AN IEEE PRESS CLASSIC REISSUE
ELECTROMAGNETIC THEORY
BY
JULIUS ADAMS STRATTON Profeccor of Physics Mu rsuchusetis Ins...
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AN IEEE PRESS CLASSIC REISSUE
ELECTROMAGNETIC THEORY
BY
JULIUS ADAMS STRATTON Profeccor of Physics Mu rsuchusetis Institute of Technology
IEEE Anfennas and Propagation Society, Sponsor
IEEE Press Series on Electromagnetic Wave Theory Donald G. Dudley, Series Editor
IEEE IEEE PRESS @
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" I c I x I I * * I a L
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FOREWORD TO THE REISSUED EDITION The purpose of the IEEE Press Series on Electromagnetic Wave Theory is to publish books of longterm archival significance in electromagnetics. Included are new titles as well as reissues and revisions of recognized classics. The book Electromagnetic Theory, by J. A. Stratton is one such classic. Originally published in 194I , Stratton’s book has formed an integral part of the electromagnetic education of both physics and electrical engineering graduate students for over sixty years. In addition, virtually every electromagnetic researcher I know has a copy in hidher library. Unfortunately, the book is out of print. It is our purpose to rectify this situation. When I consult my copy of this classic book, I never cease to be amazed at its timeliness. The equivalence principle is strongly rooted in the StrattonChu inversion of the vector wave equations, contained on pages 464470. The Hansen vectors are introduced and exploited in Chapter VII. The Sommerfeld problem is discussed in Section 9.28. Natural modes along circular cylinders embedded in a lossy medium are developed in Section 9.15. I could go on and on. Suffice it to say that the development of the theory and the inclusion of examples remain of current interest today. Professor Stratton had an amazing career, culminating with his succession to the presidency of the Massachusetts Institute of Technology (MIT). Another former president, Professor Paul E. Gray, has kindly agreed to write an introduction to this reissued edition, highlighting Stratton’s achievements. I am indebted to Professor Stratton’s surviving spouse, Kay Stratton, who kindly agreed to our reissue plans and then assisted in making the project possible. In addition, I should like to thank Carol Fleishauer, Associate Director for Collection Services of the MIT Libraries who made available from their archives a pristine copy of the book for our use in the reissue. Over the past twelve years, I have received many requests worldwide to reissue this book. It is with great pleasure that I welcome it to the IEEE Press Series on Electromagnetic Wave Theory. DONALD G . DUDLEY University of Arizona
V
Professor Julius Adams Stratton. Courtesy MIT Museum.
INTRODUCTION Julius Adams Stratton served the Massachusetts Institute of Technology, The Radiation Laboratory at MIT, the Federal Government, the National Academies of Science and Engineering and the Ford Foundation during his long and productive life. His work at MIT, both as a member of the faculty, and as provost and subsequently president, was important in the development of both research and education during intervals of rapid growth and change in the Institute. Stratton was born on May 18, 1901 in Seattle, Washington. His father, Julius A. Stratton, was an attorney who founded a law firm well known and respected throughout the Northwest. He also served as a judge. His mother, Laura Adams Stratton, was an accomplished pianist. Following his father’s retirement, the family moved to Germany in 1906 where Stratton attended school through age nine and developed fluency in the German language. The family returned to Seattle in 1910 where he completed his public school education. Stratton came to MIT, with which he was associated for 74 years, as the result of an accident at sea and the advice of a student friend. He had an early interest in finding out how things worked and in building things, particularly those that involved electricity. In high school he became very interested in radio in those early days of sparkgap transmitters and galena crystal detectors. These interests, coupled with the shutdown of amateur radio operations during World War I and the desire to serve the nation, led him to study and qualify as a commercial radio operatorsecond gradeand to sign on during summer vacations as a shipboard radio operator. Stratton had been admitted to Stanford for matriculation in September 1919, and signed on for that summer as radio operator on the SS Western Glen out of Seattle for a trip to Japan and Manchuria. The ship encountered a typhoon near Kobe, Japan and participated in the rescue of another American ship in distress. Also, it experienced an engine failure at the start of the return voyage that required a return to port in Japan. These accidents made him late in returning to the UStoo late to enroll at Stanford that year. As an alternative, he succeeded in enrolling late at the University of Washington in Seattle. During that year, in which he pursued his interests in electricity and mathematics, a conversation with a UW classmate persuaded him to apply for transfer to MIT, where he was admitted in 1920. He traveled to the Institute as a radio operatorthis time first gradeon the SS Eastern Pilot via Balboa, Panama and New York City, arriving in August 1920, a week before the start of classes. At MIT Stratton enrolled in the Electrical Communications; Telegraph, Telephone and Radio option of the Department of Electrical Engineering, which he described in a letter home as “. . . far more interesting than that of ordinary dynamoelectric machinery. Line telegraphy and telephony involve some of the most complex mathematics known.” He received his SB Degree in June 1923 with a thesis entitled “The Absolute Calibration of Wavemeters.” The equipment he developed generated harmonics up to 30 megahertz from a one kilohertz tuning fork. During his senior year, Stratton determined to continue his studies in Europe. He vii
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traveled to Paris via Cherbourg (this time as a passenger), with the dual goals of continuing his engineering studies and becoming fluent in the French language. During the ’23‘24 year he traveled to and studied in Nancy, Grenoble, Toulouse and Italy returning to the US in August 1924. From September 1924 through June 1926 he was, as a research assistant in communications, enrolled at MIT in a master’s degree program, graduating with a thesis entitled “A High Frequency Bridge.” Upon completion of his graduate studies at MIT Stratton received a traveling fellowship that enabled him to again return to Europe, where many universities seethed with excitement about quantum theory and atomic structure. He enrolled for a doctor of science degree at the Eidgenossische Technische Hochschule (ETH) in Zurich, Switzerland where he studied with Debye, graduating in March 1928 with a thesis entitled “Streuungskoeffzient von Wasserstoft nach der Wellenmechanic” (The Scattering Coefficient of Hydrogen According to Wave Mechanics). He was invited to return to MIT as an assistant professor in electrical engineeringa modern physicist embedded in the engineering department. This appointment marked the beginning of 38 years of continuous active association with the Institute. Stratton’s desire to see the world was obviously very strong. During the summers he traveled to Africa, the Yukon and to Ecuador. His interest in other cultures and other nations was deep. On June 14, 1935, in Saint Paul’s Chapel at Ivy Depot, Virginia, Julius Adams Stratton and Catherine Nelson Coffman were married. From this fortunate marriage came three daughters: Catherine, Cary, and Laura. Their mother, known to all as Kay, is active in the MIT community, as a founding member of the Council for the Arts and as the guiding force behind two annual panel discussions: one each fall in a topic of current interest; and one each spring on some aspect of ageing gracefully. His experiences at the ETH changed Stratton’s career interests and further developed his passion for mathematics and physics. As he put it “In the years 19231924 I was thinking of a doctorate in literature or philosophy. This was to be the subject: The Influence of Science on 19th Century French Literature. I decided to go into pure physics. The years 1925 through 1928 changed my mind.” In 1930 his appointment was moved to the Department of Physics. Karl Taylor Compton, the newly arrived president, set out to strengthen the sciences at MIT generally, and to give greater emphasis to modern physics. Stratton became part of that transformation. He was promoted to professor in 1941, the same year this book, Electromugnetic Theory, was published by McGraw Hill. His book, although long out of print, is still widely used and referenced by 21st century writers. The decision of the IEEE to republish it as part of their series of electrical engineering classics will be much appreciated by engineers Much of his research in the 1930s was done at the Round Hill Experiment Station in South Dartmouth, Massachusetts. Stratton’s research there involved the propagation of very short radio waves and light through rain and fog. He also studied the possibility of using intense electromagnetic radiation to disperse fog, and made measurements of the field of an antenna over the open sea, employing the
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Mqjlower, a dirigible loaned to him by the Goodyear Zeppelin Company. He prepared and published, through the National Academy of Sciences, tables of spheroidal functionssolutions of differential equations that arose in his studies of antennas. Between 1927 and 1942 Stratton published eleven technical papers in refereed journals. The German invasion of Poland in 1939 changed everything for Stratton. Prior to the start of the war, British scientists employed highfrequency radio waves (ca 100 megahertz) as an early form of radar by exploiting reflections from aircraft. They realized that higher frequencies would both enable much smaller antennas and yield greater precision in target location, but were unable to generate radiation of sufficient intensity at frequencies in the gigahertz range. Their invention of the microwave magnetron, in the months before the war started, enabled the creation of radar systems that would be of much greater effectiveness in the war. Unsure that they would be able to pursue the essential development in Britain under wartime conditions, they sent the Tizard Mission with the magnetron and its developers to the US, where American engineers and scientists could develop militarily useful microwave radar systems quickly. The Federal government established the Radiation Laboratory at MIT in October 1940 and Stratton was one of many who took on the tasks of making microwave radar useful to the military on land and on sea as well as in the air. He was a natural for this work given his understanding of electromagnetic radiation and the applications of Maxwell’s equations. He was appointed in November 1940 as a volunteer consultant to the Microwave Section of Division D of the National Defense Research Council, and seconded by MIT to the Rad Lab (as it was universally called). In August 1942 Stratton was appointed Expert Consultant to the Secretary of War, Henry G. Stimson. He served in that capacity until December 1945. In this role he made frequent visits to the theaters of war. In October 1942, as a member of a committee investigating communication problems in the North Atlantic he traveled by air to England with extended stops in Presque Isle, Labrador, Greenland and Iceland. In 1943, soon after the Allied invasion of North Africa, he traveled to Algiers, Tunis, Italy, and London to assess radar utility and communications effectiveness. Robert Buderi wrote in The Invention that Changed the World (Touchstone, 1997): “The Atomic Bomb only ended the war. Radar won it.” Stratton was a very significant part of that critically important development. In August 1945 the Office of Scientific Research and Development, that had overseen all of the laboratories created to aid the war effort, was shut down, and the Radiation Laboratory was told to wind up its affairs. On January 1, 1946 Stratton took over administration of the Division of Basic Research of the Lab, which had been created in August 1945 following the surrender of the Japanese government. At the suggestion of John Slater, head of the physics department, he named the Division the MIT Research Laboratory of Electronics (RLE). Research support in the early days of RLE came from the Department of Defense through a multiservices contract, much of it in the form of a block grant at the level of $600,000 per year. RLE was “responsible for extending the useful range
of the electromagnetic spectnim . . . to shorter wavelengths, approaching ultimately that of infrared.” Title to the temporary buildings in the heart of the MIT campus that housed the Rad Lab and to all the equipment contained therein was transferred to RLE in July 1946. The largest of those “temporary buildings” became “Building 20.” Until its demise in 1998 at age 55, is was cherished prime research space at the Institute. Stratton, as the founding director, led RLE during its formative years as the nation’s first universitybased interdepartmental research laboratory. Its history, now spanning more than sixty years of scientific and engineering accomplishments, has its origins in Stratton’s vision and leadership. In 1949 James Rhyne Killian, Jr. succeeded Karl Taylor Compton as president of the Institute, and Killian appointed Stratton as the Institute’s first Provost. The fifties and sixties were years of extraordinary growth at MIT. Vannevar Bush’s landmark report, Science, the Endless Frontier led to the creation of the National Science Foundation and the National Institutes of Health. Cold War tensions, much increased in 1957 by the Soviet’s launch of Sputnik, caused enrollments in engineering and science to grow rapidly throughout this period. The Federal Government greatly expanded financial support for research and for students in the fields of science and engineering. Charles Stark Draper’s Instrumentation Laboratory (later to become the independent Draper Laboratory) expanded greatly to add the Apollo mission to its development of inertial navigation systems for the military services. The compound annual growth rate of sponsored research at the Institute was in double digits until 1969. Stratton was primarily responsible for management of the physical and intellectual growth of MIT in these decades and for the thoughtful development and implementation of necessary structures, policies and procedures that became the foundations of, and the models for, the modem MIT. During his time as provost, two new schools were created, the School of Humanities and Social Sciences in 1950 and the Sloan School of Management in 1952. In 1957 Killian was called to Washington to serve as President Eisenhower’s Science Advisor, and Stratton, who had been appointed Chancellor in 1956, became acting president and an ex oficio member of the governing board. In 1958 he was elected eleventh president of the Institute. His presidency was a time of physical expansion for MIT. New buildings for Chemistry, Earth Sciences, Biology and the Center for Materials Science and Engineering filled former parking lots. McCormick Hall, the first dormitory for undergraduate women, was completed in 1965. This was the crucial first step in increasing the number of women at MIT, who now comprise 45% of the undergraduate student body. Stratton was liked and respected by MIT students, who suggested that the new student center, completed in 1966 be named for him. As founding director of RLE, as provost, and as chancellor and president, Jay Stratton deserves, along with Compton and Killian, a large share of credit for the transformation of MIT from the premier school of engineering to a modern research university. Stratton was elected a member of the National Academy of Sciences (NAS) in 1950. Although there was a section of the NAS for distinguished engineers, there
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were very few such members. The national engineering societies affiliated with the Engineers’ Joint Council suggested in 1963 that a new organization to be called the National Academy of Engineering (NAE) be created. Stratton, who was at the time a member of the NAS Council, chaired a committee that worked through the complex issues of the idea and that led to the creation in 1964 of thc NAE as an affiliate of the NAS. The two academies, joined some years later by the Institute of Medicine, comprise the premier organization for responding to questions or requests for studies that come from the Federal government. Stratton’s clarity of vision and persuasiveness were very important in shaping the expansion of the enterprise, which is now commonly referred to simply as “The National Academies.” Following his retirement from MIT in 1966, Stratton accepted appointment as Chairman of the Board of the Ford Foundation in New York City. At that time the Ford Foundation was the nation’s largest grantmaking charitable foundation. Stratton’s interest in MIT affairs continucd during his time in New York. Hc remained a member of the Institute’s governing board and served on several of its committees. During his years at the Ford Foundation Stratton accepted presidential appointment in 1967 as Chairman of the Committee on Marine Sciences, Engineering and Resources (COMSER). The charge to the committee was, in part, to recommend “National Policy to develop, encourage and maintain a coordinated, comprchensive, and long range program in marine science for the benefit of mankind . . . expanding scientific knowledge of the marine environment and of developing an ocean engineering capability to accelerate exploration and development of marine resources. . . .” COMSER’s report: Our Nation and the Seu. A Plan for Nutional Action was presented to a different president and published two years later in January 1969. An outcome of the study was the creation of NOAA, the National Oceanic and Atmospheric Agency. Stratton returned to MIT in 1971 full time when his term as chairman of the Ford Foundation ended. His affection and concern for the university that was integral to his professional life for more than 50 years was undiminished by his years in New York, and his renewed engagement in the life of MIT was immediately evident. Jay Stratton died on June 22, 1994. During the last twenty years of his life, he had the opportunity to see upclose the transformation from premier engineering school to important worldclass research university that he, together with Compton and Killian, had commenced sixty years earlier. In the preface to Electromugnetic Theory, Stratton noted that his wife Catherine had assisted him in its preparation by proofreading the galleys with him. In the preparation of this edition she has worked enthusiastically and closely with the IEEE to enable the book’s republication.
PAULE. GRAY Professor of Electrical Engineering President Emeritus MIT October 2006
PREFACE
The pattern set nearly 70 years ago by Maxwell’s Treatise on Electricity and Magnetism has had a dominant influence on almost every subsequent English and American text, persisting to the present day. The Treatise was undertaken with the intention of presenting a connected account of the entire known body of electric and magnetic phenomena from the single point of view of Faraday. Thus it contained little or no mention of the hypotheses put forward on the Continent in earlier years by Riemann, Weber, Kirchhoff, Helmholtz, and others. It is by no means clear that the complete abandonment of these older theories was fortunate for the later development of physics. So far as the purpose of the Treatise was to disseminate the ideas of Faraday, i t was undoubtedly fulfilled; as an exposition of the author’s own contributions, i t proved less successful. By and large, the theories and doctrines peculiar t o Maxwellthe concept of displacement current, the identity there in scarcely of light and electromagnetic vibrationsappeared greater completeness and perhaps in a less attractive form than in the original memoirs. We find that all of the first volume and a large part of the second deal with the stationary state. In fact only a dozen pages are devoted to the general equations of the electromagnetic field, 18 to the propagation of plane waves and the electromagnetic theory of light, and a score more t o magnetooptics, all out of a total of 1,000. The mathematical completeness of potential theory and the practical utility of circuit theory have influenced English and American writers in very nearly the same proportion since that day. Only the original and solitary genius of Heaviside succeeded in breaking away from this course. For an exploration of the fundamental content of Maxwell’s equations one must turn again to the Continent. There the work of Hertz, Poincar6, Lorentz, Abraham, and Sommerfeld, together with their associates and successors, has led t o a vastly deeper understanding of physical phenomena and to industrial developments of tremendous proportions. The present volume attempts a more adequate treatment of variable electromagnetic fields and the theory of wave propagation. Some attention is given to the stationary state, but for the purpose of introducing fundamental concepts under simple conditions, and always with a view to later application in the general case. The reader must possess a general knowledge of electricity and magnetism such as may be acquired from an elementary course based on the experimental laws of Coulomb, ...
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PREFACE
AmpBre, and Faraday, followed by an intermediate course dealing with the more general properties of circuits, with thermionic and electronic devices, and with the elements of electromagnetic machinery, terminating in a formulation of Maxwell’s equations. This book takes up a t that point. The first chapter contains a general statement of the equations governing fields and potentials, a review of the theory of units, reference material on curvilinear coordinate systems and the elements of tensor analysis, concluding with a formulation of the field equations in a spacetime continuum. The second chapter is also general in character, and much of i t may be omitted on a first reading. Here one will find a discnssion of fundamental field properties that may be deduced without reference to particular coordinate systems. A dimensional analysis of Maxwell’s equations leads to basic definitions of the vectors E and B, and an investigation of the energy relations results in expressions for the mechanical force exerted on elements of charge, current, and neutral matter. I n this way a direct connection is established between observable forces and the vectors employed t o describe the structure of a field. In Chaps. 111 and IV stationary fields are treated as particular cases of the dynamic field equations. The subject of wave propagation is taken up first in Chap. V, which deals with homogeneous plane waves. Particular attention is given to the methods of harmonic analysis, and the problem of dispersion is considered in some detail. Chapters VI and VII treat the propagation of cylindrical and spherical waves in unbounded spaces. A necessary amount of auxiliary material on Bessel functions and spherical harmonics is provided, and consideration is given to vector solutions of the wave equation. The relation of the field to its source, the general theory of radiation, and the outlines of the KirchhoffHuygens diffraction theory are discussed in Chap. VIII. Finally, in Chap. IX, we investigate the effect of plane, cylindrical, and spherical surfaces on the propagation of electromagnetic fields. This chapter illustrates, in fact, the application of the general theory established earlier to problems of practical interest. The reader will find here the more important laws of physical optics, the basic theory governing the propagation of waves along cylindrical conductors, a discussion of cavity oscillations, and an outline of the theory of wave propagation over the earth’s surface. It is regrettable that numerical solutions of special examples could not be given more frequently and in greater detail. Unfortunately the demands on space in a book covering such a broad field made this impractical. The primary objective of the book is a sound exposition of electromagnetic theory, and examples have been chosen with a view t o illustrating its principles. No pretense is made of an exhaustive treat
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ment of antenna design, transmissionline characteristics, or similar topics of engineering importance. It is the author’s hopc that, the present volume will provide the fundamental background necessary for a critical appreciation of original contributions in special fields and satisfy the needs of those who are unwilling t o accept engineering formulas without knowledge of their origin and limitations. Each chapter, with the exception of the first two, is followed by a set of problems. There is only one satisfactory way t o study a theory, and that is by application t o specific examples. The problems have been chosen with this in mind, but they cover also many topics which it was necessary t o eliminate from the text. This is particularly true of the later chapters. Answers or references are provided in most cases. This book deals solely with largescale phenomena. It is a sore temptation t o extend the discussion to that fruitful field which Frenkel terms the “quasimicroscopic state,” and to deal with the many beautiful results of the classical electron theory of matter. I n the light of contemporary developments, anyone attempting such a program must soon be overcome with misgivings. Although many laws of classical electrodynamics apply directly t o submicroscopic domains, one has no basis of selection. The author is firmly convinced that the transition must be made from quantum electrodynamics toward classical theory, rather than in the reverse direction. Whatever form the equations of q u a n t u n electrodynamics ultimately assume, their statistical average over large numbers of atoms must lead t o Maxwell’s equations. The m.k.s. system of units has been employed exclusively. There is still the feeling among many physicists t h a t this system is being forced upon them by a subversive group of engincers. Perhaps i t is, although i t was Maxwell himself who first had the idea. At all events, i t is a good system, easily learned, and one that avoids endless confusion in practical applications. At the moment there appears t o be no doubt of its universal adoption in the near future. Help for the tories among us who hold to the Gaussian system is offered on page 241. I n contrast t o the stand taken on the m.k.s. system, the author has no very strong convictions on the matter of rationalized units. Rationalized units have been employed because Maxwell’s equations are taken as the starting point rather than Coulomb’s law, and it seems reasonable to make the point of departure as simple as possible. As a result of this choice all equations dealing with energy or wave propagation are free from the factor 47F. Such relations are becoming of far greater practical importance than those expressing the potentials and field vectors in terms of their sources. The use of the time factor eciUL instead of e+wt is another point of mild controversy. This has been done because the time factor is invar
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iably discarded, and it is somewhat more convenient t o retain the positive exponent e+ikRfor a positive traveling wave. To reconcile any formula with its engineering counterpart, one need only replace i by +j. The author has drawn upon many sources for his material and is indebted to his colleagues in both the departments of physics and of electrical engineering at the Massachusetts Institute of Technology. Thanks are expressed particularly t o Professor M. F. Gardner whose advice on the practical aspects of Laplace transform theory proved invaluable, and to Dr. S. Silver who read with great care a part of the manuscript. In conclusion the author takes this occasion to express his sincere gratitude t o Catherine N. Stratton for her constant encouragement during the preparation of the manuscript and untiring aid in the revision of proof.
JULIUSADAMSSTRATTON. CAMBRIDQE, MASS., January, 1941.
CONTENTS
PAGE
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PREFACE
...
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CHAPTER I T H E FIELD EQUATIONS
E ~ U A T K .I N. S. . . . . . . . . . . . . . . . . . MAXWELL’S 1.1 The Field Vectors . . . . . . . . . . . . . . . . . . . 1.2 Charge and Current . . . . . . . . . . . . . . . . . . . 1.3 Divergence of the Field Vectors . . . . . . . . . . . . 1.4 Integral Form of the Field Equations . . . . . . . . . . PROPERTIES MACROSCOPIC . . OF MATTER 1.5 The Inductive Capacities t and p . 1.6 Electric and Magnetic Polarization 1.7 Conducting Media . . . . . . .
. . . . . . . . . . . . . . .
1 i
. . . . . .
2 6
. . . . . .
6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10 10
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
DIMENSIONS. . . . . . . . . . . . . . . . . . . . . . . . . 1.8 M.K.S. or Giorgi System . . . . . . . . . . . . . . . . . . . . .
UNITS AND
. . . . . . . . . . . . . . . . THE ELECTROMAGNETIC POTENTIALS 1.9 Vector and Scalar Potentials . . . . . . . . . . . . . . . . . 1.10 Conducting Media . . . . . . . . . . . . . . . . . . . . . 1.11 Hertz Vectors, or Polarization Potentials . . . . . . . . . . 1.12 Complex Field Vectors and Potentials . . . . . . . . . . .
. . . . . . . . . .
11
16 16
. . . .
23 23 26 28 32
BOUNDARY CONDITIONS. . . . . . . . . . . . . . . . . . . . . . . . . 1.13 Discontinuities in the Field Vectors . . . . . . . . . . . . . . . . .
34 34
COORDINATE SYSTEMS . . . . . . . . . . . . . . . . . . . 1.14 Unitary and Reciprocal Vectors . . . . . . . . . . . 1.15 Differential Operators . . . . . . . . . . . . . . . . . 1.16 Orthogonal Systems . . . . . . . . . . . . . . . . . . 1.17 Field Equations in General Orthogonal Coordinates . . . 1.18 Properties of Some Elementary Systems . . . . . . .
. . . .
. . . . . . . 38 38 44 47 . . . . . . . . . . . . . 50 . . . . . . . 51
. . . . . . .
. . . . . .
TENSORS . . . . . . . . . . . . . . . . . . . . . . . . . . . THEFIELD 1.19 Orthogonal Transformations and Their Invariants . . . . . . . . . . 1.20 Elements of Tensor Analysis . . . . . . . . . . . . . . . . . . . . 1.21 Spacetime Symmetry of the Field Equations . . . . . . . . . . . . 1.22 The Lorentz Transformation . . . . . . . . . . . . . . . . . . . . 1.23 Transformation of the Field Vectors to Moving Systems . . . . . . . .
xvii
59 59 64 69 74 78
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CON TENTS PAon
CHAPTER I1 STRESS AND ENERGY STRESS AND
ELASTICMEDIA . . . . . . . . . . . . . . . . . . .
83
2.1 Elastic Stress Tensor . . . . . . . . . . . . . . . . . . . . . . 2.2 Analysis of Strain . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Elastic Energy and the Relations of Stress to Strain . . . . . . . . .
83 87 93
STRAIN I N
FORCES ON CHARGES A N D CURRENTS ELECTROMAGNETIC 2.4 Definition of the Vectors E and B . . . . . . . 2.5 Electromagnetic Stress Tensor in Free Space . . 2.6 Electromagnetic Momentum . . . . . . . . .
96 96 . . . . . . . . . . . 97 . . . . . . . . . . 103 . . . . . . . . . . . . . . . . . . . . .
ELECTROSTATIC ENERGY. . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Electrostatic Energy as a Function of Charge Density . . . . . . . . . 2.8 Electrostatic Energy as a Function of Field Intensity . . . . . . . . . 2 3 A Theorem on Vector Fields . . . . . . . . . . . . . . . . . . . . 2.10 Energy of a Dielectric Body in an Electrostatic Field . . . . . . . . . 2.11 Thomson’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 2.12 Earnshaw’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 2.13 Theorem on the Energy of Uncharged Conductors . . . . . . . . . . MAGNETOSTATIC ENERGY . . . . . . . . . . . . . . . . . 2.14 Magnetic Energy of Stationary Currents . . . . . . 2.15 Magnetic Energy as a Function of Field Intensity . . 2.16 Ferromagnetic Materials . . . . . . . . . . . . . . . 2.17 Energy of a Magnetic Body in a Magnetostatic Field . 2.18 Potential Energy of a Permanent Magnet . . . . . .
104 104 107 111 112 114 116 117
. . . . . . . . 118 . . . . . . . . 118 . . . . . . . . 123 . . . . . . . 125 . . . . . . . . 126 . . . . . . . . 129
ENERGYFLOW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 2.19 Poynting’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 131 135 2.20 Complex Poynting Vector . . . . . . . . . . . . . . . . . . . . .
FIELD . . FORCESON A DIELECTRIC IN A N ELECTROSTATIC 2.21 Body Forces in Fluids . . . . . . . . . . . . . 2.22 Body Forces in Solids . . . . . . . . . . . . . 2.23 The Stress Tensor . . . . . . . . . . . . . . 2.24 Surfaces of Discontinuity . . . . . . . . . . . 2.25 Electrostriction . . . . . . . . . . . . . . . . 2.26 Force on a Body Immersed in a Fluid . . . . . . FORCES IN THE MAGNETOSTATIC FIELD . . . . . . . . . . . . 2.27 Nonferromagnetic Materials . . . . . . . . . 2.28 Ferromagnetic Materials . . . . . . . . . . . FORCESIN THE ELECTROMAGNETIC FIELD 2.29 Force on a Body Immersed in a Fluid . . . . . .
. . . . . . . . . . 137 . . . . . . . . . . 137 . . . . . . . . . 140 . . . . . . . . . .
146
. . . . . . . . . . 147 . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . .
149 151 153 153 155 156 156
CHAPTER I11 T H E ELECTROSTATIC FIELD
PROPERTIES OF AN ELECTROSTATIC FIELD. . . . . . . . . . . . . 160 GENERAL 3.1 Equations of Field and Potential . . . . . . . . . . . . . . . . . . 160 3.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 163
( ‘ONTENTS
XIX
PAQE
CALCULATION OF THE FIELD FROM THE CHARGE DISTRIBUTION . . 3.3 Green’s Theorem . . . . . . . . . . . . . . . . . . . . 3.4 Integration of Poisson’s Equation . . . . . . . . . . . . 3.5 Behavior a t Infinity . . . . . . . . . . . . . . . . . . . 3.6 Coulomb Field . . . . . . . . . . . . . . . . . . . . . 3.7 Convergence of Integrals . . . . . . . . . . . . . . . .
. . . . . . 165 . . . . . 165 . . . . . . 166 . . . . . 167 . . . . . 169 . . . . . 170
OF THE POTENTIAL IN SPHERICAL HARMONICS . . . . . . . . . . . EXPANSION 3.8 Axial Distributions of Charge . . . . . . . . . . . . . . . . . . . 3.9 TheDipole . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Axial Multipoles . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Arbitrary Distributions of Charge . . . . . . . . . . . . . . . . . 3.12 General Theory of Multipoles . . . . . . . . . . . . . . . . . . .
172 172 175 176 178 179
183 DIELECTRIC POLARIZATION. . . . . . . . . . . . . . . . . . . . . . . . 3.13 Interpretation of the Vectors P and n . . . . . . . . . . . . . . . 183 OF INTEGRALS OCCURRING IN POTENTIAL THEORY . . DISCONTINUITIES 3.14 Volume Distributions of Charge and Dipole Moment . . . . . 3.15 Singlelayer Charge Distributions . . . . . . . . . . . . . . 3.16 Doublelayer Distributions . . . . . . . . . . . . . . . . . . 3.17 Interpretation of Green’s Theorem . . . . . . . . . . . . . 3.18 Images . . . . . . . . . . . . . . . . . . . . . . . . . .
BOUNDARYVALUE PROBLEMS . . . . . . . . 3.19 Formulation of Electrostatic Problems 3.20 Uniqueness of Solution . . . . . . . 3.21 Solution of Laplace’s Equation . . . .
. . . . 185 . . . . 185
. . . . 187 . . . 188 . . . . 192 . . . 193
. . . . . . . . . . . . . . . 194 . . . . . . . . . . . . . . . 194
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
PROBLEM OF THE SPHERE . . . . . . . . . . . . . . . . . 3.22 Conducting Sphere in Field of a Point Charge . . . 3.23 Dielectric Sphere in Field of a Point Charge . . . . 3.24 Sphere in a Parallel Field . . . . . . . . . . . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
196 197
. 201 . . 201 . . 204 . 205
PROBLEM OF THE ELLIPSOID. . . . . . . . . . . . . . . . . . . . . . . 207 3.25 Free Charge on a Conducting E:llipsoid . . . . . . . . . . . . . . . 207 3.26 Conducting Ellipsoid in a Parallel Field . . . . . . . . . . . 3.27 Dielectric Ellipsoid in a Parallel Field . . . . . . . . . . . 3.28 Cavity Definitims of E and D . . . . . . . . . . . . . . . . . 3.29 Torque Exerted on an Ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PROBLEMS
. . . . 209 . . . . 211 . . . 213 . . . . 215 . . . . 217
CHAPTER IV T H E MAGNETOSTATIC FIELD GENERAL PROPERTIES OF A MAGNETOSTATIC FIELD. . . . . . . . . . . . . 225 4.1 Field Equations and the Vector Potential . . . . . . . . . . . . . . 225 226 4.2 Scalar Potential . . . . . . . . . . . . . . . . . . . . . . . . . 228 4.3 Poisson’s Analysis . . . . . . . . . . . . . . . . . . . . . . . . CALCULATION OF THE FIELD OF A CURRENT DISTRIBUTION . . . . . . . . . . 230 4.4 BiotSavart Law . . . . . . . . . . . . . . . . . . . . . . . . . 230 4.5 Expansion of the Vector Potential . . . . . . . . . . . . . . . 233
xx
CONTENTS
4.6 The Magnetic Dipole . 4.7 Magnetic Shells . . .
PAQ~
. . . . . . . . . . . . . . . . . . . . . .
236
. . . . . . . . . . . . . . . . . . . . . . 237 A DIGRESSION O N UNITS AND DIMENSIONS . . . . . . . . . . . . . . . . . 238 4.8 Fundamental Systems . . . . . . . . . . . . . . . . . . . . . . . 238 . . . . . . . . . . . . . . . . POLARIZATION . . . . . . . . . . . . . . . . . . . . . . . . . MAGNETIC 4.9
Coulomb's Law for Magnetic Matter
241 242
4.10 Equivalent Current Distributions . . . 4.11 Field of Magnetized Rods and Spheres
. . . . . . . . . . . . . . . 242 . . . . . . . . . . . . . . . 243 DISCONTINUITIES OF THE VECTORS A AND B . . . . . . . . . . . . . . . . . 245 4.12 Surface Distributions of Current . . . . . . . . . . . . . . . . . . 245 4.13 Surface Distributions of Magnetic Moment . . . . . . . . . . . . . 247 INTEGRATION OF THE EQUATION v X v X A = pJ . . . . . . . . . . . . . 250 4.14 Vector Analogue of Green's Theorem . . . . . . . . . . . . . . . . 250 4.15 Application to the Vector Potential . . . . . . . . . . . . . . .' . . 250 BOUNDARYVALUE PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . 4.16 Formulation of the Magnetostatic Problem . . . . . . . . . . . 4.17 Uniqueness of Solution . . . . . . . . . . . . . . . . . . . . . .
. .
254 254 256
ELLIPSOID. . . . . . . . . . . . . . . . . . . . . . . 257 4.18 Field of a Uniformly Magnetized Ellipsoid . . . . . . . . . . . . . . 257 4.19 Magnetic Ellipsoid in a Parallel Field . . . . . . . . . . . . . . . . 258
PROBLEM O F THE
CYLINDER IN A PARALLEL FIELD. . . . . . . . . . . . . . . . . . . . . 4.20 Calculation of the Field . . . . . . . . . . . . . . . . . . . . . . 4.21 Force Exerted on the Cylinder . . . . . . . . . . . . . . . . . . .
PROBLEMS . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
258 258 261 262
CHAPTER V PLANE WAVES I N UNBOUNDED. ISOTROPIC MEDIA
PROPAQATION OF PLANEWAVES . . . . . . . . . . . . 5.1 Equations of a Onedimensional Field . . . . . 5.2 Plane Waves Harmonic in Time . . . . . . . 5.3 Plane Waves Harmonic in Space . . . . . . . 5.4 Polarization . 5.5 Energy Flow . 5.6 Impedance . .
. . . . . . . . . .
268
. . . . . . . . . . . 268 . . . . . . . . . . . 273 . . . . . . . . . . . 278
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .
279 281 282
GENERAL SOLUTIONS OF THE ONEDIMENSIONAL WAVEEQUATION . . . . . . . 284 5.7 Elements of Fourier Analysis . . . . . . . . . . . . . . . . . . . . 285 5.8 General Solution of the Onedimensional Wave Equation in a Nondissips, tive Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 5.9 Dissipative Medium; Prescribed Distribution in Time . . . . . . . . . 297 5.10 Dissipative Medium; Prescribed Distribution in Space . . . . . . . . . 301 5.11 Discussion of a Numerical Example . . . . . . . . . . . . . . . . . 304 5.12 Elementary Theory of the Laplace Transformation . . . . . . . . . . 309 5.13 Application of the Laplace Transformation to Maxwell's Equations . . . 318
xxi
PREFACE
PAGE!
DISPERSION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.14 Dispersion in Dielectrics . . . . . . . . . . . . . . . . . . . . . . 5.15 Dispersion in Metals . . . . . . . . . . . . . . . . . . . . . . . 5.16 Propagation in an Ionized Atmosphcre . . . . . . . . . . . . . .
VELOCITIESOF PROPAGATION . . . . . . . . . . . . . . . . . . . . . . .
.
321 321 325 327 330
5.17 Group Velocity . . . . . . . . . . . . . . . 5.18 Wavefront and Signal Velocities . . . . . .
. . . . . . . . . . . 330 . . . . . . . . . . . . 333 PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 CHAPTER VI CYLJ ND RIC AL WAVES EQUATIONS OF A CYLINDRICAL FIELD . . . . . . . . . . . . . . . . . . . . 349 6.1 Representation by Hertz Vectors . . . . . . . . . . . . . . . . . . 349 351 6.2 Scalar and Vector Potentials . . . . . . . . . . . . . . . . . . . . 6.3 Impedances of Harmonic Cylindrical Fields . . . . . . . . . . . . . 354 OF THE CIRCULAR CYLINDER. . . . . . . . . . . . . . . 355 WAVEFUNCTIONS 6.4 Elementary Waves . . . . . . . . . . . . . . . . . . . . . . . . 355 6.5 Properties of the Functions Z n ( p ) . . . . . . . . . . . . . . . . . . 357 6.6 The Field of Circularly Cylindrical Wave Functions . . . . . . . . . . 360
INTEGRAL REPRESENTATIONS O F WAPEFUNCTIONS . . . . . . . . . . . . 6.7 Construction from Plane Wave Solutions . . . . . . . . . . . . . . 6.8 Integral Representations of the Functions Z, ( p ) . . . . . . . . . . 6.9 FourierBessel Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Representation of a Plane Wavt 6.11 The Addition Theorem for Circularly Cylindrical Waves . . . . . . .
. 361 . 361 . 364 369
. 371 . 372 WAVE FUNCTION5 O F THE ELLIPTIC CYLINDER . . . . . . . . . . . . . . . . 375 375 6.12 Elementary Waves . . . . . . . . . . . . . . . . . . . . . . . . 6.13 Integral Representations . . . . . . . . . . . . . . . . . . . . . . 6.14 Expansion of Plane and Circular Waves . . . . . . . . . . . . .
380
. . 384 PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 CHAPTER VII SPHERICAL WAVES
THEVECTORWAVEEQUATION. . . . . . . . . . . . . . . . . . . . . . 392 7.1 A Fundamental Set of Solutions . . . . . . . . . . . . . . . . . . 392 7.2 Application to Cylindrical Coordinates . . . . . . . . . . . . . . . 395
THESCALAR WAVEEQUATION IN SPHERICAL COORDINATTS . . . . . . . . . . 399 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10
Elementary Spherical Waves . . . . . . . . . . . . . . . Properties of the Radial Functions . . . . . . . . . . . Addition Theorem for the Legendre Polynomials . . . . . Expansion of Plane Waves . . . . . . . . . . . . . . . . Integral Representations . . . . . . . . . . . . . . . . . A FourierBessel Integral . . . . . . . . . . . . . . . Expansion of a Cylindrical Wave Function . . . . . . . . Addition Theorem for zo(kR). . . . . . . . . . . . . . .
. . . . . 399 . . . . 404 . . . . 406 . . . 408 . . . 409
. . . . . . . . . . . . . .
. . . . 411 . . . . 412
. . .
413
CON7FIV 73
XXll
PAQE
THEVECTOR 7.11 7.12 7.13 7.14
WAVE
EQUATION I N SPHERICAL COORDINATES . . . . . . . . . . 414
Spherical Vector Wave Functions . . . . . . . . . . . . . . . . . Integral Representations . . . . . . . . . . . . . . . . . . . . . . Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . Expansion of a Vector Plane Wave . . . . . . . . . . . . . . . . .
PROBLEMS . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
414 416 417 418 420
CHAPTER VIII RADIATION
THEINHOMOGENEOUS SCALAR WAVEEQUATION . . . . . . . . . . . . . . . 424 8.1 Kirchhoff Method of Integration . . . . . . . . . . . . . . . . . . 424 428 8.2 Retarded Potentials . . . . . . . . . . . . . . . . . . . . . . . . 430 8.3 Retarded Hertz Vector . . . . . . . . . . . . . . . . . . . . . .
A MULTIPOLE EXPANSION . . . . . . 8.4 Definition of the Moments . . . 8.5 Electric Dipole . . . . . . . . 8.6 Magnetic Dipole . . . . . . .
. . . . . . . . . . . . . . . . . .
431
431 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
. . . . . . . . . . . . . . . . . .
RADIATION THEORY OF LINEAR ANTENNA SYSTEMS. . . . 8.7 Radiation Field of a Single Linear Oscillator . . . . 8.8 Radiation Due to Traveling Waves . . . . . . . . 8.9 Suppression of Alternate Phases . . . . . . . . . 8.10 Directional Arrays . . . . . . . . . . . . . . . . 8.11 Exact Calculation of the Field of a Linear Oscillator 8.12 Radiation Resistance by the E.M.F. Method . . . .
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. . . . . . . . . . . .
437 438 438 445 446 448 454 457
. . .
.
. . . . . . .
460 THEKIRCHHOFFHUYGENS PRINCIPLE 8.13 Scalar Wave Functions . . . . . . . . . . . . . . . . . . . . . . 460 8.14 Direct Integration of the Field Equations . . . . . . . . . . . . . . 464 8.15 Discontinuous Surface Distributions . . . . . . . . . . . . . . . . 468 470 O F THE RADIATION PROBLEM . . . . 8.16 Integration of the Wave Equation . . . . . . . . . . . . . . . . . 470 8.17 Field of a Moving Point Charge . . . . . . . . . . . . . . . . . . 473 477 PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . .
FOURDIMENSIONAL FORMUL.4TION
CHAPTER I X BOUNDARYVALUE PROBLEMS
THEOREMS . . . . . . . . . GENERAL 9.1 Boundary Conditions . . . . . 9.2 Uniqueness of Solution . . . . 9.3 Electrodynamic Similitude . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
REFLECTION AND REFRACTION AT A PLANE SURFACE . . 9.4 Snell’s Laws . . . . . . . . . . . . . . . . . 9.5 Fresnel’s Equations . . . . . . . . . . . . . . 9.6 Dielectric Media . . . . . . . . . . . . . . 9.7 Total Reflection . . . . . . . . . . . . . . 9.8 Refraction in a Conducting Medium . . . . . 9.9 Reflection a t a Conducting Surface . . . . . .
. . . . . . .
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483 483 486 488
. . . . . . . . 490 . . . . . . . 490 . . . . . . . 492 . . . .
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. . . .
. 494 . 497 . 500 . 505
Cob TEiL TS
XXlll
PAW
PLANESHEETS. . . . . . . . . . . . . . . 9.10 Reflection and Transmission Coefficients . 9.11 Application to Dielectric Media . . . . 9.12 Absorbing Layers . . . . . . . . . . .
. . . . . . . . . . . . . 511 . . . . . . . . . . . . . . 511 . . . . . . . . . . . . . . 513 . . . . . . . . . . . . . .515
SURFACE WAVES. . . . . . . . . 9.13 Complex Angles of Incidence . 9.14 Skin Effect . . . . . . . .
516 518 520
PROPAGATION A L O N G A CIRCULAR CYLINDER . . . 9.15 Natural Modes . . . . . . . . . . . . 9.16 Conductor Embeddcd in a Dielectric . . 9.17 Further Discussion of the Principal Wave . 9.18 Waves in Hollow Pipes . . . . . . . . . COAXIAL LINES. . . . . . . . . . . . . 9.19 Propagation Constant . . . . . . . 9.20 Infinite Conductivity . . . . . . . 9.21 Finite Conductivity . . . . . . . .
. . . . . . . . . . . . 524 .
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.
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. 524
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. 527
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. 531
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537
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545 545 548 551
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.
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OF A SPHERE . . . . . . . . . . . . . OSCILLATIONS 9.22 Natural Modes . . . . . . . . . . . . . . . 9.23 Oscillations of a Conducting Sphere . . . . . 9.24 Oscillations in a Spherical Cavity . . . . . .
. . . . . . . . . . . 554 . . . . . . . . . . . 554 . . . . . . . . . . . . 558 . . . . . . . . . . . . 560
DIFFRACTION O F A P L A N E WAVE BY A SPHERE . . . 9.25 Expansion of the Diffracted Field . . . . . . 9.26 Total Radiation . . . . . . . . . . . . . . 9.27 Limiting Cases . . . . . . . . . . . . . . .
. . . . . . . . . . . . 563 . . . . . . . . . . . . 563 . . . . . . . . . . . 568 . . . . . . . . . . . 570
EFFECT OF TRC EARTH ON THE PROPAGATION OF RADIO WAVES. . . . . . . . 573 9.28 Sommerfeld Solution . . . . . . . . . . . . . . . . . . . . . . . 573 9.29 Weyl Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 577 9.30 van der Pol Solution . . . . . . . . . . . . . . . . . . . . . . . 582 9.31 Approximation of the Integrals . . . . . . . . . . . . . . . . . . . 583 PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588 APPENDIX I A . NUMERICAL VALUESOF FUNDAMENTAL CONSTANTS . . . . . . . . . . . . 601 B. DIMENSIONS OF ELECTROMAQNETIC QUANTITIES . . . . . . . . . . . . . 601 C . CONVERSIONTABLES . . . . . . . . . . . . . . . . . . . . . . . . . 602
F O R M U L A S FROM VECTOR
APPENDIX 11 ANALYSIS . . . . . . . .
APPENDIX I11 CONDUCTIVITY O F V A R I O U S h’f ATERIALS . . . . . . SPECIFIC INDUCTWE CAPACITY O F DIELECTRICS . .
. . . . . . . . . . . .
604
. . . . . . . . . . . . . 605 . . . . . . . . . . . . . 606
APPENDIX IV ASSOCIATEDLEGENDRE FUNCTIONS . . . . . . . . . . . . . . . . . . . .
608
INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
609
ELECTROMAGNETIC THEORY CHAPTER I
THE FIELD EQUATIONS
A vast wealth of experimental evidence accumulated over the past century leads one to believe that largescale electromagnetic phenomena are governed by Maxwell’s equations. Coulomb’s determination of the law of force between charges, the researches of Ampere on the interaction of current elements, and the observations of Faraday on variable fields can be woven into a plausible argument to support this view. The historical approach is recommended to the beginner, for it is the simplest and will afford him the most immediate satisfaction. In the present volume, however, we shall suppose the reader to have completed such a preliminary survey and shall credit him with a general knowledge of the experimental facts and their theoretical intcrpretation. Electromagnetic theory, according to the standpoint adopted in this book, is the theory of Maxwell’s equations. Consequently, we shall postulate these equations at the outset and proceed to deduce the structure and properties of the field together with its relation to the source. No single experiment constitutes proof of a theory. The true test of our initial assumptione will appear in the persistent, uniform correspondence of deduction with observation. In this first chapter we shall be occupied with the rather dry business of formulating equations and preparing the way for our investigation. MAXWELL’S EQUATIONS
1.1. The Field Vectors.By
an electromagnetic field let us understand the domain of the four vectors E and B, D and H. These vectors are assumed to be finite throughout the entire field, and at all ordinary points to be continuous functions of position and time, with continuous derivatives. Discontinuities in the field vectors or their derivatives may occur, however, on surfaces which mark an abrupt change in the physical properties of the medium. According to the traditional usage, E and H are known as the intensities respectively of the electric and magnetic field, D is called the electric displacement and B, the magnetic induction. Eventually the field vectors must be defined in terms of the experiments by which they can be measured. Until these experiments 1
2
THE FIELD EQUATIONS
[CHAP.
I
are formulated, there is no reason to consider one vector more fundamental than another, and we shall apply the word intensity t o mean indiscriminately the strength or magnitude of any of the four vectors at a point in space and time. The source of an electromagnetic field is a distribution of electric charge and current. Since we are concerned only with its macroscopic effects, it may be assumed that this distribution is continuous rather than discrete, and specified as a function of space and time by t.he density of charge p , and by tJhevector current density J. We shall now postulate that a t every ordinary point in space the field vectors are subject to the Maxwell equations: aB V X E +  =atO ,
f3D VXH=J. dt
By an ordinary point we shall mean one in whose neighborhood the physical properties of the medium are continuous. It has been noted that the transition of the field vectors and their derivatives across a surface bounding a material body may be discontinuous; such surfaces must, therefore, be excluded until the nature of these discontinuities can be investigated. 1.2. Charge and Current.Although the corpuscular nature of electricity is well established, the size of the elementary quantum of charge is too minute to be taken into account as a distinct entity in a strictly macroscopic theory. Obviously the frontier that marks off the domain of largescale phenomena from those which are microscopic is an arbitrary one. To be sure, a macroscopic element of volume must contain an enormous number of atoms; but that condition alone is an insufficient criterion, for many crystals, including the metals, exhibit frequently a microscopic “grain” or “mosaic” structure which will be excluded from our investigation. We are probably well on the safe side in imposing a limit of onetenth of a millimeter as the smallest admissible element of length. There are many experiments, such as the scattering of light by particles no larger than lop3 mm. in diameter, which indicate that the macroscopic theory may be pushed well beyond the limit suggested. Nonetheless, we are encroaching here on the proper domain of quantum theory, and it is the quantum theory which must eventually determine the validity of our assumptions in microscopic regions. Let us suppose that the charge contained within a volume element Av is Aq. The charge density a t any point within Av will be defined by the relation (3) Aq = p Av.
SEC.
1.21
CHARGE A N D CURRENT
3
Thus by the charge density a t a point we mean the average chaage per unit volume in the neighborhood of t h a t point. I n a strict sense (3) does not define a continuous function of position, for Au cannot approach zero without limit. Nonetheless we shall assume t h a t p can be represented by a function of the coordinates and the time which a t ordinary points is continuous and has continuous derivatives. The value of the total charge obtained by integrating that function over a largescale volume will then differ from the true charge contained therein by a microscopic quantity a t most. Any ordered motion of charge constitutes a current. A current distribution is characterized by a vector field which specifies at each point not only the intensity of the flow but also its direction. As in the study of fluid motion, it is convenient to imagine streamlines traced through the distribution and everywhere tangent to the direction of flow. Consider a surface which is orthogonal t o a system of streamlines. The current density at any point on this surface is then defined as a vector J directed along the streamline through the point and equal in magnitude t o the charge which in unit time crosses unit area of the surface in the vicinity of the point. On the other hand the current I across any surface S is equal t o the rate at which charge crosses that surface. If n is the positive unit normal t o an element A a of X,we have (4)
AI
=

