 V (c) No. Unless fi xed in place, the electron would move in the opposite direction, increasing its distance from points A and B and lowering the potential difference between them. (a) 2.67  10 6 V (b) 2.13  10 6 V (a) 103 V (b)
3.85  10
7 J; positive work must be done to separate the charges.
11.0 kV (a) 3.84  10
14 J (b) 2.55  10
13 J (c)
2.17  10
13 J (d) 8.08  10 6 m/s (e) 1.24  107 m/s 0.719 m, 1.44 m, 2.88 m. No. The equipotentials are not uniformly spaced. Instead, the radius of an equipotenial is inversely proportional to the potential. 2.74  10
14 m (a) 1.1  10
8 F (b) 27 C (a) 1.36 pF (b) 16.3 pC (c) 8.00  103 V/m (a) 11.1 kV/m toward the negative plate (b) 3.74 pF (c) 74.7 pC and
74.7 pC (a) 5.90  10
10 F (b) 3.54  10
9 C (c) 2.00  103 N/C (d) 1.77  10
8 C/m2 (e) All the answers are reduced. (a) 10.7 mC on each capacitor (b) 15.0 mC on the 2.50-mF capacitor and 37.5 mC on the 6.25-mF capacitor (a) 2.67 mF (b) 24.0 mC on each 8.00-mF capacitor, 18.0 mC on the 6.00-mF capacitor, 6.00 mC on the 2.00-mF capacitor (c) 3.00 V across each capacitor (a) 3.33 mF (b) 180 mC on the 3-mF and the 6-mF capacitors, 120 mC on the 2.00-mF and 4.00-mF capacitors (c) 60.0 V across the 3-mF and the 2-mF capacitors, 30.0 V across the 6-mF and the 4-mF capacitors Q 1  16.0 mC, Q 5  80.0 mC, Q 8  64.0 mC, Q 4  32.0 mC (b) Equilibrium: x  2.18 cm; A  2.18 cm (c) DV 5 2

MULTIPLE-CHOICE QUESTIONS 1. 3. 5. 7. 9. 11.

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17. 19. 21.

23. 25. 27. 29. 31. 33. 35.

37.

39.

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Answers to Quick Quizzes, Example Questions, Odd-Numbered Multiple-Choice Questions, Conceptual Questions, and Problems

41. (a) Q 25  1.25 mC, Q 40  2.00 mC (b) Q 25  288 mC, Q 40  462 mC, V  11.5 V 43. Q 1  3.33 mC, Q 2  6.67 mC 45. 3.24  10
4 J 47. (a) 54.0 mJ (b) 108 mJ (c) 27.0 mJ 49. (a) k  3.4. The material is probably nylon (see Table 16.1). (b) The voltage would lie somewhere between 25.0 V and 85.0 V. 51. (a) 8.13 nF (b) 2.40 kV 53. (a) volume 9.09  10
16 m3, area 4.54  10
10 m2 (b) 2.01  10
13 F (c) 2.01  10
14 C, 1.26  105 electronic charges 59. 4.29 mF 61. 6.25 mF 63. 4.47 kV 65. 0.75 mC on C 1, 0.25 mC on C 2 67. Sphere A: 0.800 mC; sphere B: 1.20 mC

CHAPTER 17 QUICK QUIZZES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

(d) (b) (c), (d) (b) (b) (b) (a) (b) (a) (c)

EX A MPLE QUESTIONS 1. No. Such a current corresponds to the passage of one electron every 2 seconds. The average current, however, can have any value. 2. True 3. Higher 4. (b) 5. (a) 6. (c)

MULTIPLE-CHOICE QUESTIONS 1. 3. 5. 7. 9. 11. 13.

(c) (d) (b) (b) (c) (a) (d)

CONCEPTUAL QUESTIONS

1. Charge. Because an ampere is a unit of current (1 A  1 C/s) and an hour is a unit of time (1 h  3 600 s), then 1 A  h  3 600 C. 3. The gravitational force pulling the electron to the bottom of a piece of metal is much smaller than the electrical repulsion pushing the electrons apart. Thus, free electrons stay distributed throughout the metal. The concept of charges residing on the surface of a metal is true for a metal with an excess charge. The number of free electrons in an electrically neutral piece of metal is the same as the number of positive ions—the metal has zero net charge. 5. A voltage is not something that “surges through” a completed circuit. A voltage is a potential difference that is applied across a device or a circuit. It would be more correct to say “1 ampere of electricity surged through the victim’s body.” Although this amount of current would have disastrous

results on the human body, a value of 1 (ampere) doesn’t sound as exciting for a newspaper article as 10 000 (volts). Another possibility is to write “10 000 volts of electricity were applied across the victim’s body,” which still doesn’t sound quite as exciting. 7. We would conclude that the conductor is nonohmic. 9. Once the switch is closed, the line voltage is applied across the bulb. As the voltage is applied across the cold filament when it is first turned on, the resistance of the filament is low, the current is high, and a relatively large amount of power is delivered to the bulb. As the filament warms, its resistance rises and the current decreases. As a result, the power delivered to the bulb decreases. The large current spike at the beginning of the bulb’s operation is the reason that lightbulbs often fail just after they are turned on. 11. The drift velocity might increase steadily as time goes on, because collisions between electrons and atoms in the wire would be essentially nonexistent and the conduction electrons would move with constant acceleration. The current would rise steadily without bound also, because I is proportional to the drift velocity.

PROBLEMS

1. 3.00  1020 electrons move past in the direction opposite to the current. 3. 1.05 mA 5. (a) n is unaffected (b) vd is doubled 7. 27 yr 9. (a) 55.85  10
3 kg/mol (b) 1.41  105 mol/m3 (c) 8.49  1028 iron atoms/m3 (d) 1.70  1029 conduction electrons/m3 (e) 2.21  10
4 m/s 11. 32 V is 200 times larger than 0.16 V 13. (a) 13.0  (b) 17.0 m 15. (a) 30  (b) 4.7  10
4   m 17. silver (r  1.59  10
8   m) 19. 256  23. 1.98 A 25. 2 200 °C 27. 26 mA 29. (a) 3.0 A (b) 2.9 A 31. (a) 1.2  (b) 8.0  10
4 (a 0.080% increase) 33. (a) 8.33 A (b) 14.4  35. 2.1 W 37. 11.2 min 39. 34.4  41. 1.6 cm 43. 15 mW 45. 23 cents 47. $1.2 49. 2.16  105 C 51. 6.7 W 53. 1.1 km 55. 1.47  10
6   m; differs by 2.0% from value in Table 17.1 57. (a) $3.06 (b) No. The circuit must be able to handle at least 26 A. 59. (a) 667 A (b) 50.0 km 61. 3.77  1028/m3 63. 37 M 65. 0.48 kg/s 67. (a) 470 W (b) 1.60 mm or more (c) 2.93 mm or more

CHAPTER 18 QUICK QUIZZES 1. True

Answers to Quick Quizzes, Example Questions, Odd-Numbered Multiple-Choice Questions, Conceptual Questions, and Problems 2. Because of internal resistance, power is delivered to the battery material, raising its temperature. 3. (b) 4. (a) 5. (a) 6. (b) 7. Parallel: (a) unchanged (b) unchanged (c) increase (d) decrease 8. Series: (a) decrease (b) decrease (c) decrease (d) increase 9. (c)

EXAMPLE QUESTIONS It would reduce the final current. (b) The 8.00- resistor The answers would be negative, but they would have the same magnitude as before. 5. Yes 6. (a) 7. (a)

1. 2. 3. 4.

MULTIPLE-CHOICE QUESTIONS 1. 3. 5. 7. 9. 11. 13.

(c) (c) (e) (a) (d) (b), (e) (d)

CONCEPTUAL QUESTIONS 1. No. When a battery serves as a source and supplies current to a circuit, the conventional current flows through the battery from the negative terminal to the positive one. However, when a source having a larger emf than the battery is used to charge the battery, the conventional current is forced to flow through the battery from the positive terminal to the negative one. 3. The total amount of energy delivered by the battery will be less than W. Recall that a battery can be considered an ideal, resistanceless battery in series with the internal resistance. When the battery is being charged, the energy delivered to it includes the energy necessary to charge the ideal battery, plus the energy that goes into raising the temperature of the battery due to I 2r heating in the internal resistance. This latter energy is not available during discharge of the battery, when part of the reduced available energy again transforms into internal energy in the internal resistance, further reducing the available energy below W. 5. The starter in the automobile draws a relatively large current from the battery. This large current causes a significant voltage drop across the internal resistance of the battery. As a result, the terminal voltage of the battery is reduced, and the headlights dim accordingly. 7. Connecting batteries in parallel does not increase the emf. A high-current device connected to two batteries in parallel can draw currents from both batteries. Thus, connecting the batteries in parallel increases the possible current output and, therefore, the possible power output. 9. She will not be electrocuted if she holds onto only one highvoltage wire, because she is not completing a circuit. There is no potential difference across her body as long as she clings to only one wire. However, she should release the wire immediately once it breaks, because she will become part of a closed circuit when she reaches the ground or comes into contact with another object. 11. The bird is resting on a wire of fixed potential. In order to be electrocuted, a large potential difference is required between

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the bird’s feet. The potential difference between the bird’s feet is too small to harm the bird. 13. The junction rule is a statement of conservation of charge. It says that the amount of charge that enters a junction in some time interval must equal the charge that leaves the junction in that time interval. The loop rule is a statement of conservation of energy. It says that the increases and decreases in potential around a closed loop in a circuit must add to zero.

PROBLEMS

1. 4.92  3. 73.8 W. Your circuit diagram will consist of two 0.800- resistors in series with the 192- resistance of the bulb. 5. (a) 17.1  (b) 1.99 A for 4.00  and 9.00 , 1.17 A for 7.00 , 0.818 A for 10.0  7. 7R/3 9. (a) 0.227 A (b) 5.68 V 11. 55  13. 0.43 A 15. (a) Connect two 50- resistors in parallel, and then connect this combination in series with a 20- resistor. (b) Connect two 50- resistors in parallel, connect two 20- resistors in parallel, and then connect these two combinations in series with each other. 17. 0.714 A, 1.29 A, 12.6 V 19. (a) 3.00 mA (b)
19.0 V (c) 4.50 V 21. (a) I1  0.492 A, I2  0.148 A, I3  0.639 A (b) 6.78 W to the 28.0- resistor, 1.78 W to the 12.0- resistor, 6.53 W to the 16.0- resistor 23. (a) 0.385 mA, 3.08 mA, 2.69 mA (b) 69.2 V, with c at the higher potential 25. (a) No. The only simplification is to note that the 2.0- and 4.0- resistors are in series and add to a resistance of 6.0 . Likewise, the 5.0- and 1.0- resistors are in series and add to a resistance of 6.0 . The circuit cannot be simplified any further. Kirchhoff’s rules must be used to analyze the circuit. (b) I1  3.5 A, I2  2.5 A, I3  1.0 A 27. (a) No. The multiloop circuit cannot be simplified any further. Kirchhoff’s rules must be used to analyze the circuit. (b) I30  0.353 A directed to the right, I5  0.118 A directed to the right, I20  0.471 A directed to the left 29. V 2  3.05 V, V 3  4.57 V, V 4  7.38 V, V 5  1.62 V 31. (a) 1.88 s (b) 1.90  10
4 C 33. 1.3  10
4 C 35. (a) 0.43 s (b) 6.0 mF 37. 48 lightbulbs 39. (a) 6.25 A (b) 750 W 41. (a) 1.2  10
9 C, 7.3  109 K ions. Not large, only 1e/290 Å 2 (b) 1.7  10
9 C, 1.0  1010 Na ions (c) 0.83 mA (d) 7.5  10
12 J 43. 11 nW 45. 7.5  47. (a) 15  (b) I 1  1.0 A, I 2  I 3  0.50 A, I 4  0.30 A, and I 5  0.20 A (c) (V )ac  6.0 V, (V )ce  1.2 V, (V )ed  (V )fd  1.8 V, (V )cd  3.0 V, (V )db  6.0 V (d)
ac  6.0 W,
ce  0.60 W,
ed  0.54 W,
fd  0.36 W,
cd  1.5 W,
db  6.0 W 49. (a) 12.4 V (b) 9.65 V 51. (a) R

open eq

5 3R, R

closed eq

5 2R (b)
open 5

e2

,
closed 5

e2

3R 2R (c) Lamps A and B increase in brightness, lamp C goes out.

53. 112 V, 0.200  55. (a) R x 5 R 2 2 14 R 1 (b) R x  2.8  (inadequate grounding) 1 144 V 2 2 R 59.
5 1 R 1 10.0 V 2 2

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Answers to Quick Quizzes, Example Questions, Odd-Numbered Multiple-Choice Questions, Conceptual Questions, and Problems


load 3.6 W

5.

10 

61. 63. 65. 67.

R load

(a) 5.68 V (b) 0.227 A 0.395 A; 1.50 V (a) 2.41  10
5 C (b) 1.61  10
5 C (c) 1.61  10
2 A (a) 6.0  102 W (b) 1.2 J (c) 37 m

CHAPTER 19

7.

9.

QUICK QUIZZES 1. 2. 3. 4. 5. 6.

(b) (c) (c) (a) (a), (c) (b)

EX A MPLE QUESTIONS 1. The force on the electron is opposite the direction of the force on the proton, and the acceleration of the electron is far greater than the acceleration on the proton due to the electron’s lower mass. 2. A magnetic field always exerts a force perpendicular to the velocity of a charged particle, so it can change the particle’s direction but not its speed. 3. Zero 4. As the angle approaches 90°, the magnitude of the force increases. After going beyond 90°, it decreases. 5. The magnitude of the momentum remains constant. The direction of momentum changes, unless the particle’s velocity is parallel or antiparallel to the magnetic field. 6. 20 m 7. It would have the same magnitude and point to the right. 8. Earth’s magnetic field is extremely weak, so the current carried by the car would have to be correspondingly very large. Such a current would be impractical to generate, and if generated, it would heat and melt the wires carrying it. 9. The proton would accelerate downward due to gravity while circling.

MULTIPLE-CHOICE QUESTIONS 1. 3. 5. 7. 9. 11. 13. 15.

(d) (c) (d) (e) (d), (e) (a) (c), (d) (a)

CONCEPTUAL QUESTIONS 1. The set should be oriented such that the beam is moving either toward the east or toward the west. 3. The magnetic force on a moving charged particle is always perpendicular to the particle’s direction of motion. There is no magnetic force on the charge when it moves parallel to the direction of the magnetic field. However, the force on a charged particle moving in an electric field is never zero and

11.

13.

is always parallel to the direction of the field. Therefore, by projecting the charged particle in different directions, it is possible to determine the nature of the field. The magnetic field produces a magnetic force on the electrons moving toward the screen that produce the image. This magnetic force deflects the electrons to regions on the screen other than the ones to which they are supposed to go. The result is a distorted image. Such levitation could never occur. At the North Pole, where Earth’s magnetic field is directed downward, toward the equivalent of a buried south pole, a coffin would be repelled if its south magnetic pole were directed downward. However, equilibrium would be only transitory, as any slight disturbance would upset the balance between the magnetic force and the gravitational force. If you were moving along with the electrons, you would measure a zero current for the electrons, so they would not produce a magnetic field according to your observations. However, the fi xed positive charges in the metal would now be moving backwards relative to you, creating a current equivalent to the forward motion of the electrons when you were stationary. Thus, you would measure the same magnetic field as when you were stationary, but it would be due to the positive charges presumed to be moving from your point of view. A compass does not detect currents in wires near light switches, for two reasons. The first is that, because the cable to the light switch contains two wires, one carrying current to the switch and the other carrying it away from the switch, the net magnetic field would be very small and would fall off rapidly with increasing distance. The second reason is that the current is alternating at 60 Hz. As a result, the magnetic field is oscillating at 60 Hz also. This frequency would be too fast for the compass to follow, so the effect on the compass reading would average to zero. There is no net force on the wires, but there is a torque. To understand this distinction, imagine a fi xed vertical wire and a free horizontal wire (see the figure below). The vertical wire carries an upward current and creates a magnetic field that circles the vertical wire, itself. To the right, the magnetic field of the vertical wire points into the page, while on the left side it points out of the page, as indicated. Each segment of the horizontal wire (of length ) carries current that interacts with the magnetic field according to the equation F  BI  sin u. Apply the right-hand rule on the right side: point the fingers of your right hand in the direction of the horizontal current and curl them into the page in the direction of the magnetic field. Your thumb points downward, the direction of the force on the right side of the wire. Repeating the process on the left side gives a force upward on the left side of the wire. The two forces are equal in magnitude and opposite in direction, so the net force is zero, but they create a net torque around the point where the wires cross. I F I B

B

F

15. Each coil of the Slinky® will become a magnet, because a coil acts as a current loop. The sense of rotation of the current is

Answers to Quick Quizzes, Example Questions, Odd-Numbered Multiple-Choice Questions, Conceptual Questions, and Problems the same in all coils, so each coil becomes a magnet with the same orientation of poles. Thus, all of the coils attract, and the Slinky® will compress. 17. (a) The field is into the page. (b) The beam would deflect upwards.