J n Au.
Since Au is a macroscopic element of area, Eq. (4)docs not define the current density with mathematical rigor as a continuous function of position, but again one may represent the distribution by such a function without incurring an appreciable error. The total current through S is, therefore, (5)
I
=
Jnda.
Since electrical charge may be either positive or negative, a convention must be adopted as t o what constitutes a positive current. If the flow through an element of area consists of positive charges whose velocity vectors form an angle of less than 90 deg. with the positive normal n, the current is said t o be positive. If the angle is greater than 90 deg., the current is negative. Likewise if the angle is less than 90 deg. but the charges are negative, the current through the element is negative. I n the case of metallic conductors the carriers of electricity are presumably negative electrons, and the direction of the current density vector is therefore opposed t o the direction of electron motion. Let us suppose now that the surface S of Eq. ( 5 ) is closed. We shall adhere t o the customary convention that the positive normal to a closed
4
[CHAP.I
THE FIELD EQUATIONS
surface i s drawn outward. I n virtue of the definition of current as the flow of charge across a surface, it follows that the surface integral of the normal component of J over S must measure the loss of charge from the region within. There is no experimental evidence to indicate that under ordinary conditions charge may be either created or destroyed in macroscopic amounts. One may therefore write
where V is the volume enclosed by S , as a relation expressing the conservation of charge. The flow of charge across the surface can originate in two ways. The surface S may be fixed in space and the density p be some function of the time as well as of the coordinates; or the charge density may be invariable with time, while the surface moves in some prescribed manner. I n this latter event the righthand integral of ( 6 ) is a function of time in virtue of variable limits. If, however, the surface is fixed and the integral convergent, one may replace d / d t by a partial derivative under the sign of integration.
We shall have frequent occasion to make use of the divergence theorem of vector analysis. Let A(z, y, Z) be any vector function of position which together with its first derivatives is continuous throughout a volume V and over the bounding surface S. The surface S is regular but otherwise arbitrary.l Then it can be shown that
A an da =
sv
V A dv.
As a matter of fact, this relation may be advantageously used as a definition of the divergence. To obtain the value of V A at a point P within V , we allow the surface S to shrink about P. When the volume V has become sufficiently small, the integral on the right may be replaced by VV A, and we obtain

(9) ‘ A regular element of arc is represented in parametric form by the equations z = z(t), y = y ( t ) , z = z ( t ) such that in the interval a S t 5 b z,y, z are continuous, singlevalued functions of t with continuous derivatives of all orders unless otherwise restricted. A regular curve is constructed of a finite number of such arcs joined end t o end but such that the curve does not cross itself. Thus a regular curve has no double points and is piecewise differentiable. A regular surface element is a portion of surface whose projection on a properly oriented plane is the interior of a regular closed curve. Hence it does not intersect itself. Cf.Kellogg, “Foundations of Potential Theory,” p. 97, Springer, 1929.
SEC.
1.21
CHARGE A N D CURRENT
5
The divergence of a vector at a point is, therefore, to be interpreted as the integral of its normal component over an infinitesimally small surface enclosing that point, divided by the enclosed volume. The flux of a vector through a closed surface is a measure of the sources within; hence the divergence determines their strength a t a point. Since S has been shrunk close about P , the value of A a t every point on the surface may be expressed analytically in terms of the values of A and its derivatives at P, and consequently the integral in (9) may be evaluated, leading in the case of rectangular coordinates to
On applying this theorem to (7) the surface integral is transformed to the volume integral
Now the integrand of (11) is a continuous function of the coordinates and hence there must exist small regions within which the integrand does not change sign. If the integral is to vanish for arbitrary volumes V , it is necessary that the integrand be identically zero. The differential equation V  J + a=P O at
expresses the conservation of charge in the neighborhood of a point. By analogy with an equivalent relation in hydrodynamics, (12) is freGently referred to as the equation of continuity. If at every point within a specified region the charge density is constant, the current passing into the region through the bounding surface must at all times equal the current passing outward. Over the bounding surface S we have
$s J  n d a = 0, and at every interior point (14)
V J = 0.
Any motion characterized by vector or scalar quantities which are independent of the time is said to be steady, or stationary. A steadystate flow of electricity is thus defined by a vector J which at every point within the region is constant in direction and magnitude. In virtue of the divergenceless character of such a current distribution, it follows
6
THE FIELD EQUATIONS
[CHAP.I
that in the steady state all streamlines, or current filaments, close upon themselves. The field of the vector J is solenoidal. 1.3. Divergence of the Field Vectors.Two further conditions satisfied by the vectors B and D may be deduced directly from Maxwell’s equations by noting that the divergence of the curl of any vector vanishes identically. We take the divergerxe of Eq. (1) and obtain
The commutation of the operators V and a / d t is admissible, for at an ordinary point B and all its derivatives are assumed to be continuous. It follows from (15) that at every point in the field the divergence of B is constant. If ever in its past history the field has vanished, this constant must be zero and, since one may reasonably suppose that the initial generation of the field was at a time not infinitely remote, we conclude that (16) V * B = 0, and the field of B is therefore solenoidal. Likewise the divergence of Eq. (2) leads to (17)
d at
V .J +  V . D
=O,
or, in virtue of (12), to
(18)
d ( V  D p ) = 0. at