PROBLEMS 1. (a) The negative z-direction (b) The positive z-direction (c) The magnetic force is zero in this case. 3. (a) into the page (b) toward the right (c) toward the bottom of the page 5. (a) 1.44  10
12 N (b) 8.62  1014 m/s2 (c) A force would be exerted on the electron that had the same magnitude as the force on a proton, but in the opposite direction because of its negative charge. The acceleration of the electron would be much greater than that of the proton because the mass of the electron is much smaller. 7. 2.83  107 m/s west 9. 1.5  10
2 T into the page (the negative z-direction) 11. Fg  8.93  10
30 N (downward), Fe  1.60  10
17 N (upward), Fm  4.80  10
17 N (downward) 13. 8.0  10
3 T in the z-direction 15. (a) into the page (b) toward the right (c) toward the bottom of the page 17. 7.50 N 19. 0.131 T (downward) 21. (a) The magnetic force and the force of gravity both act on the wire. When the magnetic force is upward and balances the downward force of gravity, the net force on the wire is zero, and the wire can move upward at constant velocity. (b) 0.20 T out of the page. (c) If the field exceeds 0.20 T, the upward magnetic force exceeds the downward force of gravity, so the wire accelerates upward. 23. ab: 0, bc: 0.040 0 N in
x-direction, cd: 0.040 0 N in the
z-direction, da: 0.056 6 N parallel to the xz-plane and at 45° to both the x- and the z-directions 25. 4.9  10
3 N  m 27. 9.05  10
4 N  m, tending to make the left-hand side of the loop move toward you and the right-hand side move away. 29. (a) 0.56 A (b) 0.063 N  m 31. (a) 3.97° (b) 3.39  10
3 N  m 33. (a) 4.3 cm (b) 1.8  10
8 s 37. 1.77 cm 39. r  3R/4 41. (a) 2.08  10
7 kg/C (b) 6.66  10
26 kg (c) Calcium 43. 20.0 mT 45. 2.0  10
10 A 47. 2.4 mm 49. 20.0 mT toward the bottom of page 51. 0.167 mT out of the page 53. (a) 4.00 m (b) 7.50 nT (c) 1.26 m (d) zero 55. 3.0  10
5 N/m; attractive 57. 4.5 mm 59. 3.2 A 61. (a) 920 turns (b) 12 cm 63. (a) 2.8 mT (b) 0.89 mA 65. (a) 5.3 mT into the page (b) 7.2 cm 67. (a) 0.500 mT out of the page (b) 3.89 mT parallel to xy-plane and at 59.0° clockwise from x-direction 69. (a) 1.33 m/s (b) the sign of the emf is independent of the charge 71. 53 mT toward the bottom of the page, 20 mT toward the bottom of the page, and 0 73. (a)
8.00  10
21 kg  m/s (b) 8.90° 75. 1.41  10
6 N

A.57

CHAPTER 20 QUICK QUIZZES 1. 2. 3. 4. 5. 6.

b, c, a (a) (b) (c) (b) (b)

EX A MPLE QUESTIONS 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

False 5.06  10
3 V; the current would be in the opposite direction. (a), (e) A magnetic force directed to the right will be exerted on the bar. Doubling the frequency doubles the maximum induced emf. (b) (a) 72.6 A (b) False

MULTIPLE-CHOICE QUESTIONS 1. 3. 5. 7. 9. 11.

(c) (c) (d) (d) (b), (e) (a)

CONCEPTUAL QUESTIONS 1. According to Faraday’s law, an emf is induced in a wire loop if the magnetic flux through the loop changes with time. In this situation, an emf can be induced either by rotating the loop around an arbitrary axis or by changing the shape of the loop. 3. As the spacecraft moves through space, it is apparently moving from a region of one magnetic field strength to a region of a different magnetic field strength. The changing magnetic field through the coil induces an emf and a corresponding current in the coil. 5. If the bar were moving to the left, the magnetic force on the negative charges in the bar would be upward, causing an accumulation of negative charge on the top and positive charges at the bottom. Hence, the electric field in the bar would be upward, as well. 7. If, for any reason, the magnetic field should change rapidly, a large emf could be induced in the bracelet. If the bracelet were not a continuous band, this emf would cause highvoltage arcs to occur at any gap in the band. If the bracelet were a continuous band, the induced emf would produce a large induced current and result in resistance heating of the bracelet. 11. As the aluminum plate moves into the field, eddy currents are induced in the metal by the changing magnetic field at the plate. The magnetic field of the electromagnet interacts with this current, producing a retarding force on the plate that slows it down. In a similar fashion, as the plate leaves the magnetic field, a current is induced, and once again there is an upward force to slow the plate. 13. If an external battery is acting to increase the current in the inductor, an emf is induced in a direction to oppose the increase of current. Likewise, if we attempt to reduce the current in the inductor, the emf that is set up tends to support the current. Thus, the induced emf always acts to oppose the change occurring in the circuit, or it acts in the “back” direction to the change.

A.58

Answers to Quick Quizzes, Example Questions, Odd-Numbered Multiple-Choice Questions, Conceptual Questions, and Problems

PROBLEMS 1. 3. 5. 7. 9. 11. 13. 15.

17. 19. 21. 23. 25. 27. 29. 31. 33. 35. 37. 39. 41. 43. 45. 47. 49. 51. 53. 55. 57. 59. 61. 63. 65.

4.8  10
3 T  m2 (a) 0.177 T (b) 0 (a) B,net  0 (b) 0 (a) 3.1  10
3 T  m2 (b) B,net  0 1.5 mV 94 mV 2.7 T/s (a) 3.77  10
3 T (b) 9.42  10
3 T (c) 7.07  10
4 m2 (d) 3.99  10
6 Wb (e) 1.77  10
5 V; the average induced emf is equal to the instantaneous in this case because the current increases steadily. (f) The induced emf is small, so the current in the 4-turn coil and its magnetic field will also be small. 10.2 mV 2.6 mV (a) toward the east (b) 4.58  10
4 V 0.763 V (a) from left to right (b) from right to left into the page (a) from right to left (b) from right to left (c) from left to right (d) from left to right (a) F  N 2B 2w 2v/R to the left (b) 0 (c) F  N 2B 2w 2v/R to the left 13.3 V 1.9  10
11 V (a) 60 V (b) 57 V (c) 0.13 s (a) 18.1 mV (b) 0 4.0 mH (a) 2.0 mH (b) 38 A/s (a) 4.0 ms (b) 0.45 V (a) 2.4 V (b) 75 mH (c) 5.1 A (d) V R 
1.5 V; V L 
0.9 V 1.92  (a) 0.208 mH (b) 0.936 mJ (a) 18 J (b) 7.2 J negative (Va V b ) (a) 20.0 ms (b) 37.9 V (c) 1.52 mV (d) 51.8 mA (a) 0.500 A (b) 2.00 W (c) 2.00 W 115 kV (a) 0.157 mV (end B is positive) (b) 5.89 mV (end A is positive) (a) 9.00 A (b) 10.8 N (c) b is at the higher potential (d) No

CHAPTER 21 QUICK QUIZZES 1. 2. 3. 4. 5. 6. 7. 8.

(c) (b) (b) (a) (a) (b) (b), (c) (b), (d)

EX A MPLE QUESTIONS 1. 2. 3. 4. 5. 6. 7. 8.

9.

False False True True The average power will be zero if the phase angle is 90° or
90°, i.e. when R  0. False (d) Yes. When the roof is perpendicular to the sun’s rays the amount of energy intercepted by the roof is a maximum. At other angles the amount is smaller. (c)

MULTIPLE-CHOICE QUESTIONS 1. 3. 5. 7. 9. 11. 13.

(c) (a) (b) (d) (a) (c) (c)

CONCEPTUAL QUESTIONS 1. For best reception, the length of the antenna should be parallel to the orientation of the oscillating electric field. Because of atmospheric variations and reflections of the wave before it arrives at your location, the orientation of this field may be in different directions for different stations. 3. An antenna that is a conducting line responds to the electric field of the electromagnetic wave—the oscillating electric field causes an electric force on electrons in the wire along its length. The movement of electrons along the wire is detected as a current by the radio and is amplified. Thus, a line antenna must have the same orientation as the broadcast antenna. A loop antenna responds to the magnetic field in the radio wave. The varying magnetic field induces a varying current in the loop (by Faraday’s law), and this signal is amplified. The loop should be in the vertical plane containing the line of sight to the broadcast antenna, so the magnetic field lines go through the area of the loop. 5. The flashing of the light according to Morse code is a drastic amplitude modulation—the amplitude is changing from a maximum to zero. In this sense, it is similar to the on-andoff binary code used in computers and compact disks. The carrier frequency is that of the light, on the order of 1014 HZ. The frequency of the signal depends on the skill of the signal operator, but it is on the order of a single hertz, as the light is flashed on and off. The broadcasting antenna for this modulated signal is the filament of the lightbulb in the signal source. The receiving antenna is the eye. 7. The sail should be as reflective as possible, so that the maximum momentum is transferred to the sail from the reflection of sunlight. 9. Suppose the extraterrestrial looks around your kitchen. Lightbulbs and the toaster glow brightly in the infrared. Somewhat fainter are the back of the refrigerator and the back of the television set, while the television screen is dark. The pipes under the sink show the same weak glow as the walls, until you turn on the faucets. Then the pipe on the right gets darker and that on the left develops a gleam that quickly runs up along its length. The food on the plates shines, as does human skin, the same color for all races. Clothing is dark as a rule, but your seat and the chair seat glow alike after you stand up. Your face appears lit from within, like a jack-o’-lantern; your nostrils and the openings of your ear canals are bright; brighter still are the pupils of your eyes. 11. No. The wire will emit electromagnetic waves only if the current varies in time. The radiation is the result of accelerating charges, which can occur only when the current is not constant. 13. The resonance frequency is determined by the inductance and the capacitance in the circuit. If both L and C are doubled, the resonance frequency is reduced by a factor of two. 15. It is far more economical to transmit power at a high voltage than at a low voltage because the I 2R loss on the transmission line is significantly lower at high voltage. Transmitting power at high voltage permits the use of step-down transformers to make “low” voltages and high currents available to the end user.

Answers to Quick Quizzes, Example Questions, Odd-Numbered Multiple-Choice Questions, Conceptual Questions, and Problems

PROBLEMS 1. 3. 5. 9. 11. 15. 17. 19. 21. 23. 25. 27. 29. 31. 33. 35. 37.

39. 41. 43. 45. 47. 49. 51. 53. 55. 57. 59. 61. 63. 65. 67. 69. 71. 73. 75.

(a) 96.0 V (b) 136 V (c) 11.3 A (d) 768 W 70.7 V, 2.95 A 6.76 W 4.0  102 Hz 17 mF 0.750 H 0.450 T  m2 (a) 0.361 A (b) 18.1 V (c) 23.9 V (d)
53.0° (a) 1.43 k (b) 0.097 9 A (c) 51.1° (d) voltage leads current (a) 89.6 V (b) 108 V 1.88 V (a) 78.5  (b) 1.59 k (c) 1.52 k (d) 138 mA (e)
84.3° (f) V R  20.7 V, V L  10.8 V, VC  2.20  102 V (a) 103 V (b) 150 V (c) 127 V (d) 23.6 V (a) 208  (b) 40.0  (c) 0.541 H 4.30 W (a) 1.8  102  (b) 0.71 H (a) 5.81  107 Hz (b) Yes; the resistance of the circuit is not needed. The circuit’s resonance frequency is found by equating the inductive reactance to the capacitive reactance, which leads to Equation 21.19. C min  4.9 nF, C max  51 nF 0.242 J 721 V 0.18% is lost (a) 1.1  103 kW (b) 3.1  102 A (c) 8.3  103 A 1 000 km; there will always be better use for tax money. 1.10  10
6 T fred  4.55  1014 Hz, f IR  3.19  1014 Hz, E max,f /E max,i  0.57 E max  1.01  103 V/m, B max  3.35  10
6 T 2.94  108 m/s 5.45  1014 Hz (a) 188 m to 556 m (b) 2.78 m to 3.4 m 5.2  1013 Hz, 5.8 mm 4.299 999 84  1014 Hz;
1.6  107 Hz (the frequency decreases) (a) 184  (b) 653 mA (c) 1.44 H 1.7 cents 99.6 mH (a) 6.7  10
16 T (b) 5.3  10
17 W/m2 (c) 1.7  10
14 W (a) 0.536 N (b) 8.93  10
5 m/s2 (c) 33.9 days

CHAPTER 22 QUICK QUIZZES 1. 2. 3. 4.

(a) Beams 2 and 4 are reflected; beams 3 and 5 are refracted. (b) (c)

EX A MPLE QUESTIONS 1. 2. 3. 4. 5. 6. 7.

(a) (b) False (a) (c) (b) (b)

MULTIPLE-CHOICE QUESTIONS 1. 3. 5. 7. 9.

(e) (c) (b) (b) (a)

A.59

CONCEPTUAL QUESTIONS 1. (a) Away from the normal (b) increases (c) remains the same 3. No, the information in the catalog is incorrect. The index of refraction is given by n  c/v, where c is the speed of light in a vacuum and v is the speed of light in the material. Because light travels faster in a vacuum than in any other material, it is impossible for the index of refraction of any material to have a value less than 1. 5. There is no dependence of the angle of reflection on wavelength, because the light does not enter deeply into the material during reflection—it reflects from the surface. 7. On the one hand, a ball covered with mirrors sparkles by reflecting light from its surface. On the other hand, a faceted diamond lets in light at the top, reflects it by total internal reflection in the bottom half, and sends the light out through the top again. Because of its high index of refraction, the critical angle for diamond in air for total internal reflection, namely uc  sin
1(n air/n diamond), is small. Thus, light rays enter through a large area and exit through a very small area with a much higher intensity. When a diamond is immersed in carbon disulfide, the critical angle is increased to uc  sin
1(n carbon disulfide/n diamond). As a result, the light is emitted from the diamond over a larger area and appears less intense. 9. The index of refraction of water is 1.333, quite different from that of air, which has an index of refraction of about 1. The boundary between the air and water is therefore easy to detect, because of the differing diffraction effects above and below the boundary. (Try looking at a glass half full of water.) The index of refraction of liquid helium, however, happens to be much closer to that of air. Consequently, the defractive differences above and below the helium-air boundary are harder to see. 11. The diamond acts like a prism, dispersing the light into its spectral components. Different colors are observed as a consequence of the manner in which the index of refraction varies with the wavelength. 13. Light travels through a vacuum at a speed of 3  108 m/s. Thus, an image we see from a distant star or galaxy must have been generated some time ago. For example, the star Altair is 16 lightyears away; if we look at an image of Altair today, we know only what Altair looked like 16 years ago. This may not initially seem significant; however, astronomers who look at other galaxies can get an idea of what galaxies looked like when they were much younger. Thus, it does make sense to speak of “looking backward in time.”

PROBLEMS 1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21. 23. 25. 27. 29. 31.

3.00  108 m/s (a) 2.07  103 eV (b) 4.14 eV (a) 2.25  108 m/s (b) 1.97  108 m/s (c) 1.24  108 m/s (a) 29° (b) 26° (c) 32° 19.5° above the horizontal (a) 1.52 (b) 417 nm (c) 4.74  1014 Hz (d) 1.98  108 m/s 111° (a) 1.559  108 m/s (b) 329.1 nm (c) 4.738  1014 Hz five times from the right-hand mirror and six times from the left u  30.4°, u  22.3° 6.39 ns (b) 4.7 cm u  tan
1(ng) 3.39 m ured  48.22°, ublue  47.79° ured
uviolet  0.31°

A.60

Answers to Quick Quizzes, Example Questions, Odd-Numbered Multiple-Choice Questions, Conceptual Questions, and Problems

33. (a) u1i  30°, u1r  19°, u2i  41°, u2r  77° (b) First surface: ureflection  30°; second surface: ureflection  41° 35. (a) 66.1° (b) 63.5° (c) 59.7° 37. (a) 40.8° (b) 60.6° 39. 1.414 41. (a) 10.7° (b) air (c) Sound falling on the wall from most directions is 100% reflected. 43. 27.5° 45. 22.0° 47. 22.5° 49. (a) 38.5° (b) 1.44 53. 24.7° 55. 1.93 57. (a) 1.20 (b) 3.40 ns 59. (a) 53.1° (b) u1  38.7°

CHAPTER 23 QUICK QUIZZES 1. 2. 3. 4. 5. 6.

At C. (c) (a) False (b) False (c) True (b) An infinite number (a) False (b) True (c) False

EX A MPLE QUESTIONS No (b) True (b) (a) (c) The screen should be placed one focal length away from the lens. 8. No. The largest magnification would be 1, so a diverging lens would not make a good magnifying glass. 9. Upright

1. 2. 3. 4. 5. 6. 7.

MULTIPLE-CHOICE QUESTIONS 1. 3. 5. 7. 9.

(b), (c), (e) (d) (c) (d), (e) (c)

CONCEPTUAL QUESTIONS 1. You will not be able to focus your eyes on both the picture and your image at the same time. To focus on the picture, you must adjust your eyes so that an object several centimeters away (the picture) is in focus. Thus, you are focusing on the mirror surface. But your image in the mirror is as far behind the mirror as you are in front of it. Thus, you must focus your eyes beyond the mirror, twice as far away as the picture to bring the image into focus. 3. A single flat mirror forms a virtual image of an object due to two factors. First, the light rays from the object are necessarily diverging from the object, and second, the lack of curvature of the flat mirror cannot convert diverging rays to converging rays. If another optical element is first used to cause light rays to converge, then the flat mirror can be placed in the region in which the converging rays are present, and it will change the direction of the rays so that the real image is formed at a different location. For example, if a real image is formed by a convex lens, and the flat mirror is placed between the lens and the image position, the image formed by the mirror will be real.