If again we admit that at some time in its past or future history the field may vanish, it is necessary that (19)
V  D = p.
The charges distributed with a density p constitute the sources of the vector D. The divergence equations (16) and (19) are frequently included as part of Maxwell’ssystem. I t must be noted, however, that if one assumes the conservation of charge, these are not independent relations. 1.4. Integral Form of the Field Equations.The properties of an electromagnetic field which have been specified by the differential equations (l),( Z ) , (16), and (19) may also be expressed by an equivalent system of integral relations. To obtain this equivalent system, we apply a second fundamental theorem of vector analysis. According to Stokes’ theorem the line integral of a vector taken about a closed contour can be transformed into a surface integral extended
SEC.1.41
INTEGRAL FORM OF T H E FIELD EQUATIONS
7
over a surface bounded by the contour. The contour C must either be regular or be resolvable into a finite number of regular arcs, and it is assumed that the otherwise arbitrary surface S bounded by C is twosided and may be resolved into a finite numbcr of regular elements. The positive side of the surface S is related to the positive direction of circulation on the contour by the usual convention that an observer, moving in a positive sense along C, will have the positive side of S on his left. Then if A(x, y, z ) is any vector function of position, which together with its first derivatives is continuous a t all points of S and C, it may be shown that (20)
LA.ds=L(VxA)nda,
where ds is an element of length along C and n is a unit vector normal t o the positive side of the element of area da. This transformation can also be looked upon as an equation defining the curl. T o determine the value of V x A a t a point P on S , we allow the contour t o shrink about P until the enclosed area S is reduced t o an infinitesimal element of a plane whose normal is in the direction specified by n. The integral on the right is then equal t o (V x A) nS, plus infinitesimals of higher order. The projection of the vector V x A in the direction of the normal is, therefore,

The curl of a vector at a point is to be interpreted as the line integral of that vector about an infinitesimal path on a surface containing the point, per unit of enclosed area. Since A has been assumed analytic in the neighborhood of P , its value a t any point on C may be expressed in terms of the values of A and its derivatives a t P , so t h a t the evaluation of the line integral in (21) about the infinitesimal path can actually be carried out. I n particular, if the element X is oriented parallel t o the yzcoordinate plane, one finds for the xcomponent of the curl
Proceeding likewise for the y and zcomponents we obtain
8
T H E FIELD EQUATIONS
[CHAP.
I
Let us now integrate the normal component of the vector d B / d t over any regular surface S bounded by a closed contour C. From (1) and (20) it follows that
If the contour is fixed, the operator a / a t may be brought out from under the sign of integration. (25)
By definition, the quantity (26)
a
=
LBncia
is the magnetic flux, or more specifically the flux of the vector B through the surface. According to (25) the line integral of the vector E about any closed, regular curve in the field is equal to the time rate of decrease of the magnetic flux through any surface spanning that curve. The relation between the direction of circulation about a contour and the positive normal to a surface bounded by it is C illustrated in Fig. 1. A positive direction FIG. 1.Convention relating direction of the positive normal about c is chosen arbitrarily and the flux a n t o the direction of circulation is then positive or negative according to about a contour C . the direction of the lines of B with respect to the normal. The time rate of change of Q, is in turn positive or n e g a tive as the positive flux is increasing or decreasing. We recall that the application of Stokes’ theorem t o Eq. (1) is valid only if the vector E and its derivatives are continuous a t all points of S and C. Since discontinuities in both E and B occur across surfaces marking sudden changes in the physical properties of the medium, the question may be raised as to what extent (25) represents a general law of the electromagnetic field. One might suppose, for example, that the contour linked or pierced a closed iron transformer core. To obviate this difficulty it may be imagined that at the surface of every material body in the field the physical properties vary rapidly but continuously within a thin boundary layer from their values just inside to their values just outside the surface. I n this manner all discontinuities are eliminated from the field and (25) may be applied to every closed contour. The experiments of Faraday indicated that the relation (25) holds whatever the cause of flux variation. The partial derivative implies a,
&
SEC.1.41
INTEGRAL FORM OF T H E FIELD EQUATIONS
9
variable flux density threading a fixed contour, but the total flux can likewise be changed by a deformation of the contour. To take this into account the Faraday law is written generally in the form
It can be shown that (27) is in fact a consequence of the differential field equations, but the proof must be based on the electrodynamics of moving bodies which will be touched upon in Sec. 1.22. In like fashion Eq. (2) may be replaced by an equivalent integral relation,
where I is the total current linking the contour as defined in (5). I n the steady state, the integral on the right is zero and the conduction current I through any regular surface is equal to the line integral of the vector H about its contour. If, however, the field is variable, the vector dD/dt has associated with it a field H exactly equal to that which would be produced by a current distribution of density J‘ = dD
.
dt
To this quantity Maxwell gave the name (‘displacement current,” a term which we shall occasionally employ without committing ourselves as yet to any particular interpretation of the vector D. The two remaining field equations (16) and (19) can be expressed in an equivalent integral form with the help of the divergence theorem. One obtains (30)
SsB.nda
=
0,
stating that the total flux of the vector B crossing any closed, regular surface is zero, and
according to which the flux of the vector D through a closed surface is equal to the total charge q contained within. The circle through the sign of integration is frequently employed to emphasize the fact that a contour or surface is closed.
10
THE FIELD EQUATIONS
[CHAP.I
MACROSCOPIC PROPERTIES OF MATTER
1.5. The Inductive Capacities E and p.No other assumptions have been made thus far than that an electromagnetic field may be characterized by four vectors E, B, D, and H, which a t ordinary points satisfy Maxwell’s equations, and that the distribution of current which gives rise to this field is such as to ensure the conscrvation of charge. Between the five vectors E, B, D, H, J t,hcre are but two independent relations, the equations (I) and (2) of the preceding scction, and we are therefore obliged t o impose further conditions if the system is t o be made determinate. Let us begin with the assumption that a t any given point in the field, whether in free space or within matter, the vector D may be represented as a function of E and the vector H as a function of B. (1)
D
=
D(E),
H
=
H(B).
The nature of these functional relations is t o be determined solely by the physical properties of the medium in the immediate neighborhood of the specified point. Certain simple relations are of most common occurrence. 1. I n free space, D differs from E only by a constant factor, as does H from B. Following the traditional usage, we shall write (2)
D = QE,
H = 1B e PO
The values and the dimensions of the constants €0 and po will depend upon the system of units adopted. In only one of many wholly arbitrary systems does D reduce to E and H t o B in empty space. 2. If the physical properties of a body in the neighborhood of some interior point are the same in all directions, the body is said t o be isotropic. At every point in an isotropic medium D is parallel t o E and H is parallel t o B. The relations between the vectors, moreover, are linear in almost all the soluble problems of electromagnetic theory. For the isotropic, linear case we put then
(3)
D
=
eE,
H
=
1 B. P
The factors cand p will be called the inductive capacities of the medium. The dimensionless ratios
(4) are independent of the choice of units and will be referred t o as the specific inductive capacities. The properties of a homogeneous medium are constant from point t o point and in this case it is customary t o refer
SEC.1.61
ELECTRIC A N D MAGNETIC POLARIZATION
11
t o K~ as the dielectric constant and to K , as the permeability. I n general, however, one must look upon the inductive capacities as scalar functions of position which characterize the electromagnetic properties of matter in the large. 3. The properties of anisotropic matter vary in a different manner along different directions about a point. I n this case the vectors D and E, H and B are parallel only along certain preferred axes. If i t may be assumed that the relations are still linear, as is usually the case, one may express each rectangular component of D as a linear function of the three components of E.
(5) The coefficients ejk of this linear transformation are the components of a symmetric tensor. An analogous relation may be set up between the vectors H and B, but the occurrence of such a linear anisotropy in what may properly be called macroscopic problems is rare. The distinction between the microscopic and macroscopic viewpoints is nowhere sharper than in the interpretation of these parameters e and p, or their tensor equivalents. A microscopic theory must deduce the physical properties of matter from its atomic structure. It must enable one t o calculate not only the average field that prevails within a body but also its local value in the neighborhood of a specific atom. It must tell us how the atom will be deformed under the influence of that local field, and how the aggregate effect of these atomic deformations may be represented in the large by such parameters as e and p. We, on the other hand, are from the present standpoint sheer behaviorists. Our knowledge of matter is, t o use a large word, purely phenomenological. Each substance is to be characterized electromagnetically i n terms of a minimum number of parameters. The dependence of the parameters E and p on such physical variables as density, temperature, and frequency will be established by experiment. Information given by such measurements sheds much light on the internal structure of matter, but the internal structure is not our present concern. 1.6. Electric and Magnetic Polarization.To describe the electromagnetic state of a sample of matter, it will prove convenient t o introduce two additional vectors. We shall define the electric and magnetic polarization vectors b y the equations
The polarization vectors are thus definitely associated with matter and
12
[CHAP.I
THE FIELD EQUATIONS
vanish in free space. By means of these relations let us now eliminate D and H from the field equations. There results the system
(7) 1 V*E=(pV*P),
V.B=O,
€0
which we are free to interpret as follows: the presence of rigid material bodies in a n electromagnetic field m a y be completely accounted for b y a n equivalent distribution of charge of density V P, and an equivalent dP distribution of current of density  V x M. dt

+
In isotropic media the polarization vectors are parallel to the corresponding field vectors, and are found experimentally to be proportional to them if ferromagnetic materials are excluded. The electric and magnetic susceptibilities xe and xmare defined by the relations (8)
P
=
x~OE,
M = xmH.
Logically the magnetic polarization M should be placed proportional to B. Long usage, however, has associated it with H and to avoid confusion on a matter which is really of no great importance we adhere to this convention. The susceptibilities xe and x,,, defined by (8) are dimensionless ratios whose values are independent of the system of units employed. In due course it will be shown that E and B are force vectors and in this sense are fundamental. D and H are derived vectors associated with the state of matter. The polarization vector P has the dimensions of D, not E, while M and H are dimensionally alike. From (3), ( 6 ) , and (8) it follows at once that the susceptibilities are related t o the specific inductive capacities by the equations (9)
Xe =
Ke
 1,
Xm =
Km
 1.
In anisotropic media the susceptibilitirs are represented by the components of a tensor. It will be a part of our task in later chapters t o formulate experiments by means of which the susceptibility of a substance may be accurately measured. Such measurements show that the electric susceptibility is always positive. I n gases it is of the order of 0.0006 (air), but in liquids and solids it may attain values as large as 80 (water). An inherent difference in the nature of the vectors P and M is indicated by the fact that the magnetic susceptibility X n may be either positive or negative. Substances characterized by a positive susceptibility are said t o be
SEC.1.71
CONDUCTING M E D I A
13
paramagnetic, whereas those whose susceptibility is negative are called diamagnetic. The metals of the ferromagnetic group, including iron, nickel, cobalt, and their alloys, constitute a particular class of substances of enormous positive susceptibility, the value of which may be of the order of many thousands. In view of the nonlinear relation of M to H peculiar to these materials, the susceptibility xmmust now be interpreted as the slope of a tangent to the MH curve at a point corresponding to a particular value of H. To include such cases the definition of susceptibility is generalized to
The susceptibilities of all nonferromagnetic materials, whether paramagnetic or diamagnetic, are so small as to be negligible for most practical purposes. Thus far it has been assumed that a functional relation exists between the vector P or M and the applied field, and for this reason they may properly be called the induced polarizations. Under certain conditions, however, a magnetic field may be associated with a ferromagnetic body in the absence of any external excitation. The body is then said to be in a state of permanent magnetization. We shall maintain our initial assumption that the field both inside and outside the magnet is completely defined by the vectors B and H. But now the difference of these two vectors a t an interior point is a fixed vector Mo, which may be called the intensity of magnetization and which bears no functional relationship to H. On the contrary the magnetization M, must be interpreted as the source of the field. If an external field is superposed on the field of a permanent rnagnct, the intensity of magnetization will be augmented by the induced polarization M. At any interior point we have, therefore,
+ +
B = po(H M Mo). (11) Of this induced polarization we can only say for the present that it is a function of the resultant H prevailing at the same point. The relation of the resultant field within the body to the intensity of an applied field generated by external sources depends not only on the magnetization Mo but also upon the shape of the body. There will be occasion to examine this matter more carefully in Chap. IV. 1.7. Conducting Media.To Maxwell's equations there must now be added a third and last empirical relation between the current density and the field. We shall assume that at any point within a liquid or solid the current density is a function of the field E.
14
T H E F I E L D EQUATIONS
[CHAP.I
The distribution of current in a n ionized, gaseous medium may depend also on the intensity of the magnetic field, but since electromagnetic phenomena in gaseous discharges are in general governed b y a multitude of factors other than those taken into account in the present theory, we shall exclude such cases from further consideration. Throughout a remarkably wide range of conditions, in both solids and weakly ionized solutions, the relation (12) proves to be linear.
J = uE.
(13)
The factor u is called the conductivity of the medium. The distinction between good and poor conductors, or insulators, is relative and arbitrary. All substances exhibit conductivity to some degree but the range of observed values of u is tremendous. The conductivity of copper, for example, is some lo7 times as great as that of such a “good” conductor as sea water, and 1019 times that of ordinary glass. I n Appendix 111 will be found a n abbreviated table of the conductivities of representative materials. Equation (13) is simply Ohm’s law. Let us imagine, for example, a stationary distribution of current throughout the volume of any conducting medium. I n virtue of the divergenceless character of the flow this distribution may be represented by closed streamlines. If a and b are two points on a particular streamline and d s is an element of its length, we have
(14)
JbE.ds
=
J ba J .nd s .
A bundle of adjacent streamlines constitutes a current filament or tube. Since the flow is solenoidal, the current I through every cross section of the filament is the same. Let S be the crosssectional area of the filament on a plane drawn normal to the direction of flow. X need not be infinitesimal, but is assumed t o be so small that over its area the current density is uniform. Then SJ ds = I d s , and