5. Light rays diverge from the position of a virtual image just as they do from an actual object. Thus, a virtual image can be as easily photographed as any object can. Of course, the camera would have to be placed near the axis of the lens or mirror in order to intercept the light rays. 7. We consider the two trees to be two separate objects. The far tree is an object that is farther from the lens than the near tree. Thus, the image of the far tree will be closer to the lens than the image of the near tree. The screen must be moved closer to the lens to put the far tree in focus. 9. If a converging lens is placed in a liquid having an index of refraction larger than that of the lens material, the direction of refractions at the lens surfaces will be reversed, and the lens will diverge light. A mirror depends only on reflection which is independent of the surrounding material, so a converging mirror will be converging in any liquid. 11. This is a possible scenario. When light crosses a boundary between air and ice, it will refract in the same manner as it does when crossing a boundary of the same shape between air and glass. Thus, a converging lens may be made from ice as well as glass. However, ice is such a strong absorber of infrared radiation that it is unlikely you will be able to start a fire with a small ice lens. 13. The focal length for a mirror is determined by the law of reflection from the mirror surface. The law of reflection is independent of the material of which the mirror is made and of the surrounding medium. Thus, the focal length depends only on the radius of curvature and not on the material. The focal length of a lens depends on the indices of refraction of the lens material and surrounding medium. Thus, the focal length of a lens depends on the lens material.

PROBLEMS

1. on the order of 10
9 s younger 3. 10.0 ft, 30.0 ft, 40.0 ft 7. (a) 13.3 cm in front of the mirror (b)
0.333; the image is real and inverted as in Figure 23.13a. 9. (a) q 
7.50 cm, so the image is behind the mirror. (b) The magnification is 0.25, so the image has a height of 0.50 cm and is upright. 11. 5.00 cm 13. 1.0 m 15. 8.05 cm 17. (a) 5.0 cm behind the mirror (b)
10.0 cm 19. (a) concave with focal length f  0.83 m (b) Object must be 1.0 m in front of the mirror. 21. 38.2 cm below the upper surface of the ice 23. 3.8 mm 25. n  2.00 27. (a) The image is 0.790 m from the outer surface of the glass pane, inside the water tank. (b) With a glass pane of negligible thickness, the image is 0.750 m inside the tank. (c) The thicker the glass, the greater the distance between the final image and the outer surface of the glass. 29. 20.0 cm 31. (a) 20.0 cm beyond the lens; real, inverted, M 
1.00 (b) No image is formed. Parallel rays leave the lens. (c) 10.0 cm in front of the lens; virtual, upright, M  2.00 33. (a) 13.3 cm in front of the lens, virtual, upright, M  1/3 (b) 10.0 cm in front of the lens, virtual, upright, M  1/2 (c) 6.67 cm in front of the lens, virtual, upright, M  2/3 35. (a) either 9.63 cm or 3.27 cm (b) 2.10 cm 37. (a) 39.0 mm (b) 39.5 mm 39. 40.0 cm 41. 30.0 cm to the left of the second lens, M 
3.00 43. 7.47 cm in front of the second lens; 1.07 cm; virtual, upright 45. from 0.224 m to 18.2 m 47. real image, 5.71 cm in front of the mirror 49. 38.6°

Answers to Quick Quizzes, Example Questions, Odd-Numbered Multiple-Choice Questions, Conceptual Questions, and Problems 160 cm to the left of the lens, inverted, M 
0.800 q  10.7 cm 32.0 cm to the right of the second surface (real image) (a) 20.0 cm to the right of the second lens; M 
6.00 (b) inverted (c) 6.67 cm to the right of the second lens; M 
2.00; inverted 59. (a) 1.99 (b) 10.0 cm to the left of the lens (c) inverted 61. (a) 5.45 m to the left of the lens (b) 8.24 m to the left of the lens (c) 17.1 m to the left of the lens (d) by surrounding the lens with a medium having a refractive index greater than that of the lens material. 63. (a) 263 cm (b) 79.0 cm 51. 53. 55. 57.

7.

9.

CHAPTER 24 QUICK QUIZZES 1. 2. 3. 4. 5. 6.

(c) (c) (c) (b) (b) The compact disc

11.

EX A MPLE QUESTIONS 1. 2. 3. 4. 5. 6.

7. 8.

False The soap film is thicker in the region that reflects red light. The coating should be thinner. The wavelength is smaller in water than in air, so the distance between dark bands is also smaller. False Because the wavelength becomes smaller in water, the angles to the first maxima become smaller, resulting in a smaller central maximum. The separation between principal maxima will be larger. The additional polarizer must make an angle of 90° with respect to the previous polarizer.

MULTIPLE-CHOICE QUESTIONS 1. 3. 5. 7. 9.

(a) (b) (b) (c) (d)

CONCEPTUAL QUESTIONS 1. You will not see an interference pattern from the automobile headlights, for two reasons. The first is that the headlights are not coherent sources and are therefore incapable of producing sustained interference. Also, the headlights are so far apart in comparison to the wavelengths emitted that, even if they were made into coherent sources, the interference maxima and minima would be too closely spaced to be observable. 3. The result of the double slit is to redistribute the energy arriving at the screen. Although there is no energy at the location of a dark fringe, there is four times as much energy at the location of a bright fringe as there would be with only a single narrow slit. The total amount of energy arriving at the screen is twice as much as with a single slit, as it must be according to the law of conservation of energy. 5. One of the materials has a higher index of refraction than water, and the other has a lower index. The material with the higher index will appear black as it approaches zero thickness. There will be a 180° phase change for the light reflected from the upper surface, but no such phase change

13.

A.61

for the light reflected from the lower surface, because the index of refraction for water on the other side is lower than that of the film. Thus, the two reflections will be out of phase and will interfere destructively. The material with index of refraction lower than water will have a phase change for the light reflected from both the upper and the lower surface, so that the reflections from the zero-thickness film will be back in phase and the film will appear bright. Because the light reflecting at the lower surface of the film undergoes a 180° phase change, while light reflecting from the upper surface of the film does not undergo such a change, the central spot (where the film has near zero thickness) will be dark. If the observed rings are not circular, the curved surface of the lens does not have a true spherical shape. For regional communication at the Earth’s surface, radio waves are typically broadcast from currents oscillating in tall vertical towers. These waves have vertical planes of polarization. Light originates from the vibrations of atoms or electronic transitions within atoms, which represent oscillations in all possible directions. Thus, light generally is not polarized. Yes. In order to do this, fi rst measure the radar reflectivity of the metal of your airplane. Then choose a light, durable material that has approximately half the radar reflectivity of the metal in your plane. Measure its index of refraction, and place onto the metal a coating equal in thickness to onequarter of 3 cm, divided by that index. Sell the plane quickly, and then you can sell the supposed enemy new radars operating at 1.5 cm, which the coated metal will reflect with extrahigh efficiency. If you wish to perform an interference experiment, you need monochromatic coherent light. To obtain it, you must first pass light from an ordinary source through a prism or diffraction grating to disperse different colors into different directions. Using a single narrow slit, select a single color and make that light diffract to cover both slits for a Young’s experiment. The procedure is much simpler with a laser because its output is already monochromatic and coherent.

PROBLEMS 1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21. 23. 25. 27. 29. 31. 33. 35. 37. 39. 41. 43. 45. 47. 49. 51.

632 nm (a) 2.6 mm (b) 2.62 mm (a) 36.2° (b) 5.08 cm (c) 5.08  1014 Hz (a) 55.7 m (b) 124 m 515 nm 11.3 m 148 m 75.0 m 85.4 nm 550 nm 0.500 cm (a) 238 nm (b) l will increase (c) 328 nm 4.35 mm 4.75 mm No, the wavelengths intensified are 276 nm, 138 nm, 92.0 nm, . . . (a) 8.10 mm (b) 4.05 mm (a) 1.1 m (b) 1.7 mm 462 nm 1.20 mm, 1.20 mm (a) 479 nm, 647 nm, 698 nm (b) 20.5°, 28.3°, 30.7° 5.91° in first order; 13.2° in second order; and 26.5° in third order 44.5 cm 514 nm 9.13 cm (a) 25.6° (b) 19.0° (a) 1.11 (b) 42.0°

A.62 53. 55. 59. 61. 63. 65. 67. 69. 71. 73.

Answers to Quick Quizzes, Example Questions, Odd-Numbered Multiple-Choice Questions, Conceptual Questions, and Problems

(a) 56.7° (b) 48.8° (a) 55.6° (b) 34.4° 6.89 units (a) 413.7 nm, 409.7 nm (b) 8.6° One slit 0.12 mm wide. The central maximum is twice as wide as the other maxima. 0.156 mm 2.50 mm (a) 16.6 m (b) 8.28 m 127 m 0.350 mm

CHAPTER 25 QUICK QUIZZES 1. (c) 2. (a)

EX A MPLE QUESTIONS 1. True 2. True 3. A smaller focal length gives a greater magnification and should be selected. 4. True 5. Yes. Increasing the focal length of the mirror increases the magnification. Increasing the focal length of the eyepiece decreases the magnification. 6. More widely-spaced eyes increase visual resolving power by effectively increasing the aperture size, D, in Equation 25.10. The limiting angle of resolution is thereby decreased, meaning finer details of distant objects can be resolved. 7. Resolution is better at the violet end of the visible spectrum. 8. True

MULTIPLE-CHOICE QUESTIONS 1. 3. 5. 7. 9.

(c) (c) (b) (c) (c)

CONCEPTUAL QUESTIONS 1. The observer is not using the lens as a simple magnifier. For a lens to be used as a simple magnifier, the object distance must be less than the focal length of the lens. Also, a simple magnifier produces a virtual image at the normal near point of the eye, or at an image distance of about q 
25 cm. With a large object distance and a relatively short image distance, the magnitude of the magnification by the lens would be considerably less than one. Most likely, the lens in this example is part of a lens combination being used as a telescope. 3. The image formed on the retina by the lens and cornea is already inverted. 5. There will be an effect on the interference pattern—it will be distorted. The high temperature of the flame will change the index of refraction of air for the arm of the interferometer in which the match is held. As the index of refraction varies randomly, the wavelength of the light in that region will also vary randomly. As a result, the effective difference in length between the two arms will fluctuate, resulting in a wildly varying interference pattern. 7. Large lenses are difficult to manufacture and machine with accuracy. Also, their large weight leads to sagging, which produces a distorted image. In reflecting telescopes, light does not pass through glass; hence, problems associated with chromatic aberrations are eliminated. Large-diameter reflecting telescopes are also technically easier to construct.

Some designs use a rotating pool of mercury as the reflecting surface. 9. In order for someone to see an object through a microscope, the wavelength of the light in the microscope must be smaller than the size of the object. An atom is much smaller than the wavelength of light in the visible spectrum, so an atom can never be seen with the use of visible light. 11. farsighted; converging 13. In a nearsighted person the image of a distant object focuses in front of the retina. The cornea needs to be flattened so that its focal length is increased.

PROBLEMS 1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21. 23. 25. 27. 29. 31. 33. 37. 39. 41. 43. 45. 47. 49. 51. 53. 55. 57. 59. 61.

7.0 177 m 1.4 8.0 42.9 cm, 2.33 diopters 23.2 cm (a)
2.00 diopters (b) 17.6 cm 17.0 diopters
2.50 diopters, a diverging lens (a) 5.8 cm (b) m  4.3 (a) 4.07 cm (b) m  7.14 (a)
M
 1.22 (b) u/u0  6.08 2.1 cm 1.84 m m 
115 fo  90 cm, fe  2.0 cm (b)
fh/p (c)
1.07 mm 492 km 0.40 mrad 5.40 mm 9.8 km No. A resolving power of 2.0  105 is needed, and that available is only 1.8  105. 1.31  102 50.4 mm 40 (a) 8.00  102 (b) The image is inverted. (a) 1.0  103 lines (b) 3.3  102 lines (a) 2.67 diopters (b) 0.16 diopter too low (a) 44.6 diopters (b) 3.03 diopters (a) m  4.0 (b) m  3.0

CHAPTER 26 QUICK QUIZZES False: the speed of light is c for all observers. (a) False No. From your perspective you’re at rest with respect to the cabin, so you will measure yourself as having your normal length, and will require a normal-sized cabin. 5. (a), (e); (a), (e) 6. False 7. (a) False (b) False (c) True (d) False

1. 2. 3. 4.

EX A MPLE QUESTIONS 1. 2. 3. 4. 5.

9.61 s (c) (a) (a) Very little of the mass is converted to other forms of energy in these reactions because the total number of neutrons and protons doesn’t change. The energy liberated is only the energy associated with their interactions.

Answers to Quick Quizzes, Example Questions, Odd-Numbered Multiple-Choice Questions, Conceptual Questions, and Problems

MULTIPLE-CHOICE QUESTIONS 1. 3. 5. 7. 9.

(d), (e) (b), (c) (c) (e) (d)

CONCEPTUAL QUESTIONS 1. An ellipsoid. The dimension in the direction of motion would be measured to be less than D. 3. No. The principle of relativity implies that nothing can travel faster than the speed of light in a vacuum, which is equal to 3.00  108 m/s. 5. The light from the quasar moves at 3.00  108 m/s. The speed of light is independent of the motion of the source or the observer. 7. For a wonderful fictional exploration of this question, get a “Mr. Tompkins” book by George Gamow. All of the relativity effects would be obvious in our lives. Time dilation and length contraction would both occur. Driving home in a hurry, you would push on the gas pedal not to increase your speed very much, but to make the blocks shorter. Big Doppler shifts in wave frequencies would make red lights look green as you approached and make car horns and radios useless. High-speed transportation would be very expensive, requiring huge fuel purchases, as well as dangerous, since a speeding car could knock down a building. When you got home, hungry for lunch, you would find that you had missed dinner; there would be a five-day delay in transit when you watch a live TV program originating in Australia. Finally, we would not be able to see the Milky Way, since the fireball of the Big Bang would surround us at the distance of Rigel or Deneb. 9. A photon transports energy. The relativistic equivalence of mass and energy means that is enough to give it momentum.

PROBLEMS 1. (a) 1.38 yr (b) 1.31 lightyears 3.

"3 2

c

5. (a) 1.3  10
7 s (b) 38 m (c) 7.6 m 7. (a) 1.55  10
5 s (b) 7.09 (c) 2.19  10
6 s (d) 649 m (e) From the third observer’s point of view, the muon is traveling faster, so according to the third observer, the muon’s lifetime is longer than that measured by the observer at rest with respect to Earth. 9. 0.950c 11. Yes, with 19 m to spare 13. (a) 1.09  10
23 kg  m/s (b) 1.89  10
22 kg  m/s 15. 0.285c 17. (a) 1.35  1016 J (b) 1.75  1016 J (c) 3.10  1016 J 19. 0.786c 21. 18.4 g/cm3 25. (a) 3.91  104 (b) 7.67 cm 27. (a) 0.979c (b) 0.065 2c (c) 0.914c 29. (a) 3.10  105 m/s (b) 0.758c 31. (a) v/c  1
1.12  10
10 (b) 6.00  1027 J (c) $2.17  1020 33. 0.80c 35. (a) v  0.141c (b) v  0.436c 37. (a) 7.0 ms (b) 1.1  104 muons 39. 5.45 yr; Goslo is older 41. (a) 17.4 m (b) 3.30° with respect to the direction of motion

CHAPTER 27 QUICK QUIZZES 1. True 2. (b)

A.63

3. (c) 4. False 5. (c)

EX A MPLE QUESTIONS False False Some of the photon’s energy is transferred to the electron. Doubling the speed of a particle doubles its momentum, reducing the particle’s wavelength by a factor of one-half. This answer is no longer true when the doubled speed is relativistic. 5. True

1. 2. 3. 4.

MULTIPLE-CHOICE QUESTIONS 1. 3. 5. 7. 9.

(a) (c) (d) (c) (b)

CONCEPTUAL QUESTIONS 1. The shape of an object is normally determined by observing the light reflecting from its surface. In a kiln, the object will be very hot and will be glowing red. The emitted radiation is far stronger than the reflected radiation, and the thermal radiation emitted is only slightly dependent on the material from which the object is made. Unlike reflected light, the emitted light comes from all surfaces with equal intensity, so contrast is lost and the shape of the object is harder to discern. 3. The “blackness” of a blackbody refers to its ideal property of absorbing all radiation incident on it. If an observed room temperature object in everyday life absorbs all radiation, we describe it as (visibly) black. The black appearance, however, is due to the fact that our eyes are sensitive only to visible light. If we could detect infrared light with our eyes, we would see the object emitting radiation. If the temperature of the blackbody is raised, Wien’s law tells us that the emitted radiation will move into the visible range of the spectrum. Thus, the blackbody could appear as red, white, or blue, depending on its temperature. 5. All objects do radiate energy, but at room temperature this energy is primarily in the infrared region of the electromagnetic spectrum, which our eyes cannot detect. (Pit vipers have sensory organs that are sensitive to infrared radiation; thus, they can seek out their warm-blooded prey in what we would consider absolute darkness.) 7. We can picture higher frequency light as a stream of photons of higher energy. In a collision, one photon can give all of its energy to a single electron. The kinetic energy of such an electron is measured by the stopping potential. The reverse voltage (stopping voltage) required to stop the current is proportional to the frequency of the incoming light. More intense light consists of more photons striking a unit area each second, but atoms are so small that one emitted electron never gets a “kick” from more than one photon. Increasing the intensity of the light will generally increase the size of the current, but will not change the energy of the individual electrons that are ejected. Thus, the stopping potential remains constant. 9. Wave theory predicts that the photoelectric effect should occur at any frequency, provided that the light intensity is high enough. However, as seen in photoelectric experiments, the light must have sufficiently high frequency for the effect to occur. 11. (a) Electrons are emitted only if the photon frequency is greater than the cutoff frequency.

A.64

Answers to Quick Quizzes, Example Questions, Odd-Numbered Multiple-Choice Questions, Conceptual Questions, and Problems

13. No. Suppose that the incident light frequency at which you first observed the photoelectric effect is above the cutoff frequency of the first metal, but less than the cutoff frequency of the second metal. In that case, the photoelectric effect would not be observed at all in the second metal.