The factor,
(16)
R
=
.I
b l
ads,
1 It is true t h a t to a very slight degree the current distribution in a liquid or solid conductor may be modified by an impressed magnetic field, but the magnitude of this socalled Hall effect is so small that it may be ignorcd without incurring an appreciable error.
CONDUCTING M E D I A
SEC.1.71
15
is equal to the resistance of the filament bstweeil the points a and b. The resistance of a linear section of homogeneous conductor of uniform cross section S and length 1 is
a formula which is strictly valid only in the case of stationary currents. W i t h i n a region of nonvanishing conductivity there can be no permanent distribution of f r e e charge. This fundamentally important theorem cag be easily demonstrated when the medium is homogeneous and such that the relations between D and E and J and E are linear. By the equation of continuity, V.J
+ aatP = V . a E + dtdP = 0.
On the other hand in a homogeneous rnedium
(19)
V .E
1
=  p, E
which combined with (18) leads to
The density of charge at any instant is, therefore,
the constant of integration po being equal to the density a t the time t = 0. The initial charge distribution throughout the conductor decays exponentially with the time a t every point and in a manner wholly independent of the applied field. If the charge density is initially zero, it remains zero a t all times thereafter. The time
required for the charge at any point to dacay to l / e of its original value is called the relaxation time. I n all but the poorest conductors r is exceedingly small. Thus in sea water the relaxation time is about 2X sec. ; even in such a poor conductor as distilled water it is not great,er than sec. I n the best insulators, such as fused quartz, it may nevertheless assume values exceeding lo6 sec., an instance of the extraordinary range in the possible values of the parameter u.
16
T H E FIELD EQUATIONS
[CHAP.I
Let us suppose that at f = 0 a charge is concentrated within a small spherical region located somewhere in a conducting body. At every other point of the conductor the charge density is zero. The charge within the sphere now begins to fade away exponentially, but according to (21) no charge can reappear anywhere within the conductor. What becomes of i t ? Since the charge is conserved, the decay of charge within the spherical surface must be accompanied by an outward flow, or current. No charge can accumulate a t any other interior point; hence the flow must be divergenceless. It will be arrested, however, on the outer surface of the conductor and it is here that we shall rediscover the charge that has been lost from the central sphere. This surface charge makes its appearance at the exact instant that the interior charge begins to decay, for the total charge is constant. UNITS AND DIMENSIONS
1.8. The M.K.S. or Giorgi System.An electromagnetic fleld thus far is no more than a complex of vectors subject to a postulated system of differential equations. To proceed further we must establish the physical dimensions of these vectors and agree on the units in which they are to be measured. In the customary sense, an “absolute” system of units is one in which every quantity may be measured or expressed in terms of the three fundamental quantities mass, length, and time. Now in electromagnetic theory there is an essential arbitrariness in the matter of dimensions which is introduced with the factors € 0 and po connecting D and E, H and B respectively in free space. No experiment has yet been imagined by means of which dimensions may be attributed to either € 0 or PO as a n independent physical entity. On the other hand, it is a direct consequence of the field equations that the quantity c =
(1)
1 d G 0
shall have the dimensions of a velocity, and every arbitrary choice of BO and po is subject to this restriction. The magnitude of this velocity cannot be calculated a priori, but by suitable experiment it may be measured. The value obtained by the method of Rosa and Dorsey of the Bureau of Standards and corrected by Curtis1 in 1929 is (2)
c =
1
= 2.99790 X los
dGZJ
meters/sec.,
ROSAand DORSEY, A New Determination of the Ratio of the Electrostatic Unit of Electricity, Bur. Standards, Bull. 3, p. 433, 1907. CURTIS, Bur. Standards J . Research, 3, 63, 1929.
SEC.1.81
THE M.K.S. OR GIORGI SYSTEM
17
or for all practical purposes
(3)
c =3
x
108
meters/sec.
Throughout the early history of electromagnetic theory the absolute electromagnetic system of units was employed for all scientific investigations. I n this system the centimeter was adopted as the unit of length, the gram as the unit of mass, the second as the unit of time, and as a fourth unit the factor p o was placed arbitrarily equal t o unity and considered dimensionless. The dimensions of c0 were then uniquely determined by (1) and it could be shown that the units and dimensions of every other quantity entering into the theory might be expressed in terms of centimeters, grams, seconds, and po. Unfortunately, this absolute system failed t o meet the needs of practice. The units of resistance and of electromotive force were, for example, far too small. To remedy this defect a practical system was adopted. Each unit of the practical system had the dimensions of the corresponding electromagnetic unit and differed from it in magnitude by a power of ten which, in the case of voltage and resistance a t least, was wholly arbitrary. The practical units have the great advantage of convenient size and they are now universally employed for technical measurements and computations. Since they have been defined as arbitrary multiples of absolute units, they do not, however, constitute an absolutc system. Now the quantities mass, length, and time are fundamental solely because the physicist has found it expedient t o raise them t o that rank. That there are other fundamental quantities is obvious from the fact that all electromagnetic quantities cannot be expressed in terms of these three alone. The restriction of the term “absolute” to systems based on mass, length, and time is, therefore, wholly unwarranted; one should ask only that such a system be selfconsistent and that every quantity be defined in terms of a minimum number of basic, independent units. The antipathy of physicists in the past to the practical system of electrical units has been based not on any firm belief in the sanctity of mass, length, and time, but rather on the lack of selfconsistency within that system. Fortunately a most satisfactory solution has been found for this difficulty. I n 1901 Giorgi,’ pursuing an idea originally due to Maxwell, called attention t o the fact that the practical system could be converted into an absolute system by an appropriate choice of fundamental units. It is indeed only necessary to choose for the unit of length the interGIORGI:Units Razionali di Elettromagnetismo, Atti deZ2’ A.E.I., 1901. An historical review of the development of the practical system, including a report of the action taken a t the 1935 meeting of the International Electrotechnical Commission and an extensive bibliography is given by Kennelly, J. Inst. Elec. Engrs., 78, 235The M.K.S. System of Electrical Units, J . Inst. 245, 1936. See also GLAZEBROOK, Elec. Engrs., 78, pp. 245247.
I8
T H E FIELD EQUAT IO NS
[CHAP.I
national meter, for the unit of mass the kilogram, for the unit of time the second, and as a fourth unit any electrical quantity belonging to the practical system such as the coulomb, the ampere, or the ohm. From the field equations it is then possible to deduce the units and dimensions of every electromagnetic quantity in terms of these four fundamental units. Moreover the derived quantities will be related to each other exactly as in the practical system and may, therefore, be expressed in practical units. I n particular it is found that the parameter po must have the value 47r X lop7, whence from (1) the value of co may be calculated. Inversely one might equally well assume this value of po as a fourth basic unit and then deduce the practical series from the field equations. At a plenary session in June, 1935, the International Electrotechnical Commission adopted unanimously the m.k.s. system of Giorgi. Certain questions, however, still remain to be settled. No official agreement has as yet been reached as to the fourth fundamental unit. Giorgi himself recommended that the ohm, a material standard defined as the resistance of a specified column of mercury under specified conditions of pressure and temperature, be introduced as a basic quantity. If pa = 47r x 107 be chosen as the fourth unit and assumed dimensionless, all derived quantities may be expressed in terms of mass, length, and time alone, the dimensions of each being identical with those of the corresponding quantity in the absolute electromagnetic system and differing from them only in the size of the units. This assumption leads, however, to fractional exponents in the dimensions of many quantities, a direct consequence of our arbitrariness in clinging to mass, length, and time as the sole fundamental entities. I n the absolute electromagnetic system, for example, the dimensions of charge are grams? . centimeters*, an irrationality which can hardly be physically significant. These fractional exponents are entirely eliminated if we choose as a fourth unit the coulomb; for this reason, charge has been advocated at various times as a fundamental quantity quite apart from the question of its magnitude.’ I n the present volume we shall adhere exclusively to the meterkilogramsecondcoulomb system. A subsequent choice by the I.E.C. of some other electrical quantity as basic will in nowise affect the size of our units 01. the form of the equations.2 See the discussion by WALLOT:Elektrotechnische Zeilschrift, Nos. 4446, 1922. Also SOMMERFELD: “Ueber die Electromagnetischen Einheiten,” pp. 157165, Zeeman Verhandelingen, Martinus Nijhoff, The Hague, 1935; Physik. 2.36, 814820, 1935. No ruling has been made as yet on the question of rationalization and opinion seems equally divided in favor and against. If one bases the theory on Maxwell’s equations, it seems definitely advantageous t o drop the factors 47r which in unrationalired systems stand before the charge and current densities. A rationalized system will be employed in this hook.
19
T H E M.K.S. OR GIORGI S Y S T E M
SEC.1.81
To demonstrate that the proposed units do constitute a selfconsistent system let us proceed as follows. The unit of current in the m.k.s. system is t o be the absolute ampere and the unit of resistance is t o be the absolute ohm. These quantities are to be such that the work expended per second by a current of 1 amp. passing through a resistance of 1 ohm is 1 joule (absolute). If R is the resistance of a section of conductor carrying a constant current of I amp., the work dissipated in heat in t sec. is
W
(4)
=
IzRt
joules.
By means of a calorimeter the heat generated mr,y be measured and thus one determines the relation of the unit of electrical energy to the unit quantity of heat. It is desired that the joule defined by (4) be identical with the joule defined as a unit of mechanical work, so that in the electrical as well as in the mechanical case 1 joule
(5)
=
0.2389
gramcalorie (mean).
Now we shall dejine the ampere on the basis of the equation of continuity (6), page 4,as the current which transports across any surface 1 coulomb in 1 sec. Then the ohm is a derived unit whose magnitude and dimensions are determined by (4) : watt lohm=lampere2
kilogram . meter2 coulomb2 second' 1
since 1 watt is equal to 1 joule/sec. The resistivity of a medium is defined as the resistance measured between two parallel faces of a unit cube. The reciprocal of this quantity is the conductivity. The dimensions of u follow from Eq. (17), page 15. (7)
1 unit of conductivity
=
1 ohm . meter
coulomb2 * second hlogram . meter3.
I n the United States the reciprocal ohm is usually called the mho, although the name siemens has been adopted officially by the I.E.C. The unit of conductivity is therefore 1 siemens/meter. The volt will be defined simply as 1 wattlamp., or
1 volt = 1
watt ampere
~
kilogram . meter2 . coulomb . second2
Since the unit of current density is 1 amp./meter2, we deduce from the relation J = aE that (9)
1 unit of E
=
watt volt 1 ampere . meter Imeter
kilogram meter coulomb . second2'
20
[CHAP.I
T H E FIELD EQUATIONS
The power expended per unit volume by a current of density J is therefore E J watts/meter3. It will be noted furthermore that the product of charge and electric field intensity E has the dimensions of force. Let a charge of 1 coulomb be placed in an electric field whose intensity is 1 volt/meter.

(10)
joule kilogram * meter volt 1 coulomb X 1  1 1 meter meter second2
The unit of force in the m.k.s. system is called the newton, and is equivalent to 1 joulc/meter, or lo5 dynes. The flux of the vector B shall be measured in webers,
@=JB.Ilda
(11)
webers,
and the intensity of the field B, or flux density, may therefore be expressed in webers per square meter. According to ( 2 5 ) , page 8, L E  d s = d@ dt
The line integral
webers s e d '
f E  ds is measured in volts and is usually called the
electromotive force (abbreviated e.m.f.) between t,he points a and b; although its value in a nonstationary field depends on the path of integration. The induced e.m.f. around any closed contour C is, therefore, equal to the rate of decrease of flux threading that contour, so that between the units there exists the relation
1 volt
=
weber 1 1 second
or
1 weber = 1
joule ampere
=
1
kilogram . meter2 . coulomb . second
It is important to note that the product of current and magnetic flux is an energy. Note also that the product of B and a velocity is measured in volts per meter, and is therefore a quantity of the same kind as E. 1 unit of B
=
weber 1A = meter2
kilogram coulomb . second'
The units which have been deduced thus far constitute an absolute system in the sense that each has been expressed in terms of the four
21
THE M.K.S. OR GIORGI S Y S T E M
SEC.1.81
basic quantities, mass, length, time, and charge. That this system is identical with the practical series may be verified by the substitutions
(17)
1 kilogram =
lo3 grams,
lo2 centimeters, 1 coulomb = & abcoulomb.
1 meter
=
The numerical factors which now appear in each relation are observed t o be those that relate the practical units to the absolute electromagnetic units. For example, from (6), (18)
1 ohm
=
1 kilogram. meter2 coulomb2 . second
lo3 grams . lo4 centimeters2 abcoulomb2 ‘ seconds = lo9 abohms;
and again from (8), (19)
1 volt = 1
kilogram. meter2  lo3 grams . l o 4 centimeters2 10I abcoulomb . second2 coulomb . second2 = lo8 abvolts.
The series must be completed by a determination of the units and 1 dimensions of the vectors D and H. Since D = EE,H =  B , it is P
necessary and sufficient that € 0 and PObe determined such as to satisfy Eq. (2) and such that the proper ratio of practical t o absoiute units be maintained. We shall represent mass, length, time, and charge by the letters M , L , T, and &, respectively, and employ t,he customary symbol [ A ] as meaning “the dimensions of A.” Then from Eq. (31), page 9,
JD.nda
(20)
=
q
coulombs
and, hence,
[Dl
=
Q coulombs _ L2’ meter2 coulombs  Q2T2 .
The farad, a derived unit of capacity, is defined as the capacity of ft conducting body whose potential will be raised 1 volt by a charge of 1 coulomb. It is equal, in other words, to 1 coulomb/volt. The parameter € 0 in the m.k.s. system has dimensions, and may be measured in farads per meter.
f
H ds taken along a specified path is commonly called the magnetomotive force By analogy with the electrical case, the line integral
22
THE FIELD EQUATIONS
(abbreviated m.m.f.).
In a stationary magnetic field
LHds=I
(23)
[CHAP.I
amperes,
where I is the current determined by the flow of charge through any surface spanning the closed contour C. If t h field is variable, I must include the displacement current as in (28), page 9. According to (23) a magnetomotive force has the dimensions of current. I n practice, however, the current is frequently carried by the turns of a coil or winding which is linked by the contour C. If there are n such turns carrying a current I , the total current threading C is nI ampereturns and it is customary to express magnetomotive force in these terms, although dimensionally n is a numeric. (24)
[m.m.f.]
=
ampereturns,
whence =
.
ampereturns  Q meter LT
It will be observed that the dimensions of D and those of H divided by a velocity are identical. For the parameter po we find

ML volt second  . As in the case of c,, it is convenient to express po in terms of a derived unit, in this case the henry, defined as 1 voltsecond/amp. (The henry is that inductance in which an induced e.m.f. of 1 volt is generated when the inducing current is varying a t the rate of 1amp./sec.) The parameter po may) therefore, be measured in henrys per meter. From (22) and (26) it follows now that
and hence that our system is indeed dimensionally consistent with Eq. (2). Since it is known that in the rationalized, absolute c.g.s. electromagnetic system pa is equal in magnitude to 47r, Eq. (26) fixes also its magnitude in the m.k.s. system. (28)
po =
4~
gram . centimeters = 47r abcoulombs2
kilogram lop2meter lo2 coulombs2
or (29)
=
4a
x
107
kilogram . meters coulombs2
=
1.257 X
henry meter
23
VECTOR A N D SCALAR POTENTIALS
SEC.1.91
The appropriate value of c=
(2)
€0
is then determined from
dG0
=
2.998
x
lO8meters second
to be (30)
=
8.854 X
coulomb2 . seconds2 = 8.854 X kilogram meter3
lo''.
farad meter
It is frequently convenient to know the reciprocal values of these factors>