PROBLEMS 1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21. 23. 25. 27. 29. 31. 33. 35. 37. 39. 41. 43. 45. 47. 49.

(a) 3 000 K (b) 20 000 K 5.2  103 K (a) 2.49  10
5 eV (b) 2.49 eV (c) 249 eV 2.27  1030 photons/s (a) 2.24 eV (b) 555 nm (c) 5.41  1014 Hz (a) 1.02  10
18 J (b) 1.53  1015 Hz (c) 196 nm (d) 2.15 eV (e) 2.15 V 4.8  1014 Hz, 2.0 eV 1.2  102 V and 1.2  107 V, respectively 17.8 kV 0.078 nm 0.281 nm 70° 1.18  10
23 kg  m/s, 478 eV (a) 1.46 km/s (b) 7.28  10
11 m 3.58  10
13 m (a) 15 keV (b) 1.2  102 keV 10 6 m/s 116 m/s 3  10
29 J (a) 2.21  10
32 kg  m/s (b) 1.00  1010 Hz (c) 4.14  10
5 eV 2.5  1020 photons 5 200 K; clearly, a firefly is not at that temperature, so this cannot be blackbody radiation. 1.36 eV 2.00 eV (a) 0.022 0c (b) 0.999 2c

3.

5.

7.

9.

CHAPTER 28 QUICK QUIZZES 1. (b) 2. (a) 5 (b) 9 (c) 25 3. (d)

11.

EX A MPLE QUESTIONS 1. The energy associated with the quantum number n increases with increasing quantum number n, going to zero in the limit of arbitrarily large n. A transition from a very high energy level to the ground state therefore results in the emission of photons approaching an energy of 13.6 eV, the same as the ionization energy. 2. The energy difference in the helium atom will be four times that of the same transition in hydrogen. Energy levels in hydrogen-like atoms are proportional to Z 2, where Z is the atomic number. 3. The quantum numbers n and  are never negative. 4. No. The M shell is at a higher energy; hence, transitions from the M shell to the K shell will always result in more energetic photons than any transition from the L shell to the K shell.

MULTIPLE-CHOICE QUESTIONS 1. 3. 5. 7. 9.

(a) (e) (c) (b), (e) (c)

CONCEPTUAL QUESTIONS 1. If the energy of the hydrogen atom were proportional to n (or any power of n), then the energy would become infinite

13.

15.

as n grew to infinity. But the energy of the atom is inversely proportional to n 2. Thus, as n grows to infinity, the energy of the atom approaches a value that is above the ground state by a finite amount, namely, the ionization energy 13.6 eV. As the electron falls from one bound state to another, its energy loss is always less than the ionization energy. The energy and frequency of any emitted photon are finite. The characteristic x-rays originate from transitions within the atoms of the target, such as an electron from the L shell making a transition to a vacancy in the K shell. The vacancy is caused when an accelerated electron in the x-ray tube supplies energy to the K shell electron to eject it from the atom. If the energy of the bombarding electrons were to be increased, the K shell electron will be ejected from the atom with more remaining kinetic energy. But the energy difference between the K and L shell has not changed, so the emitted x-ray has exactly the same wavelength. A continuous spectrum without characteristic x-rays is possible. At a low accelerating potential difference for the electron, the electron may not have enough energy to eject an electron from a target atom. As a result, there will be no characteristic x-rays. The change in speed of the electron as it enters the target will result in the continuous spectrum. The hologram is an interference pattern between light scattered from the object and the reference beam. If anything moves by a distance comparable to the wavelength of the light (or more), the pattern will wash out. The effect is just like making the slits vibrate in Young’s experiment, to make the interference fringes vibrate wildly so that a photograph of the screen displays only the average intensity everywhere. If the Pauli exclusion principle were not valid, the elements and their chemical behavior would be grossly different, because every electron would end up in the lowest energy level of the atom. All matter would therefore be nearly alike in its chemistry and composition, since the shell structures of each element would be identical. Most materials would have a much higher density, and the spectra of atoms and molecules would be very simple, resulting in the existence of less color in the world. The three elements have similar electronic configurations, with filled inner shells plus a single electron in an s orbital. Because atoms typically interact through their unfi lled outer shells, and since the outer shells of these atoms are similar, the chemical interactions of the three atoms are also similar. Each of the eight electrons must have at least one quantum number different from each of the others. They can differ (in ms) by being spin-up or spin-down. They can differ (in ) in angular momentum and in the general shape of the wave function. Those electrons with   1 can differ (in m ) in orientation of angular momentum. Stimulated emission is the reason laser light is coherent and tends to travel in a well-defined parallel beam. When a photon passing by an excited atom stimulates that atom to emit a photon, the emitted photon is in phase with the original photon and travels in the same direction. As this process is repeated many times, an intense, parallel beam of coherent light is produced. Without stimulated emission, the excited atoms would return to the ground state by emitting photons at random times and in random directions. The resulting light would not have the useful properties of laser light.

PROBLEMS 1. (a) 121.5 nm, 102.5 nm, 97.20 nm (b) Far ultraviolet 3. (a) 2.3  10
8 N (b)
14 eV 5. (a) 1.6  10 6 m/s (b) No, v/c  5.3  10
3 1 (c) 0.46 nm (d) Yes. The wavelength is roughly the same size as the atom.

Answers to Quick Quizzes, Example Questions, Odd-Numbered Multiple-Choice Questions, Conceptual Questions, and Problems 7. (a) 0.212 nm (b) 9.95  10
25 kgm/s (c) 2.11  10
34 J  s (d) 3.40 eV (e)
6.80 eV (f)
3.40 eV 11. E 
1.51 eV (n  3) to E 
3.40 eV (n  2) 13. (a) 2.86 eV (b) 0.472 eV 15. (a) 13.6 eV (b) 1.51 eV 17. (a) 488 nm (b) 0.814 m/s 19. 4.42  104 m/s 21. (a) 0.476 nm ( b) 0.997 nm 23. (a) 0.026 5 nm (b) 0.017 6 nm (c) 0.013 2 nm 25. 1.33 nm ms 27. n  m 3 3 3 3 3 3 3 3 3 3

2 2 2 2 2 2 2 2 2 2

2 2 1 1 0 0
1
1
2
2

1 2 2 12 1 2 2 12 1 2 2 12 1 2 2 12 1 2 2 12

29. Fifteen possible states, as summarized in the following table: n 3 3 3 3 3 3 3  2 2 2 2 2 2 2 m  2 2 2 1 1 1 0 ms 1 0
1 1 0
1 1

3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 0 0
1
1
1
2
2
2 0
1 1 0
1 1 0
1

31. (a) 30 possible states (b) 36 33. (a) n  4 and   2 (b) m   (0, 1, 2), ms  1/2 (c) 1s 22s 22p 63s 23p 63d104s 24p 64d 25s 2  [Kr] 4d 25s 2 35. 0.160 nm 37. L shell: 11.7 keV; M shell: 10.0 keV; N shell: 2.30 keV 39. (a) 10.2 eV (b) 7.88  104 K 41. (a) 5.40 keV (b) 3.40 eV (c) 5.39 keV 43. (a) 4.24  1015 W/m2 (b) 1.20  10
12 J

CHAPTER 29 QUICK QUIZZES 1. 2. 3. 4. 5.

False (c) (c) (a) and (b) (b)

EX A MPLE QUESTIONS 1. Tritium has the greater binding energy. Unlike tritium, helium has two protons that exert a repulsive electrostatic force on each other. The helium-3 nucleus is therefore not as tightly bound as the tritium nucleus. 2. Doubling the initial mass of radioactive material doubles the initial activity. Doubling the mass has no effect on the halflife. 3. 7.794  10
13 J, 4.69  1011 J 4. The binding energy of carbon-14 should be greater than nitrogen-14 because nitrogen has more protons in its nucleus. The mutual repulsion of the protons means that the nitrogen nucleus would require less energy to break up than the carbon nucleus. 5. No 6. Reactions between helium and beryllium can be found in the Sun.

A.65

MULTIPLE-CHOICE QUESTIONS 1. 3. 5. 7. 9. 11.

(e) (d) (a) (c) (b) (c), (e)

CONCEPTUAL QUESTIONS 1. Isotopes of a given element correspond to nuclei with different numbers of neutrons. This will result in a variety of different physical properties for the nuclei, including the obvious one of mass. The chemical behavior, however, is governed by the element’s electrons. All isotopes of a given element have the same number of electrons and, therefore, the same chemical behavior. 3. An alpha particle contains two protons and two neutrons. Because a hydrogen nucleus contains only one proton, it cannot emit an alpha particle. 5. In alpha decay, there are only two final particles: the alpha particle and the daughter nucleus. There are also two conservation principles: of energy and of momentum. As a result, the alpha particle must be ejected with a discrete energy to satisfy both conservation principles. However, beta decay is a three-particle decay: the beta particle, the neutrino (or antineutron), and the daughter nucleus. As a result, the energy and momentum can be shared in a variety of ways among the three particles while still satisfying the two conservation principles. This allows a continuous range of energies for the beta particle. 7. The larger rest energy of the neutron means that a free proton in space will not spontaneously decay into a neutron and a positron. When the proton is in the nucleus, however, the important question is that of the total rest energy of the nucleus. If it is energetically favorable for the nucleus to have one less proton and one more neutron, then the decay process will occur to achieve this lower energy. 9. Carbon dating cannot generally be used to estimate the age of a stone, because the stone was not alive to take up carbon from the environment. Only the ages of objects that were once alive can be estimated with carbon dating. 11. The protons, although held together by the nuclear force, are repelled by the electrostatic force. If enough protons were placed together in a nucleus, the electrostatic force would overcome the nuclear force, which is based on the number of particles, and cause the nucleus to fission. The addition of neutrons prevents such fission. The neutron does not increase the electrical force, being electrically neutral, but does contribute to the nuclear force. 13. The photon and the neutrino are similar in that both particles have zero charge and very little mass. (The photon has zero mass, but recent evidence suggests that certain kinds of neutrinos have a very small mass.) Both must travel at the speed of light and are capable of transferring both energy and momentum. They differ in that the photon has spin (intrinsic angular momentum)  and is involved in electromagnetic interactions, while the neutrino has spin /2, and is closely related to beta decays.

PROBLEMS

1. A  2, r  1.5 fm; A  60, r  4.7 fm; A  197, r  7.0 fm; A  239, r  7.4 fm 3. 1.8  102 m 5. (a) 27.6 N (b) 4.16  1027 m/s2 (c) 1.73 MeV 7. (a) 1.9  107 m/s (b) 7.1 MeV 9. (a) 8.1 MeV (b) 8.7 MeV (c) 8.6 MeV 11. 3.54 MeV

A.66

Answers to Quick Quizzes, Example Questions, Odd-Numbered Multiple-Choice Questions, Conceptual Questions, and Problems

13. 0.210 MeV/nucleon greater for 2113Na, attributable to less proton repulsion 15. 0.46 Ci 17. (a) 6.95  105 s (b) 9.98  10
7 s
1 (c) 1.9  104 decays/s (d) 1.9  1010 nuclei (e) 5, 0.200 mCi 19. 4.31  103 yr 21. (a) 5.58  10
2 h
1, 12.4 h (b) 2.39  1013 nuclei (c) 1.9 mCi 211 55 23. 1 a 2 65 28Ni* 1 b 2 82Pb 1 c 2 27Co  5 6 5 6 25. e decay, 27 Co → 26 Fe  e  n 27. (a) cannot occur spontaneously (b) can occur spontaneously 29. 18.6 keV 31. 4.22  103 yr 144  33. 1 a 2 21 10Ne 1 b 2 54Xe (c) X  e , X  n 35. 1 a 2 136C 1 b 2 105B 198 2 37. 1 a 2 197 79 Au 1 n S 80Hg 1 e 1 n (b) 7.88 MeV 1 39. 1 a 2 0n (b) Fluoride mass  18.000 953 u 41. 18.8 J 43. 24 d 45. (a) 8.97  1011 electrons (b) 0.100 J (c) 100 rad 47. (a) 3.18  10
7 mol (b) 1.92  1017 nuclei (c) 1.08  1014 Bq (d) 8.96  10 6 Bq 49. 46.5 d 51. (a) 4.0  10 9 yr (b) It could be no older. The rock could be younger if some 87Sr were initially present. 53. 54 mCi 55. 4.4  10
8 kg/h

CHAPTER 30 QUICK QUIZZES 1. (a) 2. (b)

EX A MPLE QUESTIONS 1. 2. 3. 4.

1 mg/yr 1.77  1012 J True No. That reaction violates conservation of baryon number.

MULTIPLE-CHOICE QUESTIONS 1. 3. 5. 7. 9.

(d) (b) (b) (d) (a), (e)

CONCEPTUAL QUESTIONS 1. The experiment described is a nice analogy to the Rutherford scattering experiment. In the Rutherford experiment, alpha particles were scattered from atoms and the scattering was consistent with a small structure in the atom containing the positive charge. 3. The largest charge quark is 2e/3, so a combination of only two particles, a quark and an antiquark forming a meson, could not have an electric charge of 2e. Only particles containing three quarks, each with a charge of 2e/3, can combine to produce a total charge of 2e. 5. Until about 700 000 years after the Big Bang, the temperature of the Universe was high enough for any atoms that

formed to be ionized by ambient radiation. Once the average radiation energy dropped below the hydrogen ionization energy of 13.6 eV, hydrogen atoms could form and remain as neutral atoms for relatively long period of time. 7. In the quark model, all hadrons are composed of smaller units called quarks. Quarks have a fractional electric charge and a baryon number of 13 . There are six flavors of quarks: up (u), down (d), strange (s), charmed (c), top (t), and bottom (b). All baryons contain three quarks, and all mesons contain one quark and one antiquark. Section 30.12 has a more detailed discussion of the quark model. 9. Baryons and mesons are hadrons, interacting primarily through the strong force. They are not elementary particles, being composed of either three quarks (baryons) or a quark and an antiquark (mesons). Baryons have a nonzero baryon number with a spin of either 12 or 32 . Mesons have a baryon number of zero and a spin of either 0 or 1. 11. All stable particles other than protons and neutrons have baryon number zero. Since the baryon number must be conserved, and the final states of the kaon decay contain no protons or neutrons, the baryon number of all kaons must be zero.

PROBLEMS 1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21.

23. 25.

27. 29. 31. 33. 35.

37. 39. 41. 43.

0.387 g 126 MeV (a) 16.2 kg (b) 117 g 2.9  103 km (1 800 miles) 1.01 g (a) 84Be (b) 126C (c) 7.27 MeV 3.07  1022 events/yr 14.1 MeV (a) 4.53  1023 Hz (b) 0.622 fm 67.5 MeV, 67.5 MeV/c, 1.63  1022 Hz (a) conservation of electron-lepton number and conservation of muon-lepton number (b) conservation of charge (c) conservation of baryon number (d) conservation of baryon number (e) conservation of charge nm (a) charge, baryon number, Le , L t (b) charge, baryon number, Le , L m , L t (c) charge, Le , L m , L t, strangeness number (d) charge, baryon number, Le , L m , L t, strangeness number (e) charge, baryon number, Le , L m , L t, strangeness number (f) charge, baryon number, Le , L m , L t, strangeness number 3.34  1026 electrons, 9.36  1026 up quarks, 8.70  1026 down quarks (a) " (b) p
(c) K0 (d) #
a neutron, udd 18.8 MeV (a) electron-lepton and muon-lepton numbers not conserved (b) electron-lepton number not conserved (c) charge not conserved (d) baryon and electron-lepton numbers not conserved (e) strangeness violated by 2 units 26 (b) 12 days 29.8 MeV 3.60  1038 protons/s

INDEX Page numbers followed by f and t refer to figures and tables respectively.

A Absorption, selective, polarization of light waves by, 809–811, 809f, 810f Absorption spectra, 893 AC. See Alternating current Accommodation, 825–826 Acrylic, index of refraction, 742f Actinium (Ac), radioactive series, 926t Action potentials, 613, 615f Activity. See Decay rate Adenine, 878 Air atmosphere, refraction of light by, 772–773, 772f dielectric constant, 558t dielectric strength, 558t index of refraction, 738t Aircraft, induced emf in, 671 Air pollution, removing particulate matter, 544–545, 544f Alkali metals, 904–905 Alpha (a) decay, 918, 921–922, 932 Alpha (a) particles, 891–892, 892f, 918, 921 Alpher, Ralph, 954f Alternating-current (AC) circuits, 696–710 average power delivered in, 707–708, 724 capacitors in, 699–701, 700f, 723 inductors in, 701–702, 701f, 723–724 resistors in, 696–699, 697f, 698f, 723 RL circuits, 705t RLC series circuits, 702–706, 702f, 703f, 705t, 724 applications, 708–709 average power delivered in, 707–708, 724 problem-solving strategies, 705 resonance in, 708–710, 708f, 724 Alternating-current (AC) generators, 676–680, 676f, 677f, 687 Alternating-current (AC) motors, 638 Alternating-current (AC) power source circuit symbol, 697 rms current, 697–699, 723 Alternating current (AC) transformers, 710–712, 710f, 724 Aluminum (Al) ground-state configuration, 904t as paramagnetic substance, 651 resistivity, 577t as superconductor, 584, 584t temperature coefficient of resistivity, 577t work function, 873t

Ammeter, 574–575, 575f Ampere (A), 570–571, 645 Ampère, André-Marie, 643, 643f Ampère’s circuital law for solenoid, 648, 650, 650f for wire, 643, 643f, 644f, 652 Analyzer, 809, 809f Anderson, Carl, 945 Angle(s), critical, in total internal reflection, 748, 748f, 752 Angle of deviation (d), 742, 742f Angle of incidence critical, in total internal reflection, 748, 748f, 752 in reflection, 735, 735f, 751 in refraction, 736, 737f Angle of reflection, 735, 735f, 742, 751 Angle of refraction, 736, 737f, 751 Angstrom (Å), 720 Angular magnification of simple magnifier, 829–830, 841 of telescope, 833, 841 Angular momentum (L), of electromagnetic wave, 717, 724 Antenna, electromagnetic wave production by, 714–715, 714f, 741f Antimatter, storage of, 640 Antineutrinos, 924, 932 Antiparticles, 918, 924, 944–945, 946t, 956 Antiquarks, 951–952, 951t Aperture of cameras, 781–782, 823f, 824 circular, resolution of, 836, 836f single-slit, resolution of, 835–836, 835f, 836f Apnea monitors, 669–670, 670f Aqueous humor, 824, 825f Architecture, stress models in, 813, 813f Argon (Ar) ground-state configuration, 904t as noble gas, 904 Arsenic (As), ground-state configuration, 904t Artificial radioactivity, 926 Astigmatism, 781f, 826 Astronomy. See also Sun; Telescopes atmospheric refraction, 772–773, 772f Big Bang theory, 954–955, 954f, 957 black holes, 864–865 curvature of spacetime, 864 observations, electromagnetic spectrum and, 722, 722f solar system, size of dust in, 717 stars color of, 871 surface temperature of, 871 Atmosphere, refraction of light by, 772–773, 772f

Atmospheric electric field, measurement of, 512, 512f Atom(s). See also Nucleus, atomic characteristic x-rays, 895f, 905–906, 905f, 909 emission spectra of, 892–894, 893f history of concept, 891–892, 891f, 892f, 913f magnetic effects produced by, 650–651, 650f, 651f spontaneous emission, 907, 907f, 909 stimulated absorption, 906–907, 907f, 909f stimulated emission, 907, 908f, 909 structure of Bohr model, 894–899, 894f, 908 and chemical properties, 903–905 early models, 891–892, 891f, 892f Pauli exclusion principle, 902–903, 903t, 909 quantum-mechanical model, 899–905, 908–909 shells and subshells, 900–903, 900t, 903t, 909 Atomic bomb, 879f, 939, 944f Atomic clock, 865 Atomic mass units (u), conversion to/ from rest energy, 861 Atomic number (Z), 913, 931 Atom smashers. See Cyclotron Atrial fibrillation, 586, 586f Atrioventricular (AV) node, 585, 585f Aurora australis, 894 Aurora borealis, 626f, 894 Automobiles battery, potential difference in, 537 hybrid gas-electric, 639, 639f and lightning, safety of occupants from, 515 rearview mirrors, 761–762, 761f roadway flashers, 608 side-view mirrors, 766, 766f windshield wipers, timed, 608 Average electric current (Iav), 570, 571, 588 AV (atrioventricular) node, 585, 585f Axial myopia, 826 Axon, 614, 614f

B Back side of convex mirror, 764, 764f of spherical refracting surface, 769 of thin lens, 775 Bacteria doubling time of, 608 magnetic navigation in, 629 Bakelite dielectric constant, 558t dielectric strength, 558t

I.1

I.2

Index

Balmer, Johann, 892 Balmer series, 892–893, 893f, 897, 897f Barely resolved image, 835, 836f Baryon(s), 945, 946t, 950, 950f, 951, 951t, 952, 952f Baryon number, conservation of, 947 Battery in automobile, 537 circuit symbol for, 548, 549f history of, 570 internal resistance, 595 open-circuit voltage, 595 terminal voltage, 595, 615–616 Becquerel (Bq), 919 Becquerel, Antoine Henri, 918, 918f Bednorz, J. Georg, 584 Bees, navigation in, 812 Benzene, index of refraction, 738t Beryllium (Be), ground-state configuration, 904t Beta (b) decay, 918, 922–924, 923f, 932 Beta (b) particles, 918 Biconcave lens, 773, 773f Biconvex lens, 773f Big Bang theory, 954–955, 954f, 957 Binding energy, 916–917, 917f, 931 Birds, magnetic navigation in, 629 Blackbody radiation, and quantum theory, 871–872, 871f, 872f, 886 Black holes, 864–865 Bohr, Niels, 892, 894, 895f Bohr correspondence principle, 898, 908 Bohr radius (a 0), 895f Boron (B), ground-state configuration, 904t Bosons, 944, 944t, 950, 952–953 Bottom quark (b), 950–951, 951t Bragg, W. L., 877 Bragg’s law, 877, 886 Bremsstrahlung, 876 Brewster, David, 812 Brewster’s angle, 812 Brewster’s law, 812, 815 Bright fringes conditions for, 792–793, 792f, 815 in thin films, 796–797, 815 location of, 793, 815 Bromine (Br), ground-state configuration, 904t Bunsen, Robert, 604f

C Calcium (Ca) ground-state configuration, 904t as paramagnetic substance, 651 Cameras, 823–824, 823f, 841 aperture of, 781–782, 823f, 824 digital, 823, 823f flash attachment, 547 f-number, 824, 841 Cancer treatment, electric field therapy, 699 Capacitance (C) definition of, 546 equivalent for parallel capacitors, 550–551, 563 for series capacitors, 552–553, 563

Capacitance (C) (Continued ) of parallel-plate capacitor with dielectric, 557–558, 557f, 560–561, 563 without dielectric (air), 547–548 problem-solving strategy for, 553 Capacitive reactance (XC), 700, 723 Capacitor(s) in AC circuits, 699–701, 700f, 723 impedance, 705t phase angle, 705t RC circuits, 705t RL circuits, 705t RLC series circuits, 702–706, 702f, 703f, 724 applications, 547, 547f, 556, 559 circuit symbol for, 548 commercial, structure of, 558–559, 559f in DC circuits in parallel, 549–551, 549f, 563 problem-solving strategy for, 553 RC circuits, 607–611, 607f, 616 applications, 608 in series, 551–553, 551f, 563 definition of, 546, 546f with dielectrics, 557–562, 557f, 559f, 563 electrolytic, 558–559, 559f energy stored in, 555–557, 555f, 563 maximum operating voltage, 555 parallel-plate, 523, 547–549, 547f, 562–563 variable, 558–559, 559f Carbon (C) carbon-14, radioactive decay of, 923, 924, 925–926 ground-state configuration, 904t resistivity, 577t temperature coefficient of resistivity, 577t Carbon dating, 924, 925–926 Carbon dioxide (CO2), index of refraction, 738t Carbon disulfide, index of refraction, 738t Carbon tetrachloride, index of refraction, 738t Cardiac pacemaker, 586 Carlson, Chester, 545 Case ground, 612–613, 613f Cat, eye, resolution of, 836 CCD. See Charge-coupled device CDs. See Compact discs Center of curvature, of spherical mirror, 762–763, 762f CERN. See European Council for Nuclear Research Chadwick, Robert, 927 Chain reaction, nuclear, 939–940, 940f Characteristic x-rays, 876f, 905–906, 905f, 909 Charge carriers definition of, 571 microscopic view of, 572–574, 572f, 588 Charge-coupled device (CCD), 823, 823f Charging by conduction, 499, 499f by induction, 499–500, 500f

Charmed quark (c), 950–951, 951t Charmonium, 951 Cheney, Dick, 586 Chlorine (Cl), ground-state configuration, 904t Choroid, 825f Christmas lights in series, 596–597 Chromatic aberration, 782, 782f, 783 Chromium (Cr) ground-state configuration, 904t as radioactive tracer, 931 Ciliary muscle, 825, 825f Circuit breakers, 600–601, 611–612, 611f, 612f Circular aperture, resolution of, 836, 836f Clocks, atomic, 865 Closed surface, definition of, 517 CMOS. See Complementary metal-oxide semiconductor Coal, as fuel, 940 Cobalt (Co) as ferromagnetic substance, 651 ground-state configuration, 904t radioactive, 930 Coherent light source, definition of, 790 Color (color charge), of quark, 952, 952f, 957 Color force, 952, 957 Coma (optical aberration), 781f Commutators, 638, 638f, 678, 678f Compact discs (CDs) multicolored surface of, 807 reading of, 800–801, 801f tracking mechanism, 807, 807f Compasses, 626, 628–629, 629f, 630 Complementary metal-oxide semiconductor (CMOS), 823 Compound microscopes, 830–832, 831f, 841 Compton, Arthur H., 879, 879f Compton shift, 879–880, 879f, 886 Compton wavelength (h/mec), 879 Computer(s) hard drives, 675 keys, capacitors in, 547, 547f Concave mirrors, 762, 766–769, 782 Conduction. See Electrical conduction Conductors. See Electrical conductors Cones, of eye, 824–825, 825f Conservation of baryon number, 947 of charge, 603, 616 of Coulomb force, 532 of electric charge, 498–499, 499f of energy in change of electric potential energy, 534–535 in electric circuits, 603–604, 616 of lepton number, 948 of strangeness, 949 Constructive interference, of light waves conditions for, 792–793, 792f, 796–797, 815 location of, 793, 815 Control rods, of nuclear reactor, 940, 940f

Index Converging (positive) thin lenses, 773, 773f, 774f, 775–778, 776f Conversions atomic mass units to/from rest energy, 861 electron volt to/from joules, 860–861 gauss to/from tesla, 630 Convex-concave lens, 773f Convex mirrors, 764, 766, 768, 782 Copper (Cu) ground-state configuration, 904t resistivity, 577t temperature coefficient of resistivity, 577t work function, 873t Cornea, 824, 825f Correspondence principle (Bohr), 898, 908 Cosmic Background Explorer (COBE), 955 Coulomb (C), 499, 645 Coulomb, Charles, 500, 501f Coulomb force, 502. See also Coulomb’s Law conservation of, 532 Coulomb’s constant (ke), 501, 523 Coulomb’s Law, 500–505, 502f, 523 Crick, F. H. C., 878 Critical angle, 748, 748f, 752 Critical condition, of nuclear reactor, 940 Critical temperature, of superconductor, 584, 584f, 584t Crystalline lens, 824, 825, 825f Crystalline solids, diffraction of x-rays by, 875, 875f, 876–878, 876f, 877f, 886 Crystallization of molecules, 878 Curie (Ci), 919 Curie, Marie, 918, 918f Curie, Pierre, 918, 918f Curie temperature, 651 Current. See Electric current Curvature of spacetime, 864 Cutoff frequency ( fc), in photoelectric effect, 873–874, 874f Cutoff wavelength (lc), in photoelectric effect, 874 Cyclotron(s) (accelerators), 537–538, 927, 953f Cyclotron equation, 639–642 Cytosine, 878

D Daughter nucleus, 921 Davisson-Germer experiment, 881 DC. See Direct current de Broglie, Louis, 880–881, 881f, 887 de Broglie wavelength, 881–882, 887, 894 Decay constant (l), 918 Decay rate (activity; R), 918–919, 919f, 932 Defibrillators external, 556 Implanted Cardioverter Defibrillators (ICDs), 586–587, 587f, 587t Dendrites, 614, 614f

Depolarization wave, 585, 585f Depth of field, of camera, 824 Destructive interference, of light waves conditions for, 792, 792f, 793, 796–797, 815 location of, 793 in single-slit diffraction, 803, 815 Deuterium binding energy, 916 as fusion fuel, 937f, 941–942, 956 as isotope, 913 Dialysis machines, 635 Diamagnetic material, 651, 651f Diamond index of refraction, 738t sparkling of, 749 Dielectric(s), 557–562, 563 atomic description of, 561–562, 561f and capacitance, 557–558, 557f, 560–561, 563 definition of, 557 Dielectric breakdown, 531f, 558, 558f Dielectric constant (k), 557–558, 558t, 563 Dielectric strength, 558, 558t Diffraction of light waves, 802, 815 diffraction gratings, 805–808, 805f, 806f, 815, 838–840, 841 Fraunhofer diffraction, 802, 802f, 803–804, 803f history of research on, 733 and resolution, 835–840 single-slit diffraction, 803–805, 803f, 815 of sound waves, 804 of x-rays by crystals, 875, 875f, 876–878, 876f, 877f, 886 Diffraction gratings, 805–808, 805f, 806f, 815 resolution of, 838–840, 841 Diffraction grating spectrometer, 806, 806f Diffuse reflection, 734–735, 734f Digital cameras, 823, 823f Digital multimeter, 575, 575f Digital videodiscs. See DVDs Diode, as nonohmic device, 576 Diopters, 826, 841 Dip angle, 629 Dirac, Paul A. M., 900, 944, 945f Dirac equation, 900 Direct-current (DC) circuits, 594–615 capacitors in parallel, 549–551, 549f, 563 capacitors in series, 551–553, 551f, 563 emf sources, 594–595, 615–616 Kirchhoff’s rules, 603–606, 603f, 604f, 616 problem-solving strategies capacitors, 553 Kirchhoff’s rules, 604 resistors, 601–602 RC circuits, 607–611, 607f, 616 applications, 608 resistors in parallel, 598–603, 598f, 616 resistors in series, 595–597, 596f, 616 RL circuits, 683–687, 683f, 684f, 688

I.3

Direct-current (DC) motors, 638–639, 638f Dispersion, of light, 742–746, 742f, 743f, 745f, 746f, 752 Diverging mirrors. See Convex mirrors Diverging (negative) thin lenses, 773, 773f, 774f, 775–776, 776f, 778–779 Diving masks, with vision-correcting lens, 776 DNA, structure of, determining, 877–878, 878f Domains, magnetic, 651, 651f Doppler effect, electromagnetic waves, 722–723 Double-helix structure of DNA, 878, 878f Double-slit experiment (Young), 791–795, 815 Douglas, Melissa, 937f Down quark (d), 950–951, 951t Drift speed, 572–574, 572f, 588 DVDs (digital videodiscs), reading of, 741–742, 800–801, 801f

E e (fundamental unit of electric charge), 499, 515–516 Earth atmosphere, refraction of light by, 772–773, 772f magnetic field of, 628–629, 628f, 629f, 635–636, 665 Eddington, Arthur, 864 Eightfold way, 949–950, 950f Einstein, Albert, 733, 850, 850f, 860, 863–864, 873, 886 EKGs (electrocardiograms), 585–586, 586f Electrical conduction charging by, 499, 499f in nervous system, 613–614, 614f Electrical conductors current-carrying electric potential in, 571 magnetic force on, 633–636, 633f, 634f, 652 defined, 499, 523 in electrostatic equilibrium (isolated) electric potential on, 542–543, 542f properties of, 513–515, 513f, 514f, 523 irregularly-shaped, charge accumulated on, 513–514, 514f parallel, magnetic force between, 645–646, 645f, 652 resistance. See Resistance Electrical resistivity. See Resistivity Electrical safety, 612–613, 613f Electric charge of atomic particles, 498, 499, 501, 501t, 914, 914t charging by conduction, 499, 499f charging by induction, 499–500, 500f conservation of, 498–499, 499f fundamental unit of (e), 499, 515–516 grounded, defined, 499–500 negative, 498, 498f

I.4

Index

Electric charge (Continued ) polarization of, 500 positive, 498, 498f properties of, 497–499, 498f, 499f, 523 SI units of, 499, 645 transfer of, 498, 499–500 Electric circuits. See also Alternatingcurrent (AC) circuits; Directcurrent (DC) circuits definition of, 574 electron flow in, 572–574, 572f, 588 energy transfer in, 580–581, 581f household circuits, 600–601, 611–612, 612f symbols AC power source, 697 battery, 548, 549f capacitor, 548 ground, 581, 581f inductor, 683, 683f lightbulbs, 549, 549f resistors, 548, 549f, 576 wires, 549, 549f Electric current, 570–572, 571f average (Iav), 570, 571, 588 definition of, 570 direction of, 571, 588 induced, 666 instantaneous (I), 571, 588 junction rule, 603–606, 603f, 616 measurement of, 574–575, 575f microscopic view of, 572–574, 572f, 588 Ohm’s law, 576, 588 in RL circuit, 683–685, 688 rms, 697–699, 723 SI unit of, 645 Electric current loop in magnetic field, torque on, 636–638, 636f, 652 magnetic field of, 646–648, 647f, 652 Electric dipole electric field lines, 511, 511f electric potential, 539, 539f Electric field, 505–510, 523 atmospheric, measurement of, 512, 512f definition of, 505–506, 506f direction of, 506, 506f, 507f, 523 of equipotential surface, 543, 562 on Gaussian surface (Gauss’s law), 517, 519–523 magnitude of, 506, 523 problem-solving strategy for, 508–509 SI units of, 506, 536 superposition principle, 507, 509–510 Electric field lines, 510–512, 511f, 512f, 523 equipotentials, 543, 544f Electric field therapySfor cancer, 699 Electric field vector (E ), 510 Electric flux (E), 517–519, 517f flux lines, sign of, 517 Gauss’s law, 517, 519–523 Electric force (Fe) Coulomb’s Law, 500–505, 502f problem-solving strategy for, 508–509 properties, 500 superposition principle for, 503–504 vs. gravitational force, 502–503