(31)
PO
 _ 0.1129 x 1012 meters Ae0 farad '
meters henry '
 0.7958 X lo6
and the quantities
dg
(32)
=
376.6 ohms,
=
2.655 X
mho,
recur constantly throughout the investigation of wave propagation. In Appendix I there will be found a summary of the units and dimensions of electromagnetic quantities in terms of mass, length, time, and charge. THE ELECTROMAGNETIC POTENTIALS
1.9. Vector and Scalar Potentials.The analysis of an electromagnetic field is often facilitated by the use of auxiliary functions known as potentials. A t every ordinary point of space, the field vectors satisfy the system
+ f3B = 0,
(111) V * B = 0,
f3D (11) V X H   = J,
(IV) v * D= p.
(I) V X E
at
According to (111) the field of the vector B is always solenoidal. quently B can be represented as the curl of another vector Ao. B
(1)
=
Conse
V x Ao.
However A. is not uniquely defined by (1); for B is equal also to the curl of some vector A,
B=VXA,
(2) where
(3) and
A
= Ao
 V#J,
9 is any arbitrary scalar function of position.
24
T H E FIELD EQUATIONS
[CHAP.1
If now B is replaced in (I) by either (1) or (2), we obtain, respectively, (4) Thus the fields of the vectors E
dA are irrotational and + aAo and E + dt
equal to the gradients of two scalar functions 40and 4.
The functions #I and
+o
are obviously related by
4
(7)
=
a+ + Z'
$0
The functions A are vector potentials of the field, and the 4 are scalar potentials. A0 and 40 designate one specific pair of potentials from which the field can be derived through (1) and (5). An infinite number of potentials leading to the same field can then be constructed from (3) and (7). Let us suppose that the medium is homogeneous and isotropic, and that e and p are independent of field intensity.
D = eE,
(8)
B
= pH.
I n terms of the potentials (9)
which upon substitution into (11) and (IV) give (10)
V
x
V X A
+~
a4
+
d2A
V d t ' L E= XPJ,
dA
V24+V'dt=
1
p.
All particular solutions of (10) and (11) lead to the same electromagnetic field when subjected to identical boundary conditions. They differ among themselves by the arbitrary function +. Let us impose now upon A and 4 the supplementary condition
V*A+~ (12) To do this it is only necessary that
Ea4 =O. at
+ shall satisfy
SEC.
25
VECTOR A N D SCALAR POTENTIALS
1.91
where +,, and A. are particular solutions of (10) and (1 1). The potentials $ and A are now uniquely defined and are solutions of the equations (14)
V x V x A  VVA
d2A +/J E = pJ, T
Equation (14) reduces to the same form as (15) when use is made of the vector identity (16)
V X
V x A=VVAV*VA.
The last term of (16) can be interpreted as the Laplacian operating on the rectangular components of A. I n this case
The expansion of the operator V VA in curvilinear systems will be discussed in Sec. 1.16, page 50. ? The relations (2) and (6) for the vectors B and E are by no means general, To them may be added any particular solution of the homogeneous equations (Ia) V X E
+ aB
(IIa) V x H 
aD at

1
0,
(IIIu) V * B
=
0,
=
0,
(IVU)V  D
=
0.
From the symmetry of this system it is at once evident that it can be satisfied identically by (1%
D
=
V X A*,
dA *
H = V+*   7
at
from which we construct 1 E=vxA*,
B = 
The new potentials are subject only to the conditions
26
T H E FIELD EQUATIONS
[CHAP.
1
A general solution of the inhomogeneous system (I) to ( I V ) is, therefore, B
=
aA V X A  p __  pV+', 2%
at
provided 1.1 and e are constant. The functions +* and A* are potentials of a source distribution which is entirely external to the region considered. Usually +* and A* are put equal to zero and the potentials of all charges, both distant and local, are represented by 4 and A. At any point where the charge and current densities are zero a possible field is 40 = 0, A, = 0. The function I) is now any solution of the homogeneous equation @I)  pr 2 = 0. at
V'I)
+
Since at the same point the scalar potential satisfies the same equation, $ may be chosen such that vanishes. I n this case the field can be expressed in terms of a vector potential alone.
+
B=VxA,
(24)
aA E = 
at '
V*A=O.
(25)
Concerning the units and dimensions of these new quantities, we note first that E is measured in volts/meter and that the scalar potential 4 is therefore to be measured in volts. I f q is a charge measured in coulombs, it follows that the product q+ represents an energy expressed in joules. From the relation B = v x A it is clear that the vector potential A may be expressed in webers/meter, but equally well in either voltseconds/meter or in joules/ampere. The product of current and vector potential is therefore an energy. The dimensions of A* are found t o be coulombs/meter, while +* will be measured in ampereturns. 1.10. Homogeneous Conducting Media.In view of the extreme brevity of the relaxation time it may be assumed that the density of free charge is always zero in the interior of a conductor. The field equations for a homogeneous, isotropic medium then reduce to
(Ib) V X E
+ aB at
=
0,
aD (IIb) V x H    UE = 0, at
( I I I b ) V * B = 0, ( I V b ) V D = 0.
SEC. 1.101
HOMOGENEOUS CONDUCTING M E D I A
27
We are now free to express either B or D in terms of a vector potential. In the first alternative we have
aA
B=VXA,
(26)
E=V+*
at
If the vector and scalar potentials are subjected to the relation
a possible electromagnetic field may be constructed from any pair of solutions of the equations
a2A aA V2A  p~ 7 p~  = 0 , at
at
a2(b a4 = 0. v2+  p c  puat2 at
As in the preceding paragraph one will note that the field vectors are invariant to changes in the potentials satisfying the relations
4
(30)
=
40
+ a+Z J
A
=
Ao  v$,
+
where $ 0 , A0 are the potentials of a possible field and is an arbitrary scalar function. In order that A and (b satisfy (27) it is only necessary that be subjected to the additional condition
+
To a particular solution of (31) one is free to add any solution of the homogeneous equation
+
Frequently it is convenient to choose such that the scalar potential vanishes. The field within the conductor is then determined by a single vector A. aA B=VxA, E=, (33) at a2A aA V2Ap€Tp~X=0, V*A=0. (34) at The field may also be defined in terms of potentials
(35)
I> = v
x
A*,
H
=
v4*
+* and A * by
 aA*  !?A*. at
28
T H E FZELD EQUATIONS
[CHAP.I
If 4 * and A * are to satisfy (28) and (29)' it is necessary that they be related by
The field defined by (35) is invariant to all transformations of the potentials of the type (37)
+* = 4;
a+* c + at + ;+*,
A*
=
At

V+*,
where as above 4: and A t are the potentials of any possible electromagnetic field. To ensure the relation (36) it is only nccessary that +* be chosen such as to satisfy
Finally, by a proper choice of to vanish.
+* the scalar
V X A*,
potential 4* may be made
a A*,
1.11. The Hertz Vectors, or Polarization Potentials.We have seen that the integration of Maxwell's equations may be reduced t o the determination of a vector and a scalar potential, which in homogeneous media satisfy one and the same differential equation. It was shown by Hertz' that it is possible under ordinary conditions to define an electromagnetic field in terms of a single vector function. Let 11s confine ourselves for the present to regions of an isotropic, homogeneous medium within which there are neither conduction currents nor free charges. The field equations then reduce to the homogencous system (1a)(IVa). We assume, for reasons which will become apparent, that the vector potential A is proportional to the time derivative of a vector n.
HERTZ,Ann. Physik, 36, 1, 1888. The general solution is due to Righi: Bologna Mem., (5) 0, 1, 1901, and Zl Nuovo Cimento, (5) 2, 2, 1901.
SEC.1.111 THE HERTZ VECTORS, OR POLARIZATION POTENTIALS
29
and, when in turn this expression for E is introduced into ( I I a ) , it is found that
(43)
5at (v
x v xn
+ vc#J+ p €
We recall that at points where there is no charge, the scalar function 4 is wholly arbitrary so long as it satisfies an equation such as ( 2 3 ) . In the present instance it will be chosen such that
(44)
=
vn.
Then upon integrating (43)with respect to the time, we obtain
(45)
 + pe
V x V X II  V V Tz
a2n
__ =
at2
constant.
The particular value of the constant does not affect the determination of the field and we are therefore free to place it equal to zero. Equation (IVa)is also satisfied, for the divergence of the curl of any vector vanishes identically. Then we may state that every solution of the vector equation
(46)
v x v x n  v v . n + p € =a m
=
0
determines an electromagnetic jield through (47) The condition that 4 shall satisfy (23) is fulfilled in virtue of (46). One may replace (46) by
provided V 2is understood to operate on the rectangular components of m. Since the vector D as well as B is solenoidal in a chargefree region, an alternative solution can be constructed of the form
* A* = p c a ,n at where n*is any solution of (46)or (48). From these results we conclude that the electromagnetic field within a region throughout which e and p are constant, p and J equal to zero, may be resolved into two partial fields, the one derived from the vector
30
[CHAP.I
THE FIELD EQUATIONS
and the other from the vector n*. The origin of these fields lies exterior to the region. To determine the physical significance of the Hertz vectors it is now necessary to relate them to their sources; in other words, we must find the inhomogeneous equations from which (48) is derived. Let us express the vector D in terms of E and the electric polarization P. According to (6), page 11, D = QE P. Then in place of (IIa) and (IVa), we must now write
+
V
X
H  €0aE at
aP
=
at’
1
V * E = VP. €0
It may be verified without difficulty that these two equations, as well as (Ia) and (IIIa), are still identically satisfied by (47), provided only that E be replaced by c0 and that ][I be now any solution of
T h e source of the vector n and the electromagnetic jield derived f r o m it i s a distribution of electric polarization P. In due course we shall interpret the vector P as the electric dipole moment per unit volume of the medium. Since n is associated with a distribution of electric dipoles, the partial field which it defines is sometimes said to be of electric type, and n itself may be called the electric polarization potential. In like manner it can be shown that the field associated with ][I* is set up by a distribution of magnetic polarization. According to (6), page 11, the vector B is related to H by B = po(H M), which when introduced into (Ia) and (IIIa) gives
+
Then these equations, as well as (IIa) and (IVa), are satisfied identically by (50) if we replace there p by PO and prescribe that n* shall be a solution of
(54) We shall show later that the polarization M may be interpreted as the density of a distribution of magnetic moment. The partial field derived from n*may be imagined to have its origin in magnetic dipoles and is said to be a field of magnetic type. The electric polarization P may be induced in the dielectric by the field E, but it may also contain a part whose magnitude iscontrolled by wholly external factors. In the practical application of the theory one is interested usually only in this independent part Po,which will be
Snc. 1.111 THE HERTZ VECTORS, OR POLARIZATION POTENTIALS
31
shown to represent the electric moment of dipole oscillators activated by external power sources. The same is true for the magnetic polarization. To represent these conditions we shall write ( 6 ) , page 11, in the modified form D = EE Po, H = 1 B  Mo, (55)
+
P
in which Po and Mo are prescribed and independent of E and H,and where the induced polarizations of the medium have again been absorbed into the parameters e and p. Then the electromagnetic field due to these distributions of Poand Mo is determined by
when n and
n*are solutions of 1
a2n
V 2 n  p~  = Po,
(58)
at2
a m*
V211*  p~ at2 = Mo.
In virtue of the second of Eqs. (58) and of the identity (16) we may aIso write (57) as H=eV X
(59)
an + V dt
X
V X III*Mo.
Since B = V x A, it is evident from this last relation that the vector potential A may be derived from the Hertzian vectors by putting
A
(60) where
with
an + pV
= pe
at
X 11*
+ is an arbitrary scalar function.
 V+,
The associated potential
+ is
+ subject only to the condition that it satisfy
The extension of these equations to a homogeneous conducting medium follows without difficulty. The reader will verify by direct substitution that the system (1b)(IVb), in a medium which is free Gf
32
[CHAP.I
THE FIELD EQUATIONS
fixed polarization Po and MO, is satisfied by
(63)
an * at ’ + v x v x n*,
E = V X V XIIpV X H
=
vx
( c % + u ~ )

1.12. Complex Field Vectors and Potentials.It has been shown by Silberstein, Bateman, and others that the equations satisfied by the fields and potentials may be reduced to a particularly compact form by the construction of a complex vector whose real and imaginary parts are formed from the vectors defining the magnetic and electric fields.’ The procedure has no apparent physical significance but frequently facilitates analysis. Consider again a homogeneous, isotropic medium in which D = EE, B = pH. If now we define Q as a complex field vector by (66)
Q=B+~&E,
the Maxwell equations (I)(IV) reduce to (67)
V X Q + i G dQ X
= pJ,
V.Q
=i$p.
The vector operation V X Q may be eliminated from (67) by the simple expedient of taking the curl of both members. By the identity (16) we obtain
which, on replacing the curl and divergence of Q by their values from (67), reduces to
When this last equation is resolved into its real and imaginary com‘SILBERSTEIN, Ann. phys., 22, 24, 1907. Also Phil. Mag. (6) 23, 790, 1912. BATEMAN, “Electrical and Optical Wave Motion,” Chap. I, Cambridge University Press.
SEC.1.121
COMPLEX FIELD VECTORS A N D POTENTIALS
33
ponents, one obtains the equations satisfied individually by the vectors E and H. 82H
V'HepS
(70)
=
V x J,
Next, let us define Q in terms of complex vector and scalar potentials L and CP by the equation (72)
Q
=
vxL
subject to the condition

aL at
v L + r p a+ at
=
*
(73)
 iz/GV+,
idep
0.
It will be verified without difficulty that (72) is an integral of (67) provided the complex potentials satisfy the equations
PL
(74)
V2L  ' 1 . = 1 pJ, 7 dt
(75)
v2CP €
8% p i = at
1

p.
E
If the real and imaginary parts of these potentials are written in the form
and substituted into (72), one finds again after separation of reals and imaginaries the general expressions for the field vectors deduced in Eqs. (21) and (22). If the free currents and charges are everywhere zero in the region under consideration, Eq. (67) reduces to
(77)
V x Q
+i
d ~aQp = x 0,
V  Q = 0.
The electromagnetic field may now be expressed in terms of a single complex Hertzian vector r.
(78)
Q
=
~ E XV
a r fifiev
X V X
where r is any solution of
(79)
v2r
a2r
 Ep
at2
=
0.
r,
34
T H E FIELD EQUATIONS
If, finally,
[CHAP.
1
r is defined as
r = n  i$n*
(80)
and substituted into (781, one finds again after separation into real and imaginary parts exactly the expressions (47) and (50) for the electric and magnetic field vectors. When the mcdium is conducting, the field equations are no longer symmetrical and the method fails. The difficulty may be overcome if the field varies harmonically. The time then enters explicitly as a factor such as ekiwt. After differentiating with respect to time, the system (16)(IVb) may be made symmetrical by introducing a complex
inductive capacity
E’
=
u
E
C i. w
BOUNDARY CONDITIONS
1.13. Discontinuities in the Field Vectors.The
validity of the field equations has been postulated only for ordinary points of space; that is t o say, for points in whose neighborhood the physical properties of the medium vary continuously. However, across any surface which bounds one body or medium from another there occur sharp changes in the parameters E , p , and u. On a macroscopic scale these changes may usually be considered discontinuous and hence the field vectors themselves may be expected to exhibit corresponding discontinuities. Let us imagine at the start that the surface S which bounds medium (1) from medium ( 2 ) has been replaced by a very thin transition layer within which the parameters e, p, u vary rapidly but continuously from their values near S in (1) to their values near S in (2). Within this layer, as within the media (1) and ( Z ) , the field vectors and their first derivatives are continuous, bounded functions of position and time. Through the layer we now draw a small right cylinder, as indicated in Fig. 2a. The elcments of the cylinder are normal to S and its ends lie in the surfaces of the layer so that they are separated by just the layer thickness Al. Fixing our attention first on the field of the vector B, we have (1)
$Bnda=O,
when integrated over the walls and ends of the cylinder. If the base, whose area is Aa, is made sufficiently small, i t may be assumed that B has a constant value over each end. Neglecting differentials of higher order we may approximate (1) by (2)
 + B  n2)Aa + contributions of the walls
(B nl
=
0.
SEC.1.131
D I S C O N T I N U I T I E S I N T H E FIELD VECTORS
35
The contribution of the walls to the surface integral is directly proportional t o AZ. Now let the transition layer shrink into the surface S. I n the limit, as A1 to, the cnds of the cylinder lie just on either side of S and the contribution from the walls becomes vanishingly small. The value of B a t a point on X in medium (1) will be denoted by B1,while
FIG.2a.For
the normal boundary condition.
the corresponding value of B just across the surface in (2) will be denoted b y Bz. We shall also indicate the positive normal t o S by a unit vector n drawn from (1) into (2). According t o this convention medium (1) lies on the negative side of S , medium (2) on the positive side, and nl = n. Then as A1 + 0, Aa 3 0,