Electric guitar, sound production in, 669, 669f Electricity and heart function, 585–587, 585f, 586f human utilization of, 497, 570 safety, 612–613, 613f transmission over long distances, 711 Electric motors back emf in, 679–680, 680f structure and function, 638–639, 638f, 679 Electric potential (voltage; V) applications, 544–546 on charged conductors, 542–543, 542f in conductors carrying current, 571 difference in (V ) definition of, 535–536, 562 in uniform electric field, 536–538, 562 due to point charges, 538–542, 538f, 539f, 562 equipotential surfaces, 543, 562 loop rule, 603–606, 604f, 616 measurement of, 574–575, 575f Ohm’s law, 576, 588 problem-solving strategy for, 540 superposition principle for, 538, 540–541 Electric potential difference. See Electric potential Electric potential energy, 531–535, 532f change in, in constant electric field, 532–535, 562 due to point charges, 538–542, 538f, 562 and power, 580–581 vs. gravitational potential energy, 533, 533f Electrocardiograms (EKGs), 585–586, 586f Electrocardiographs, 585 Electrolytic capacitors, 558–559, 559f Electromagnet. See Solenoid Electromagnetic force, as fundamental force, 943, 944t, 953, 953f, 956 Electromagnetic pumps, 634f, 635 Electromagnetic spectrum, 720–722, 721f, 724 Electromagnetic waves Doppler effect in, 722–723 frequency vs. wavelength, 720 Hertz’s research on, 713–714, 713f, 714f, 724 Maxwell’s predictions regarding, 712–713, 724 production by antenna, 714–715, 714f, 741f properties of, 715–720, 724 quantization of, 873 (See also Quantum physics) spectrum of, 720–722, 721f, 724 speed of, 715 Electron(s) antiparticle of, 918, 924 in atomic structure, 498 charge, 498, 499, 501, 501t, 914, 914t

Electron(s) (Continued ) in electric circuits, drift speed of, 572–574, 572f, 588 electron clouds, 902, 902f as lepton, 946, 946t mass, 501t, 914, 914t number of, in gram of matter, 498 rest energy, 861 spin, 900–902, 900f Electron microscopes, 882, 883f Electron neutrino (ve), 946, 946t Electron volt (eV), 543 conversion to/from Joules, 860–861 Electrostatic air cleaners, 545 Electrostatic confinement, 547 Electrostatic equilibrium conductors in electric potential on, 542–543, 542f properties of, 513–515, 513f, 514f, 523 definition of, 513 Electrostatic forces, 501 Electrostatic precipitators, 544–545, 544f Electroweak force, 953, 954, 954f Electroweak theory, 953 Elements chemical properties, 903–905 periodic table, 903–905, 909 emf induced applications, 675–680 back emf, 679–680, 679f definition of, 663–664, 664f direction of, 667–668, 667f, 674–675, 674f, 675f Faraday’s law of magnetic induction, 666–670, 687 in generators, 676–680 Lenz’s law, 667–668, 674–675, 674f, 675f motional emf, 670–674, 670f, 671f, 687 self-induced emf, 680–683, 680f, 681f, 687 motional, 670–674, 670f, 671f, 687 sources of, 594–595, 615–616 Emission spectra, 892–894, 893f fine structure, 900 hydrogen, 892, 893f, 894, 896 mercury, 893f neon, 893f Endothermic nuclear reactions, 928, 932 Energy. See also Electric potential energy in capacitor, 555–557, 555f, 563 carried by electromagnetic wave, 716, 724 of characteristic x-rays, 905–906 in circuit, transfer of, 580–581, 581f conservation of in change of electric potential energy, 534–535 in electric circuits, 603–604, 616 from fission, 938–939 from fusion, 942, 943 gravitational potential energy vs. electric potential energy, 533, 533f in inductor, 686–687, 688

Index Energy (Continued ) kinetic energy, relativistic expression for, 859, 861, 865 mass equivalent of, 860, 862, 914 of photon, 733, 751 quantization of. See Quantum physics rest, 859, 865 conversion of atomic mass units to/ from, 861 of electron, 861 total, 859–860, 861, 866 and relativistic momentum, 860–861, 866 Energy level diagrams, hydrogen, 895–896, 896f Equilibrium, electrostatic conductors in electric potential on, 542–543, 542f properties of, 513–515, 513f, 514f, 523 definition of, 513 Equipotential(s), 543, 544f Equipotential surfaces, 543, 562 Eta (h), 946t Ether, luminiferous, 849–850, 849f, 850f Ether wind, 849–850, 849f, 850f Ethyl alcohol, index of refraction, 738t Euler’s constant, 607 European Council for Nuclear Research (CERN), 854 Event horizon, 864–865 Excited states, 907, 907f, 909 Exclusion principle (Pauli), 902–903, 903t, 909 Exothermic nuclear reactions, 928, 932 Exponential decay, 919, 919f Eye cat, resolution of, 836 fish, view from under water, 749–750 human, 824–829, 825f accommodation, 825–826 anatomy, 824–825, 825f conditions of, 825–829, 826f, 827f, 841 diving masks with vision-correcting lens, 776 image formation in, 824–825 red-eye in photographs, 735 and sunglasses UV protection, 720f underwater vision, 770 vision correction, 826–828, 826f, 827f wavelength sensitivity of, 722 Eyepiece, of microscope, 830, 831f

F Farad (F), 546 Faraday, Michael, 505, 510, 514, 663, 664f Faraday’s ice-pail experiment, 514, 514f Faraday’s law of magnetic induction, 666–670, 687 Farnsworth, Philo, 942 Far point, of eye accommodation, 825 Farsightedness (hyperopia), 825–827, 826f, 841 Femtometer (fm), 914–915 Fermi (fm), 914–915

Fermi, Enrico, 923, 924f, 939–940 Fermions, 950 Ferromagnetic substances, 651 Feynman, Richard, 870, 944, 944f Feynman diagram, 944, 944f Fiber optics, 750–751, 750f, 751f Field forces. See Electric force; Gravitational force; Magnetic force Field mill, 512, 512f Field particles, 944, 944t Films. See Thin films Fine structure, of emission spectra, 900 Fingerprints, dusting for, 628 Fireworks, 891f First-order maximum, 793, 806 Fish, view from under water, 749–750 Fission, 860, 862, 937–940, 938f, 956 Fission fragments, 937 Flashers, 608 Flat mirrors, 759–762, 759f, 760f, 782 Flat surface, images formed by refraction at, 770, 770f, 771–772 Fluorine (F), ground-state configuration, 904t Fluorite, index of refraction, 738t Flux. See Electric flux; Magnetic flux f-number of camera, 824, 841 range, of eye, 824 Focal length (f) of spherical mirror, 764, 765t, 782 of thin lens, 773, 774f, 775 Focal point (F) of concave mirror, 764 of thin lens, 773, 774f Force(s). See also Electric force; Gravitational force; Magnetic force color, 952, 957 Coulomb, 502 (See also Coulomb’s Law) conservation of, 532 electromagnetic, 943, 944t, 953, 953f, 956 electrostatic, 501 electroweak, 953, 954, 954f fundamental, 943–944, 944t, 954, 954f, 956 (See also specific forces) nuclear, 915, 952 strong Big Bang and, 954, 954f characteristics of, 943, 944t, 945, 956 as color force, 952 in Standard Model, 953, 953f weak Big Bang and, 954, 954f characteristics of, 944, 944t, 946, 956 and electroweak theory, 952–953 in Standard Model, 953, 953f Forensic science, fingerprints, 628 Fovea, 825f Frames of reference. See Reference frames Franklin, Benjamin, 498, 498f, 514 Franklin, Rosalind, 878, 878f Fraunhofer diffraction, 802, 802f, 803–804, 803f

I.5

Frequency (f) cutoff, in photoelectric effect, 873–874, 874f of matter waves, 881, 887 of radiation emitted by hydrogen orbit change, 894, 896–897, 908 Fresnel, Augustin, 802 Fresnel bright spot, 802, 802f Fringes bright conditions for, 792–793, 792f, 815 in thin films, 796–797, 815 location of, 793, 815 dark conditions for, 792, 792f, 793 in thin films, 796–797, 815 location of, 793 defined, 791f, 792 Fringe shift, 840 Front side of convex mirror, 764, 764f of spherical refracting surface, 769 of thin lens, 775 Fundamental forces, 943–944, 944t, 954, 954f, 956 Fundamental particles, 943 Fundamental unit of electric charge (e), 499, 515–516 Fuses, 600–601, 611 Fusion, nuclear, 941–943, 956 Fusion reactors, 937f, 941–943, 946

G Galaxy, center of, 865 Galilean relativity, 847–848, 848f Gallium (Ga), ground-state configuration, 904t Gamma (g) decay, 918, 924, 932 Gamma (g) rays, 721f, 722, 918, 924, 941 Gamow, George, 954f Gas(es), spectral analysis of, 743–744 Gauss (G), 630, 652 conversion to/from tesla, 630 Gauss, Karl Friedrich, 517 Gaussian surface, 519–520 Gauss’s law, for electric flux, 517, 519–523 Geiger, Hans, 891–892, 892f Gell-Mann, Murray, 950, 950f General relativity, 850f, 863–865 Generators, 676–680 alternating current (AC), 676–680, 676f, 677f, 687 direct current, 677–678, 678f Genetic damage, from radiation, 929 Geometric optics, 790 Germanium (Ge) ground-state configuration, 904t resistivity, 577t as semiconductor, 499 temperature coefficient of resistivity, 577t GFIs. See Ground-fault interrupters Glass Brewster’s angle for, 812 index of refraction, 738t, 742f as insulator, 499 resistivity, 577t Gluons, 944, 944t, 952

I.6

Index

Glycerine, index of refraction, 738t Goeppert-Mayer, Maria, 915f Gold resistivity, 577t temperature coefficient of resistivity, 577t Goudsmit, Samuel, 900 Grand unified theory (GUT), 947, 953–954 Gravitational force (Fg) as fundamental force, 944, 944t, 954, 954f, 956 in general relativity, 863–865 and light, bending of, 863f, 864–865, 865f in Standard Model, 953, 953f and time, 864, 865 vs. electric force, 502–503 Gravitational lenses, 865f Gravitational potential energy (PEg), vs. electric potential energy, 533, 533f Gravitons, 944, 944t Grimaldi, Francesco, 733 Ground, electrical defined, 499–500 symbol for, 581, 581f Ground-fault interrupters (GFIs), 613, 668–669, 668f, 669f Ground state, 895 Guanine, 878 GUT. See Grand unified theory

H Hadrons, 945–946, 946t, 956 quark model of, 950–952, 951t, 956–957 Hahn, Otto, 862 Hale telescope, 834f, 838 Half-life (T1/2), 919, 921, 932 Halogens, 905 Hard magnetic materials, 627, 651 Heart artificial, 635 electrical activity of, 585–587, 585f, 586f Heisenberg, Werner, 884, 884f, 899 Helium (He) ground-state configuration, 903, 904t helium-neon laser, 907, 908f as noble gas, 904 Henry (H), 681, 687 Henry, Joseph, 663, 681 Herman, Robert, 954f Herschel, William, 720 Hertz, Heinrich Rudolf, 713–714, 713f, 720, 724, 733, 872 Higgs boson, 953 Hilbert, David, 850f Household circuits, 600–601, 611–612, 612f Hubble Space Telescope, 723, 834 Human body DNA, structure of, determining, 877–878, 878f eye, 824–829, 825f accommodation, 825–826 anatomy, 824–825, 825f

Human body (Continued ) conditions of, 825–829, 826f, 827f, 841 diving masks with vision-correcting lens, 776 image formation in, 824–825 red-eye in photographs, 735 and sunglasses UV protection, 720f underwater vision, 770 vision correction, 826–828, 826f, 827f wavelength sensitivity of, 722 heart artificial, 635 electrical activity of, 585–587, 585f, 586f kidneys, dialysis, 635 nervous system, electrical signal conduction in, 613–614, 614f neurons electrical signal conduction in, 613–614, 614f structure of, 614, 614f types of, 614–615 potassium ion channels, 614–615 sodium ion channels, 614–615 Hummingbirds, feather color in, 790f Huygens, Christian, 732, 733f Huygens’ principle, 746–748, 746f, 747f, 752 Hydrogen (H) Balmer series, 897, 897f Bohr model of, 894–898, 894f, 908 emission spectrum, 892, 893f, 894, 896 energy level diagram, 895–896, 896f fusion of, in Sun, 941 ground-state configuration, 903, 904t ionization energy, 896 isotopes, 913 quantum mechanical model of, 899–902, 908–909 quantum state energies, 895, 908 Hydrogen bomb, 941 Hydrogen-like atoms, Bohr model and, 898–899, 908 Hyperopia (farsightedness), 825–827, 826f, 841

I ICDs. See Implanted Cardioverter Defibrillators Ice, index of refraction, 738t Ice-pail experiment (Faraday), 514, 514f Ideal transformers, 711, 724 Image barely resolved, 835, 836f definition of, 759 just resolved, 835, 836f, 841 not resolved, 835, 835f, 836f real, 759, 762, 765t, 782 resolved, 835, 835f virtual, 759–760, 765t, 782 Image distance, 759, 759f, 763, 769, 782 Image point, of spherical mirror, 762–763, 762f, 763f Impedance (Z), 703f, 704, 705t, 724 Implanted Cardioverter Defibrillators (ICDs), 586–587, 587f, 587t

Index of refraction, 737, 738t, 742, 742f, 751 Indium, as superconductor, 584 Induced current, 666 Induced emf applications, 675–680 back emf, 679–680, 679f definition of, 663–664, 664f direction of, 667–668, 667f, 674–675, 674f, 675f Faraday’s law of magnetic induction, 666–670, 687 in generators, 676–680 Lenz’s law, 667–668, 674–675, 674f, 675f motional, 670–674, 670f, 671f, 687 self-induced, 680–683, 680f, 681f, 687 Induced polarization, of molecules, in dielectric, 561, 561f Inductance (L), 681, 687 of solenoid, 682–683, 687 Induction, 663–664, 664f applications, 675–680 back emf, 679–680, 679f charging by, 499–500, 500f eliminating charge from houses, 514 direction of induced emf, 667–668, 667f, 674–675, 674f, 675f Faraday’s law of, 666–670, 687 in generators, 676–680 Lenz’s law, 667–668, 674–675, 674f, 675f motional emf, 670–674, 670f, 671f, 687 self-induced emf, 680–683, 680f, 681f, 687 Inductive reactance (X L), 701–702, 701f, 723–724 Inductor(s) in AC circuits, 701–702, 701f, 723–724 RL circuits, 705t RLC series circuits, 702–706, 702f, 703f, 724 circuit symbol for, 683, 683f in DC RL circuits, 683–687, 683f, 684f, 688 definition of, 683 energy stored in magnetic field of, 686–687, 688 impedance, 705t phase angle, 705t Inertial electrostatic confinement fusion, 937f, 942 Inertial laser confinement fusion, 942 Infrared waves, 721, 721f Instantaneous electric current (I), 571, 588 Insulators defined, 499, 523 induced charge on, 500, 501f Intensity (I), of electromagnetic wave, 716, 724 Interference, of light waves, 733. See also Diffraction conditions for, 790–793, 792f, 796–797, 815 constructive conditions for, 792–793, 792f, 796–797, 815 location of, 793, 815

Index Interference, of light waves (Continued ) destructive conditions for, 792, 792f, 793, 796–797, 815 location of, 793 in single-slit diffraction, 803, 815 in Michelson interferometer, 840–841, 840f Newton’s rings, 797–798, 797f in phase change due to reflection, 795–797, 795f, 796f, 815 in thin films, 796–800, 796f, 797f, 815 problem-solving strategies, 798 Young’s double-slit experiment, 791–795, 815 Interferometer, Michelson, 840–841, 840f, 849, 850f Interneurons, 613–614, 614f Iodine (I), as radioactive tracer, 930 Ionization energy, 896 Iris, 824, 825, 825f Iron (Fe) ground-state configuration, 904t and magnetism, 626, 627, 628, 651 resistivity, 577t temperature coefficient of resistivity, 577t work function, 873t Isotopes, 913, 931 I 2R loss, 581

J Jensen, Hans, 915f Joule (J), conversion to electron volt, 860–861 Junction rule, 603–606, 603f, 616 Just resolved image, 835, 836f, 841

K Kaons (K), 946t, 951t Keck telescopes, 823f, 834 Kidneys, dialysis, 635 Kilowatt-hour (kWh), 581–583 Kinetic energy (KE), relativistic expression for, 859, 861, 865 Kirchhoff, Gustav, 604f Kirchhoff’s rules, 603–606, 603f, 604f, 616 problem-solving strategies, 604 Krypton (Kr) ground-state configuration, 904t as noble gas, 904

L Lambda (l0), 946t, 951t Land, E. H., 809 Large Electron-Positron (LEP) collider, 953f Laser(s), 907, 908f, 909 Laser printers, 545–546, 545f Lateral magnification of microscope, 831, 841 of mirror, 760, 765t, 782 of spherical refracting surface, 769, 782 of thin lens, 774, 782

Laue pattern, 876–877 Law of conservation of baryon number, 947 Law of conservation of lepton number, 948 Law of refraction (Snell), 737–742, 751–752 Lawson’s criterion, 942, 956 LCDs. See Liquid crystal displays Lead resistivity, 577t as superconductor, 584, 584t temperature coefficient of resistivity, 577t work function, 873t Length contraction of, in special relativity, 856–858, 857f, 865 proper, 856, 865 Lens(es). See also Thin lenses aberrations in, 781–782, 781f, 782f, 783 angular magnification of, 829–830, 841 of eye, 824, 825, 825f gravitational, 865f nonreflective coatings, 799 Lens-maker’s equation, 775 Lenz’s law, 667–668, 674–675, 674f, 675f Lepton(s), 943, 946–947, 946t, 956 Lepton number, conservation of, 948 Light and light waves coherent source of, defi ned, 790 diffraction. See Diffraction dispersion of, 742–746, 742f, 743f, 745f, 746f, 752 dual nature of, 880 gravitational bending of, 863f, 864–865, 865f Huygens’ principle, 746–748, 746f, 747f, 752 interference. See Interference nature of, 716, 724, 732–733, 751 ordinary, incoherence of, 791 polarization. See Polarization in quantum theory, 873–874, 886 ray approximation of, 734, 734f reflection of, 733, 734–736, 734f, 747, 747f, 751 angle of reflection, 735, 735f, 742, 751 phase change due to, 795–797, 795f, 796f, 815 polarization by, 811–812, 811f total internal reflection, 748–751, 748f, 749f, 752 refraction of. See Refraction speed of as constant, 850–851 and ether wind theory, 849–850, 849f, 850f and inertial reference frames, 848 Michelson-Morley experiment, 849–850, 850f, 851 wave fronts, 734, 734f Huygens’ principle, 746–748, 746f, 747f, 752 wavelength. See Wavelength

I.7

Lightbulbs Christmas lights in series, 596–597 circuit symbol for, 549, 549f dimming of, with age, 578 failing of, 582 three-way, 600–601 Lightning, 497f and automobiles, safety of occupants in, 515 deflection by Earth’s magnetic field, 635 Lightning rods, 514 Linear accelerator. See Cyclotron Linearly polarized light, 809, 809f. See also Polarization S Linear momentum (p ) carried by electromagnetic wave, 717, 724 relativistic expression for, 858–859, 865 and total energy, 860–861, 866 Liquid crystal displays (LCDs), 813–814, 814f Lithium (Li), ground-state configuration, 903, 904t Lloyd’s mirror, 795, 795f Load resistance, 595 Loop rule, 603–606, 604f, 616 Luminiferous ether, 849–850, 849f, 850f Lyman series, 893

M Magnesium (Mg), ground-state configuration, 904t Magnet(s), 626–628, 627f permanent, 651 poles of, 626–627 Magnetic declination, 629 Magnetic domains,S651, 651f Magnetic field(s) (B ) charged particle motion in, 639–642, 639f, 652 of current loops, 646–648, 647f, 652 direction of finding, with compass, 627–628, 627f notation for, 633 right-hand rule, 642, 642f of Earth, 628–629, 628f, 629f, 635–636, 665 inductors, energy stored in, 686–687, 688 of long straight wire, 642–645, 642f, 652 magnitude of, defined, 630 SI units of, 630, 652 of solenoid, 648–650, 648f, 650f, 652 torque on current loop in, 636–638, 636f, 652 Magnetic field lines, 627–628, 627f, 628f Magnetic flux (B), 664–666, 664f, 665f, 687 of Earth’s magnetic field, 665 induced emf applications, 675–680 back emf, 679–680, 679f definition of, 663–664, 664f direction of, 667–668, 667f, 674–675, 674f, 675f

I.8

Index

Magnetic flux (B), induced emf (Continued ) Faraday’s law of magnetic induction, 666–670, 687 in generators, 676–680 Lenz’s law, 667–668, 674–675, 674f, 675f motional, 670–674, 670f, 671f, 687 self-induced, 680–683, 680f, 681f, 687 SI unit, 664–665 Magnetic force applications, 634–635, 634f on charged particle, 630–632, 652 on current-carrying conductor, 633–636, 633f, 634f, 652 direction of (right-hand rule #1), 631–632, 631f, 634, 652 torque on current loop, 636–638, 636f, 652 between two parallel conductors, 645–646, 645f, 652 S Magnetic moment (m ), 637–638, 637f, 652 Magnetic monopoles, 627 Magnetic resonance imaging (MRI), 931, 931f Magnetism applications, 626, 634–635 Earth’s magnetic field, 628–629, 628f, 629f, 635–636, 665 hard magnetic materials, 627, 651 magnetic monopoles, 627 magnets characteristics of, 626–628, 627f permanent, 651 poles of, 626–627 microscopic view of, 650–651, 650f, 651f soft magnetic materials, 627, 651 types of magnetic materials, 627, 651 Magnification. See also Microscopes; Telescopes angular of simple magnifier, 829–830, 841 of telescope, 833, 841 lateral of microscope, 831, 841 of mirror, 760, 765t, 782 of spherical refracting surface, 769, 782 of thin lens, 774, 782 simple magnifiers, 829–830, 829f, 841 Malus’s law, 810–811, 815 Manganese (Mn), ground-state configuration, 904t Marsden, Ernest, 891–892, 892f Mass (m) of atomic particles, 501t, 914, 914t energy equivalent of, 860, 862, 914 in general relativity, 863–864 and gravitational force, 863–864 unified mass unit (u), 914 Mass number (A), 913, 931 Matter, dual nature of, 880–883, 887 Maxima, 802 for diffraction grating, 805–806, 805f, 815 in single-slit diffraction, 803–804

Maximum angular magnification, 829–830, 841 Maxwell, James Clerk, 626, 712–713, 712f, 720, 724, 733 Measurement. See also SI (Système International) units; Units electric current, 574–575, 575f temperature, 580 of voltage, 574–575, 575f Mechanics Newtonian, 847 relativistic, 851–858 Medicine cancer treatment, electric field therapy, 699 defibrillators external, 556 Implanted Cardioverter Defibrillators (ICDs), 586–587, 587f, 587t fiber optic viewing scopes, 750 heart artificial, 635 electrical activity of, 585–587, 585f, 586f kidneys, dialysis, 635 lasers in, 907 magnetic resonance imaging (MRI), 931, 931f positron-emission tomography (PET), 945 radiation damage, 929–930, 930t radioactivity, uses of, 930–931, 931f, 945 Meitner, Lise, 862 Mendeleyev, Dmitry, 903–904 Mercury (Hg) emission spectrum, 893f as superconductor, 584, 584t Mercury (planet), orbit of, 864 Mesons, 945, 946t, 950, 950f, 951, 951t, 952f Metal(s) alkali, 904–905 characteristic x-rays, 876f, 905–906, 905f, 909 photoelectric effect, 872–875, 872f, 873f, 886 Metal detectors, 709 Michelson, Albert A., 840, 849 Michelson interferometer, 840–841, 840f, 849, 850f Michelson-Morley experiment, 849–850, 850f, 851 Microscopes compound, 830–832, 831f, 841 electron, 882, 883f optical, limitations of, 831–832 resolution of, 837 (See also Resolution) Microwave(s), 721, 721f polarization of, 810 Microwave background radiation, 955, 955f, 957 Millikan, Robert A., 499, 515 Millikan oil-drop experiment, 515–516, 515f, 516f Minima, 802 Mirage(s), 772f, 773

Mirror(s) aberrations in, 781–782, 781f, 782f, 783 concave, 762, 766–769, 782 convex, 764, 766, 768, 782 flat, 759–762, 759f, 760f, 782 lateral magnification, 760, 765t, 782 mirror equation, 763–764, 782 parabolic, 782, 833 ray diagram for, 764–766, 765f reflecting telescopes, 832–834, 833f sign conventions for, 765t spherical (See also Concave mirrors; Convex mirrors) definition of, 762 spherical aberration in, 762, 762f Mirror equation, 763–764, 782 Moderator, of nuclear reactor, 940, 940f Molecules crystallization of, 878 polarization of, in dielectric, 561, 561f Molybdenum, characteristic x-rays, 876f Momentum angular, of electromagnetic wave, 717, 724 linear carried by electromagnetic wave, 717, 724 relativistic expression for, 858–859, 865 and total energy, 860–861, 866 Morley, Edward W., 849 Motional emf, 670–674, 670f, 671f, 687 Motor neurons, 613–614, 614f Motors, electric back emf in, 679–680, 680f structure and function, 638–639, 638f, 679 MRI (magnetic resonance imaging), 931, 931f Müller, K. Alex, 584 Multimeter, digital, 575, 575f Muon (m
), 854, 946, 946t Muon neutrino (nm), 946, 946t Myopia (nearsightedness), 826, 827f, 828, 841

N Natural radioactivity, 926, 926f, 926t Near point, of eye accommodation, 825 Nearsightedness (myopia), 826, 827f, 828, 841 Ne’eman, Yuval, 950 Negative electric charge, 498, 498f Negative lenses. See Diverging (negative) thin lenses Neon (Ne) emission spectrum, 893f ground-state configuration, 904t helium-neon laser, 907, 908f as noble gas, 904 Neon signs, 892 Neptunium (Np), radioactive series, 926, 926t Nervous system, electrical signal conduction in, 613–614, 614f Neurons electrical signal conduction in, 613–614, 614f

Index Neurons (Continued ) structure of, 614, 614f types of, 614–615 Neutrinos, 923–924, 932, 941, 946–947, 946t Neutron(s) as baryons, 945, 946t charge, 498, 499, 501t, 914, 914t discovery of, 927 mass, 501t, 914, 914t quark composition, 951t Neutron number (N), 913, 931 Newton, Isaac, 732–733, 797–798, 797f Newtonian focus, 833 Newtonian mechanics, 847 Newton’s rings, 797–798, 797f Nichrome resistivity, 577t, 578–579 temperature coefficient of resistivity, 577t Nickel (Ni) as ferromagnetic substance, 651 ground-state configuration, 904t Niobium (Nb), as superconductor, 584t Nitrogen (N), ground-state configuration, 904t Nobel Prize, 516, 584, 862, 873, 878, 879f, 881f, 884f, 913f, 915f, 918f, 924f, 944f, 950f Noble gases, 904 Nonohmic materials, 576, 577f Nonreflective coatings, 799 Normal vector, 517 North pole of Earth, 628, 628f of magnet, 626 Notation circuit elements. See Electric circuits, symbols nucleus, atomic, 913, 931 Nuclear fission, 860, 862, 937–940, 938f, 956 Nuclear force, 915, 952 Nuclear fusion, 941–943, 956 Nuclear magnetic resonance, 931 Nuclear power plants, 860, 913f, 939–940, 940f, 956 Nuclear reactions, 927–929, 932 endothermic, 928, 932 exothermic, 928, 932 fission, 860, 862, 937–940, 938f, 956 fusion, 941–943, 956 Q values, 928–929, 932 threshold energy of, 929, 932 Nuclear reactors, 860, 913f, 939–940, 940f, 956 Nuclear weapons atomic bomb, 879f, 939, 944f hydrogen bomb, 941 Nucleus, atomic binding energy, 916–917, 917f, 931 daughter, 921 decay of. See Radioactivity density of, 915 history of concept, 891 parent, 921 properties, 913–916, 931 size of, 914–915, 931 stability of, 915–916, 915f

Nucleus, atomic (Continued ) structure of, 498, 913 symbols for, 913, 931 Nylon dielectric constant, 558t dielectric strength, 558t

O Object distance, 759, 759f, 763, 769, 782 Objective, of microscope, 830, 831f Ocular lens, of microscope, 830, 831f Oersted, Hans Christian, 642, 642f, 663 Ohm (), 576, 588 Ohm, George Simon, 576, 576f Ohmic materials, 576, 577f Ohm’s law, 576, 588 Oil-drop experiment (Millikan), 515–516, 515f, 516f Omega (
), 946t, 950, 951t, 952 Onnes, H. Kamerlingh, 584 Open-circuit voltage, of battery, 595 Optical activity, 812–813, 813f Optic disk, 825f Optic nerve, 825f Optics geometric, 790 wave, 790 (See also Diffraction; Interference; Polarization) Orbital magnetic quantum number (m ), 899–900, 900t, 902, 909 Orbital quantum number (), 899–900, 900t, 902, 908 Order number, of fringe, 793, 815 Oxygen (O), ground-state configuration, 904t

P Paintings, x-raying of, 876 Paper dielectric constant, 558t dielectric strength, 558t Parallel-plate capacitors, 523, 547–549, 547f, 562–563 Paramagnetic substances, 651 Parent nucleus, 921 Particle(s). See also specific particles antiparticles, 918, 924, 944–945, 946t, 956 charged accelerating, energy radiated by, 714 magnetic force on, 630–632, 652 motion in magnetic field, 639–642, 639f, 652 storing of, 640 classification of, 945–947, 946t, 956 conservation laws, 947–950, 956 de Broglie wavelength of, 881–882, 887 eightfold way, 949–950, 950f field, 944, 944t fundamental, 943 hadrons, 945–946, 946t, 956 leptons, 946–947, 946t, 956 wave function () for, 883–884, 887, 902 Particle accelerators. See Cyclotron(s) Paschen series, 893 Pauli, Wolfgang, 900, 903, 903f, 923

I.9

Pauli exclusion principle, 902–903, 903t, 909 Penning trap, 640 Penzias, Arno A., 955, 955f Periodic table, 903–905, 909 Periscopes, submarine, 749 Permanent magnet, 651 Permeability of free space (m0), 643, 652 Permittivity of free space (P0), 519, 523, 547 Phase angle (f), 703–706, 703f, 705t, 724 Phasor(s), 703, 703f Phasor diagrams, 703, 703f Phosphorus (P), ground-state configuration, 904t Photocells, 872, 872f, 875 Photoconductors, in xerography, 545–546, 545f Photocopiers, 545–546, 545f Photoelectric effect, 733, 872–875, 872f, 873f, 886 Photoelectrons, 872 maximum kinetic energy, 873–874, 873f, 886 Photographs, red-eye in, 735 Photon(s) in electroweak theory, 953 emitted by hydrogen orbit change, 896–897, 908 energy and momentum of, 860 energy of, 733 as field particle, 944, 944t as fundamental particles, 943 gamma (g) rays, 721f, 722, 918, 924 history of concept, 733 as quanta, 873–874, 886 spontaneous emission, 907, 907f, 909 stimulated absorption, 906–907, 907f, 909f stimulated emission, 907, 908f, 909 virtual, 944, 944f Picture tube, of television, 648, 648f Pigeons, homing, navigation in, 812 Pions (p), 945, 946t, 951t Planck, Max, 872, 872f Planck’s constant (h), 733, 751, 872 Plane of polarization, 809 Plane polarized light, 809, 809f. See also Polarization Plane waves, electromagnetic, 715, 716f Plano-concave lens, 773f Plano-convex lens, 773f Plasma, 942 Plasma confinement time (t), 942, 956 Plasma ion density (n), 942, 956 Plastic, optical activity in, 813, 813f Platinum as paramagnetic substance, 651 resistivity, 577t temperature coefficient of resistivity, 577t work function, 873t Polarization of electrical charge, 500 of light waves, 808–814, 815 applications, 813–814, 814f intensity of polarized beam, 810–811, 810f, 815

I.10

Index

Polarization (Continued ) optically-active substances, 812–813, 813f by reflection, 811–812, 811f by scattering, 812, 812f by selective absorption, 809–811, 809f, 810f of molecules, in dielectric, 561, 561f Polarizer, 809, 809f Polarizing angle, 811–812, 811f, 815 Polaroid, 809 Poles, of magnet, 626–627 Pollution, air, removing, 544–545, 544f Polonium (Po), 918, 918f Polystyrene dielectric constant, 558t dielectric strength, 558t index of refraction, 738t Population inversion, 907 Positive electric charge, 498, 498f Positive lenses. See Converging (positive) thin lenses Positron(s) (e+), 918, 924, 941, 944–945 Positron-emission tomography (PET), 945 Potassium (K), ground-state configuration, 904t Potassium ion channels, 614–615 Potential difference. See Electric potential Potential energy (PE). See Electric potential energy; Gravitational potential energy Power average, delivered in RLC series circuit, 707–708, 724 commercial units of (kilowatt-hour), 581–583 delivered to resistor/device, 581–584, 588 emf source total output, 595 of lens, in diopters, 826, 841 SI units of, 581 Power factor, 707–708, 724 Power plants fusion reactors, 937f, 941–943, 946 nuclear (fission reactors), 860, 913f, 939–940, 940f, 956 Presbyopia, 826 Principal axis, of spherical mirror, 762–763, 762f Principle quantum number (n), 872, 886, 894–895, 899–900, 900t, 902, 908 Prisms, 742–745, 742f, 743f, 752 prism spectrometer, 743–744, 743f redirection of light with, 749, 749f Probability, wave function () and, 883–884, 887, 902 Problem-solving strategies capacitor combinations, 553 direct-current circuits capacitors, 553 Kirchhoff’s rules, 604 resistors, 601–602 electric field, 508–509 electric force, 508–509 electric potential, 540

Problem-solving strategies (Continued ) Kirchhoff’s rules, 604 resistors, 601–602 RLC circuits, 705 thin-film interference, 798 Proper length, 856, 865 Proper time, 854, 865 Proteins, structure of, determining, 877 Proton(s) in atomic structure, 498 as baryons, 945, 946t charge, 498, 499, 501, 501t, 914, 914t mass, 501t, 914, 914t number of, in gram of matter, 498 quark composition, 951t Proton-proton cycle, 941 Pupil, 824, 825f Purkinje fibers, 585, 585f Pyrex® glass dielectric constant, 558t dielectric strength, 558t

Q Q values, 928–929, 932 QCD (quantum chromodynamics), 952, 953, 957 Quantum chromodynamics (QCD), 952, 953, 957 Quantum numbers orbital (), 899–900, 900t, 902, 908 orbital magnetic (m ), 899–900, 900t, 902, 909 and Pauli exclusion principle, 902–903, 903t, 909 principle (n), 872, 886, 894–895, 899–900, 900t, 902, 908 spin magnetic (ms), 900–902, 909 strangeness (S), 946t, 949 Quantum physics atom model in, 899–905, 908–909 Compton shift, 879–880, 879f, 886 Einstein and, 850f history of, 872 matter, dual nature of, 880–883, 887 and photoelectric effect, 872–875, 872f, 873f, 886 spectrum emitted by x-ray tube, 875–876, 875f, 876f, 887 uncertainty principle, 884–886, 885f, 887 wave function (), 883–884, 887, 902 Quantum state, 872 energies of, 895, 908 Quantum tunneling, 941 Quarks, 943, 950–952, 951t, 956–957 Quartz dielectric constant, 558t dielectric strength, 558t index of refraction, 738t, 742f resistivity, 577t

R Rad (radiation absorbed dose), 929–930 Radio broadcast antennas, 714 tuning circuit of, 708 Radioactive tracers, 930–931

Radioactivity, 918–927, 932 alpha (a) decay, 918, 921–922, 932 artificial, 926 beta (b) decay, 918, 922–924, 923f, 932 decay constant, 918 decay rate (activity), 918–919, 919f, 932 discovery of, 918, 918f dose limits, 930 gamma (g) decay, 918, 924, 932 half-life (T1/2), 919, 921, 932 medical uses of, 930–931, 931f, 945 natural, 926, 926f, 926t natural background, 930 radiation damage, 929–930, 930t spontaneous decay (transmutation), 921 units of, 919 uses of, 924–926 Radio telescopes, 838 Radio waves, 721, 721f, 722f Radium (Ra) discovery of, 918, 918f radioactive decay of, 920, 921, 921f, 922 Radius of curvature (R), of spherical mirror, 762 Radon (Rn) detectors, 925 discovery of, 925 as noble gas, 904 Rail guns, 671 Rainbows, 732f, 745–746, 745f, 746f Ray diagrams for mirrors, 764–766, 765f for thin lenses, 775–776, 776f Rayleigh’s criterion, 835–836 RBE (relative biological effectiveness) factor, 929, 930t RC circuits alternating-current impedance, 705t phase angle, 705t applications, 608 direct-current, 607–611, 607f, 616 Real images, 759, 762, 765t, 782 Red-eye, in photographs, 735 Red shift, 723 Reference frames. See also Relativity absolute, 849 inertial, 847–848, 848f Reflecting telescopes, 832–834, 833f Reflection, of light, 733, 734–736, 734f, 747, 747f, 751 angle of reflection, 735, 735f, 742, 751 phase change due to, 795–797, 795f, 796f, 815 polarization by, 811–812, 811f total internal reflection, 748–751, 748f, 749f, 752 Refracting telescopes, 832–833, 833f Refraction, of light, 733, 736–742, 737f, 738f, 751 angle of refraction, 736, 737f, 751 atmospheric, 772–773, 772f dispersion, 742–746, 742f, 743f, 745f, 746f, 752 images formed by flat surface, 770, 770f, 771–772 spherical surface, 768–771, 769f, 770t, 782

Index Refraction, of light (Continued ) index of refraction, 737, 738t, 742, 742f, 751 Snell’s law, 737–742, 747–748, 747f, 751–752 Refractive myopia, 826 Relativity Galilean, 847–848, 848f general, 850f, 863–865 special kinetic energy, relativistic expression for, 859, 861, 865 length contraction in, 856–858, 857f, 865 momentum, relativistic expression for, 858–859, 865 and total energy, 860–861, 866 postulates of, 850–851, 850f, 865 relativity of time in, 851–852, 857f, 865 rest energy, 859, 865 conversion of atomic mass units to/from, 861 of electron, 861 time dilation in, 852–855, 852f, 854f, 865 total energy, 859–860, 861, 866 and relativistic momentum, 860–861, 866 twin paradox, 855–856, 856f Rem (roentgen equivalent in man), 930 Reproduction constant (K), 939–940, 956 Resistance (R) conductor properties and, 577–579, 577t, 588 definition of, 575–576, 576f, 588 equivalent for parallel resistors, 598–601, 602–603 616 for series resistors, 595–597, 602–603, 616 load, 595 ohmic and nonohmic materials, 576, 577f Ohm’s law, 576, 588 problem-solving strategies, 601–602 temperature variation of, 579–580, 588 Resistance thermometers, 580 Resistivity (r), 577, 577t, 588 Resistor(s), 576f in AC circuits, 696–699, 697f, 698f, 723 impedance, 705t phase angle, 705t RC circuits, 705t RL circuits, 705t RLC series circuits, 702–706, 702f, 703f, 724 circuit symbol for, 549, 549f, 576 in DC circuits in parallel, 598–603, 598f, 616 power delivered to, 581–584, 588 problem-solving strategies, 601–602 RC circuits, 607–611, 607f, 616 RL circuits, 683–687, 683f, 684f, 688 in series, 595–597, 596f, 616 definition of, 576 Resolution, 835–840, 841 of circular aperture, 836, 836f of diffraction grating, 838–840, 841

Resolution (Continued ) of microscope, 837 of single-slit aperture, 835–836, 835f, 836f of telescope, 838 Resolved image, 835 Resolving power, of diffraction grating, 839, 841 Resonance, in RLC series circuits, 708–710, 708f, 724 Resonance frequency, in RLC series circuits, 708–709, 708f, 724 Resonators, in blackbody radiation, 872, 873 Rest energy, 859, 865 conversion of atomic mass units to/from, 861 of electron, 861 Retina, 824, 825, 825f Retroreflection, 735 Right-hand rule for electromagnetic wave propagation, 715 for magnetic field (right-hand rule #2), 642, 642f, 667 for magnetic force (right-hand rule #1), 631–632, 631f, 634, 652 for magnetic moment, 637 RL circuits alternating-current, 705t direct-current, 683–687, 683f, 684f, 688 phase angle, 705t RLC series circuits, 702–706, 702f, 703f, 705t, 724 applications, 708–709 average power delivered in, 707–708, 724 problem-solving strategies, 705t resonance in, 708–710, 708f, 724 Rms current, 697–699, 723 RNA, structure of, determining, 877 Roadway flashers, 608 Rods, of eye, 824–825, 825f Roentgen (R), 929 Röntgen, Wilhelm, 875 Root mean square (rms) current, 697–699, 723 Rubber dielectric constant, 558t dielectric strength, 558t as insulator, 499, 499f resistivity, 577t Runway markings, 629 Rutherford, Ernest, 891–892, 913f, 914, 925, 927 Rydberg constant (R H), 892–893, 896 Rydberg equation, 893

S SA (sinoatrial) node, 585, 585f Scandium (Sc), ground-state configuration, 904t Scattering, polarization of light waves by, 812, 812f Schrödinger, Erwin, 883, 884f, 899 Schrödinger wave equation, 883 Schwarzschild radius, 864–865

I.11

Schwinger, Julian S., 944f Sclera, 825f Secondary maxima, 802 Second-order maximum, 806 Selective absorption, polarization of light waves by, 809–811, 809f, 810f Selenium (Se), ground-state configuration, 904t Self-inductance, 680–683, 680f, 681f, 687 Self-sustained chain reaction, 939–940 Semiconductors properties, 499 temperature coefficient of resistivity, 579 Sensory neurons, 613–614, 614f Shells atomic, 900, 900t Pauli exclusion principle, 902–903, 903t, 909 thin, nonconducting, 521–523 SIDS. See Sudden infant death syndrome Sigma ("), 946t, 951t Silicon (Si) ground-state configuration, 904t resistivity, 577t as semiconductor, 499 temperature coefficient of resistivity, 577t Silicone oil in capacitors, 558 dielectric constant, 558t dielectric strength, 558t Silver (Ag) resistivity, 577t temperature coefficient of resistivity, 577t work function, 873t Simple magnifiers, 829–830, 829f, 841 Simultaneity, special relativity and, 851–852, 852f, 865 Single-slit aperture, resolution of, 835–836, 835f, 836f Single-slit diffraction, 803–805, 803f, 815 Singularities, 865 Sinoatrial (SA) node, 585, 585f, 586 SI (Système International) units activity (decay rate), 919 capacitance, 546 change in electric potential energy, 532 electric charge, 499, 645 electric current, 645 average, 570 instantaneous, 571 electric field, 506, 536 electric flux, 517 electric potential, 536 electric potential difference, 536, 562 emf, 594 inductance, 681, 687 length, 914–915 magnetic field, 630, 652 magnetic flux, 664–665 power, 581 resistance, 576, 588 resistivity, 577, 588 Smoke detectors, 925, 925f Snell, Willebrørd, 738

I.12

Index

Snell’s law of refraction, 737–742, 747–748, 747f, 751–752 Soddy, Frederick, 925 Sodium (Na) ground-state configuration, 904t work function, 873t Sodium chloride (NaCl) index of refraction, 738t structure of, 877f x-ray diffraction by, 875, 875f, 876–877, 877f Sodium ion channels, 614–615 Soft magnetic materials, 627, 651 Solar cells, nonreflective coatings, 799 Solar system, size of dust in, 717 Solenoid (electromagnetic) inductance of, 682–683, 687 magnetic field of, 648–650, 648f, 650f, 652 Solids, crystalline, diffraction of x-rays by, 875, 875f, 876–878, 876f, 877f, 886 Somatic radiation damage, 929 Sound system, speakers in, 634–635, 634f South pole of Earth, 628, 628f of magnet, 626 Space catapult, 671 Spacetime, curvature of, 864 Space travel, special relativity and, 857 Speakers, in sound system, 634–635, 634f Special relativity kinetic energy, relativistic expression for, 859, 861, 865 length contraction in, 856–858, 857f, 865 momentum, relativistic expression for, 858–859, 865 and total energy, 860–861, 866 postulates of, 850–851, 850f, 865 relativity of time in, 851–852, 857f, 865 rest energy, 859, 865 conversion of atomic mass units to/from, 861 of electron, 861 time dilation in, 852–855, 852f, 854f, 865 total energy, 859–860, 861, 866 and relativistic momentum, 860–861, 866 twin paradox, 855–856, 856f Spectral lines, 743–744 Specular reflection, 734–735, 734f Speed, of light as constant, 850–851 and ether wind theory, 849–850, 849f, 850f and inertial reference frames, 848 Michelson-Morley experiment, 849–850, 850f, 851 Spherical aberrations, in lens and mirrors, 762, 762f, 781–782, 781f, 783 Spherical mirrors. See also Concave mirrors; Convex mirrors definition of, 762 spherical aberration in, 762, 762f

Spherical surface, images formed by refraction at, 768–771, 769f, 770t, 782 Spin, of electron, 900–902, 900f Spin magnetic quantum number (ms), 900–902, 909 Split-ring commutators, 638, 638f, 678, 678f Spontaneous decay (transmutation), 921 Spontaneous emission, 907, 907f, 909 Standard model, 953, 953f Stars color of, 871 surface temperature of, 871 Step-down transformers, 710, 724 Step-up transformers, 710, 724 Sterilization, with radiation, 930 Stimulated absorption, 906–907, 907f, 909f Stimulated emission, 907, 908f, 909 Stopping potential, in photoelectric effect, 873, 873f Storage mite (Lepidoglyphus destructor), 870f Strangeness (S), conservation of, 946t, 949 Strange quark (s), 950–951, 951t Stress, models of, in architecture, 813, 813f Strong force Big Bang and, 954, 954f characteristics of, 943, 944t, 945, 956 as color force, 952 in Standard Model, 953, 953f Strontium titanate dielectric constant, 558t dielectric strength, 558t Stud finders, 559, 559f Submarine periscopes, 749 Subshells, atomic, 900–901, 900t Sudden infant death syndrome (SIDS), apnea monitor for, 669–670, 670f Sulfur (S) ground-state configuration, 904t resistivity, 577t Sun atmosphere, identification of gases in, 893 energy production in, 860 fusion in, 941 and human eye, wavelength sensitivity of, 722 polarization of light from, 812 Sunglasses polarized, 812 UV protection in, 720f Superconductors, 584–585, 584f, 584t, 595f Supercritical condition, of nuclear reactor, 940 Superposition principle for electric field, 507, 509–510 for electric force, 503–504 for electric potential, 538, 540–541

Symbols for circuit elements. See Electric circuits, symbols Symmetry breaking, 953

T Tape recorders, 675–676, 676f Tau (t
), 946–947, 946t Tau neutrino (v t), 946, 946t Teflon dielectric constant, 558t dielectric strength, 558t Telescopes, 832–835, 841 angular magnification of, 833, 841 Hale telescope, 834f, 838 Hubble Space Telescope, 723, 834 Keck telescopes, 823f, 834 lenses of, 782 radio, 838 reflecting, 832–834, 833f refracting, 832–833, 833f resolution of, 838 (See also Resolution) space dust and, 838 Television interference, 793 picture tube, 648, 648f Temperature (T) and electrical resistance, 579–580, 588 measurement of, 580 Temperature coefficient of resistivity (a), 577t, 579, 588 Terminal voltage, of battery, 595, 615–616 Tesla (T), 630, 652 conversion to/from gauss, 630 Tesla, Nikola, 705f Thermal radiation, and quantum theory, 870–872, 871f, 872f, 886 Thermometers, resistance, 580 Thermonuclear fusion reactions, 941 Thin films, interference of light waves in, 796–800, 796f, 797f, 815 problem-solving strategies, 798 Thin lenses, 773–781, 782–783 aberrations in, 781–782, 781f, 782f, 783 angular magnification of, 829–830, 841 combinations of, 779–781, 780f converging (positive), 773, 773f, 774f, 775–778, 776f diverging (negative), 773, 773f, 774f, 775–776, 776f, 778–779 focal point, 773, 774f lateral magnification of, 774, 782 lens-maker’s equation, 775 ray diagrams for, 775–776, 776f shapes, common, 773, 773f sign conventions, 775, 775t thin-lens equation, 775, 783 Thin shells, nonconducting, 521–523 Thomson, Joseph John, 891, 891f, 892f Thorium (Th), radioactive series, 926, 926t Threshold energy, of nuclear reaction, 929, 932 Threshold voltage, in x-ray emission, 876 Thymine, 878

Index Time in general relativity, 864, 865 gravity and, 864, 865 proper, 854, 865 in special relativity dilation of, 852–855, 852f, 854f, 865 relativity of time frames, 851–852, 852f, 865 Time constant (t) RC circuits, 607, 616 RL circuits, 683–684, 688 Tin, as superconductor, 584, 584t Titanium (Ti), ground-state configuration, 904t Tokomaks, 942, 942f Tomonaga, Shinichiro, 944f Toner, in xerography, 545f, 546 Top quark (t), 950–951, 951t Torque, on current loop in magnetic field, 636–638, 636f, 652 Total energy, 859–860, 861, 866 and relativistic momentum, 860–861, 866 Total internal reflection, 748–751, 748f, 749f, 752 Tracers, radioactive, 930–931 Transformers AC, 710–712, 710f, 724 ideal, 711, 724 Transition electron microscope, 882, 883f Transmission axis, of polarizing material, 809 Transmutation (spontaneous decay), 921 Transverse waves, 715 Tritium, 913, 942, 956 Tungsten resistivity, 577t temperature coefficient of resistivity, 577t Tuning circuit of radio, 708 Twin paradox, 855

U Uhlenbeck, George, 900 Ultraviolet (UV) light, 721–722, 721f Uncertainty principle, 884–886, 885f, 887 Unified field theory, 850f. See also Grand unified theory Unified mass unit (u), 914 Units. See also SI (Système International) units of radiation exposure, 929–930, 930t Universe microwave background radiation, 955, 955f, 957 origin of, 954, 954f

Unpolarized light, 808–809, 809f Up quark (u), 950–951, 951t Uranium (U) fission of, 862, 937–939, 938f as nuclear fuel, 939–940 radioactive series, 926, 926t UV. See Ultraviolet (UV) light

V Vanadium (V), ground-state configuration, 904t Van de Graaff generator, 514, 516–517, 516f, 531f Vector(s), normal, 517 Virtual image, 759–760, 765t, 782 Virtual photons, 944, 944f Visible light, 721, 721f Visible spectrum, 742–743, 743f Vitreous humor, 825f Voice coil, 634, 634f Volt (V), 536, 562 Volta, Alessandro, 570 Voltage. See Electric potential Voltmeter, 574–575, 575f Von Laue, Max, 875

W Water dielectric constant, 558t dielectric strength, 558t index of refraction, 738t Watson, J. D., 878 Wave(s). See Light and light waves Wave front, of light wave, 734, 734f Huygens’ principle, 746–748, 746f, 747f, 752 Wave function (), 883–884, 887, 902 Wavelength (l) of light absorption spectra, 893 cutoff, in photoelectric effect, 874 determining from diffraction grating, 806 emission spectrums, 892–894, 893f measurement of, 793–794, 838–839, 840–841 in medium, 796, 815 of matter (de Broglie wavelength), 881–882, 887 of x-rays characteristic x-rays, 905–906 Compton shift, 879–880, 879f, 886 emitted by x-ray tube, 875–876, 875f, 876f, 887 Wavelets, 746 Wave optics, 790. See also Diffraction; Interference; Polarization

I.13

W bosons, 944, 944t, 952–953 Weak force Big Bang and, 954, 954f characteristics of, 944, 944t, 946, 956 and electroweak theory, 952–953 in Standard Model, 953, 953f Weber (Wb), 664–665 Weber per square meter (Wb/m2), 630, 652 Wheeler, John, 864 Wien’s displacement law, 871, 886 Wilkins, Maurice, 878 Wilson, Charles, 879f Wilson, Robert W., 955, 955f Work (W) to charge capacitor, 555, 555f and electric potential energy, 531–535, 532f Work function, 873, 873t, 886

X Xenon (Xe), as noble gas, 904 Xerography, 545–546, 545f Xi (#), 946t, 951t X-rays applications, 722 characteristic, 876f, 905–906, 905f, 909 Compton shift in, 879–880, 879f, 886 diffraction by crystals, 875, 875f, 876–878, 876f, 877f, 887 discovery of, 875 in electromagnetic spectrum, 721f, 722, 722f spectrum emitted by x-ray tube, 875–876, 875f, 876f, 887 tissue damage from, 929, 930t X-ray tubes, spectrum emitted by, 875–876, 875f, 876f, 887

Y Young, Thomas, 733 Young’s double-slit experiment, 791–795, 815

Z Z bosons, 944, 944t, 952–953 Zeeman effect, 899, 899f Zeroth-order maximum, 793, 806 Zinc (Zn) ground-state configuration, 904t as superconductor, 584, 584t work function, 873t Zircon, index of refraction, 738t Z machine, 937f Zonules, 825