(Bz B,) n
(3)
= 0;
the transition of the normal component of B across a n y surface of discontinuity in the medium i s continuous. Equation (3) is a direct consequence of the condition V B = 0, and is sometimes called the surface divergence.
S (1) El>,u,.O1 FIG.2b.For tho tangential boundary condition.
The vector D may be treated in the same manner, but in this case the surface integral of the normal component over a closed surface is equal to the total charge contained within it. (4)
$ D  n da = (I.
The charge is distributed throughout the transition layer with a density p. As the ends of the cylinder shrink together, the total charge q remains constant, for i t c.annot be destroyed, and
(5)
q
= p A1 Aa.
I n the limit as A1 + 0, the volume density p becomes infinite. It. is then convenient to replace the product p A1 by a surface density W, defined as
36
T H E FIELD EQUATIONS
[CHAP.
I
the charge per unit area. The transition of the normal component of the vector D across any surface X is now given by

(Dz  D J n = W . (6) The presence of a layer of charge on S results in a n abrupt change i n the normal component of D, the amount of the discontinuity being equal to the surface density measured in coulombs per square meter. Turning now to the behavior of the tangential components we replace the cylinder of Fig. 2a b y a rectangular path drawn as in Fig. 2b. The sides of the rectangle of length As lie in either face of the transition layer and the ends which penetrate the layer are equal in length to its thickness Al. This rectangle constitutes a contour COabout which
(7) where So is the area of the rectangle and no its positive normal. The direction of this positive normal is determined, as in Fig. 1, page 8, b y the direction of circulation about CO. Let T I and  2 be unit vectors in the direction of circulation along the lower and upper sides of the rectangle as shown. Neglecting differentials of higher order, one rnay approximate (7) by (8)
(E
 +Er1
T,)
As
+ contributions from ends = aB  noAs Al. at
As the layer contracts to the surface S, the contributions from th e segments a t the ends, which are proportional to All become vanishingly small. If n is again the positive normal to S drawn from (1) into (2), we may define the unit tangent vector T by (9)
T
=
no x n.
Since (10)
no x n  E = n o . n x E ,
we have in the limit as A1 f 0, A s + 0,
The orientation of the rectangle  and hence also of no  is entirely arbitrary, from which i t follows th at the bracket in (11) must equal zero, or
(12)
n x (Ez El)
=
 lim AZ+0
dB 
at
AZ.
SEC.1.131
DISCONTINTJITIES I N THE FIELD VECTORS
37
The field vectors and their derivatives have been assumed to be bounded ; consequently the righthand side of (12) vanishes with A l . (13)
n X (Ez  El)
= 0.
T h e transition of the tangential components of the vector E through a surface of discontinuity i s continuous.
The behavior of H a t the boundary may be deduced immediately from (12) and the field equation
We have
n X (Hz  H1)
=
lim AZ+O
The first term on the right of (15) vanishes as A1 + 0 because D and its derivatives are bounded. If the current density J is finite, the second term vanishes as well. It may happen, however, that the current I = J noA s A1 through the rectangle is squeezed into a n infinitesimal layer on the surface S as the sides are brought together. It is convenient to represent this surface currant by a surface density K defined as the limit of the product J A 1 as A1 + 0 and J + a. Then (16)
n x (Hz  H1) = K.
When the conductivities of the contiguous media are finite, there can be no surface current, for E is bounded and hence the product aE A1 vanishes with Al. I n this case, which is the usual one, (17)
n x (Hz  HI) = 0,
(finite conductivity).
Not infrequently, however, it is necessary to assume the conductivity of a body to be infinite in order to simplify the analysis of its field. One must then apply (16) as a boundary condition rat>herthan (17). Summarizing, we are now able t o supplement the field equations by four relations which determine the transition of a n electromagnetic field from one medium to another separated by a surface of discontinuity.
(18)
n . (Bz B1) = 0, n x (El  El) = 0,
n x (Hz  HI) = K, n (D2  D1) = W .

From them follow immediately the conditions for the transition of the normal components of E and H.
38
T H E FIELD EQUATIONS
[CHAP.
1
Likewise the tangential components of D and B must satisfy
COORDINATE SYSTEMS
1.14. Unitary and Reciprocal Vectors.It is one of the principal advantages of vector calculus that the equations defining properties common to all electromagnetic fields may be formulated without reference to any particular system of coordinates. To determine the peculiarities that distinguish a given field from all other possible fields, it becomes necessary, unfortunately, to resolve each vector equation into an equivalent scalar system in appropriate coordinates. In a given region let (l)
y, z>,
u1 =
u2 = f2(2~
y,
u3 = f3(Z1
Yt z>,
be three independent, continuous, singlevalued functions of the rectangular coordinates x. y, z. These equations may be solved with respect to x, y, z, and give (2) x
=
Cpl(U1,
u2, u3),
y =
CpZ(U1,
u2, u3),
z = cp3(u1, u2, u3),
three functions which are also independent and continuous, and which are singlevalued within certain limits. In general the functions cpI as well as the functions fi are continuously differentiable, but at certain singular points this property may fail and due care must be exercised in the application of general formulas. y, z) in the region there is associated by means of With each point P(x, (1) a triplet of values ul, u2,u3;inversely (within limits depending on the boundaries of the region) thcre corresponds to each triplet ul, u2, u3 a definite point. The functions ul, u2,u3 are called general or curvilinear coordinates. Through each point P there pass three surfaces
(3)
u1 =
constant,
u2 = constant,
u3 = constant,
called the coordinate surfaces. On each coordinate surface one coordinate is constant and t w o are variablc. A surface will be designated by the coordinate which is constant. Two surfaces intersect in a curve, called a coordinate curve, along which two coordinates are constant and one is variable. A coordinate curve will be designated by the variable coordinate. Let r denote the vector from an arbitrary origin to a variable point P(z, y, 2). The point, and consequently also its position vector r, may be considered functions of the curvilinear coordinates u ] , u2,u3.
(4)
r = r(ul, u2,u3).
Next Page
SEC.1.141
39
U N I T A R Y A N D RECIPROCAL VECTORS
A differential change in r due to small displacements along the coordinate curves is expressed b y (5)
dr
dr
=  du'
aul
+ au,dr
du2
dr + au3 du3.
~
Now if one moves unit distance along the ulcurve, the change in r is directed tangentially to this curve and is equal to dr/dul.
The vectors
are known as the unitary vectors associated with the point P. constitute a base system of reference for all other vectors associated with that particular point.
+
They
+
(7) dr = al du' a2du2 a3du3. It must be carefully noted that the unitary vectors are not necessarily of unit length, and their dimensions will depend on the nature of the general coordinates. The three base vectors al, a,, a3 define a parallelepiped whose volume is
(8) V = al * (a,
x a3) = a2* (aa x al) = a3 (al x ad.

FIG.
S.Base vectors for a curvilinear coordinate system.
The three vectors of a new triplet defined by
are respectively perpendicular to the plancs determined by the pairs (az, a3), (a3, al), (al, a2). Upon forming all possible scalar products of the form a i . ai, i t is easy to see that they satisfy the condition (10)
ai. a1. =
6ii,
where 6ii is a commonly used symbol denoting unity when i = j , and zero when i # j . The unitary vectors can be expressed in terms of the system a', a2, a3by relations identical in form.
Any two sets of noneoplanar vectors related b y the Eqs. (8) to (11) are said to constitute reciprocal systems. The triplets al, a2, a3 are called reciprocal unitary vectors and they may serve as a base system quite as well as the unitary vectors themselves.
Previous Page 40
[CHAP.I
T H E FIELD EQUATIONS
If the reciprocal unitary vectors are employed as a base system, the differential dr will be written dr = a' d u l
(12)
+ a2duz + a3du3.
The differentials dul, duz, du3 are evidently components of dr in the directions defined by the new base vectors. The quantities UI, uz, u3 are functions of the coordinates ul, u2, u3,but the differentials d u l , duz, du3 are not necessarily perfect. On th e contrary they are related to the differentials of the coordinates by a set of linear equations which in general are nonintegrable. Thus equating (7) and (12), we have 3
3
dr
(13)
aidui =
= i=l
2 aidui. j=1
Upon scalar multiplication of (13) b y ai and by ai in turn, we find, thanks to (10) : 3
(14)
dui
3

ai a; dui,
=
dui
i=l

ai ai dui.
= j=1
It is customary to represent the scalar products of the unitary vectors and those of the reciprocal unitary vectors by the symbols
The components of dr in the unitary and in the reciprocal base systems are then related by
A fixed vector F a t the point P may be resolved into components either with respect to the base syst,em all azl a3, or with respect t o th e reciprocal system all a2, a3. 3
F
=
xfiai i=l
3
=
2jiai. j=1
The components of F in the unitary system are evidently related t o those in its reciprocal system by 3
(19)
jl
=
2 giifi,
i=l
c 3
fi = j=1 Siifi,
and in virtue of the orthogonality of the base vectors aj with respect to
SEC.1.141
41
U N I T A R Y A N D RECIPROCAL VECTORS
the reciprocal set ai as expressed by (lo), we may also write
fi = F . ai,
(20)
F . a+
fi =
It follows from this that (18) is equivalent to
The quantities? are said t o he the contravariant components of the vector F, while the components fi are called covariant. A small letter has been used to designate these components to avoid confusion with the components Fl, F2, F3 of F with respect t o a base syst,em coinciding with the ai but of unit length. It has been noted previously that the length and dimensions of the unitary vectors dcpend on the nature of the curvilinear coordinates. An appropriate set of unit vectors which, like the unitary set ai, are tangent to the uicurves, is defined by (22)
il
1
a1 = ____  G
l
i2 =
filal'
1
 ~
a2,
6;
1
13
= __
1/933
a3,
and, hence,
F
(23)
= Flil
+ Fzi2+ Fa&,
with (24)
Fi
=
6 f i .
The F i are of the same dimensions as the vector F itself. The vector dr represents a n infinitesimal displacement from the point P(u', u2, u3) to a neighboring point whose coordinates are u' du', u2 du2, u3 du3. The magnitude of this displacement, which constitutes a line element, we shall denote by ds. Then
+
+
+
or, in the notation of (15) and (16), 3
3
The gij and g i i appear here as coefficients of two differential quadratic forms expressing the length of a line element in the space of the general
42
T H E FIELD EQUATIONS
[CHAP.
1
coordinates ui or of its reciprocal set ui. They are commonly called the metrical coeficients. It is now a relatively simple matter to obtain expressions for elements of arc, surface, and volume in a system of curvilinear coordinates. Let dsl be an infinitesimal displacement a t P(u', u2, u3)along the ulcurve. dsl = al dul,
(27)
dsl = (dsll =
du1.
Similarly, for rlcmcnts of length along the u2 and uhcurves, we have
dsz
(28)
=
6du2,
ds3 =
dz
du3.
Consider next a n infinitesimal parallelogram in the ulsurface bounded b y intersecting u2 and u3curves as indicated in Fig. 4. The area of such an element is equal in magnitude to
(29) dual
= Ids2 X d~31 = laz X a3/du2du3

= d ( a 2x as) (az x as) du2 du3.
By a wellknown vector identity

=
FIG.4.Elernent
(31)
of area in the u*surface.


(a x b) (c x d) (a c)(b d)  ( a . d)(b c),
(30)


where a, b, c, d are any four vectors, and hcnce



(az X a,) (a2 X as) = (az az) (as .a3)  (a2 a3)(a3 az) =
922933
 9223.
For the area of an clement in the u'surface we have, therefore, dai = d g z z g s  g&
(32)
d U 2 du3,
and similarly for elements in thc u2 and u3surfaces, daz da3
(33)
d U 3 dul, = d g i i g 2 z  giz du' dU2. =
dq33g11'
g';
Finally, a volume element bounded by coordinate surfaces is written as (34)
dv
=
If now in (21) we let F (35) az x a3
=


dsl dsz X ds3 = al a2 X a3du' du2 du3. =
a2 x a3, we have
(a' az x a3)al
+ (a2 a2 x a3)az+ (a3 az x a$)a3,
SEC.1.141
U N I T A R Y A N D RECIPROCAL VECTORS
43
or, on replacing the ai by their values from (8) and (9)) (36)

a1 a2 X a3


a1 [(az x a3 az x a3)al ale az x a3 (a3 x a1 az x a3)az (al x az az x a3>a3].
= _____

+
+
The quantities within parentheses can be expanded by (30) and the terms arranged in the form (37)





.
(a1 a2 x a3)z= a1 al[(a2 az>(a3a3)  (az a3)(a3 aZ)l a1 az[(az a3) (a3 * al)  (az al)(a3 ad] a1 * ad(a2 ad (a3 az)  (az az)(aa all].
+ 



+



.
If finally the scalar products in ( 3 7 ) are replaced by their gii, we obtain as an expression for a volume element
dv
(38)
=
~du1du2du3,
in which (39)
A corresponding set of expressions for the elements of arc, area, and volume in the reciprocal base system may be obtained by replacing the g.i by the gii, but they will not be needed in what follows. Clearly the coefficients g l i are sufficient t o characterize completely the geometrical propcrties of space with respect to any curvilinear system of coordinates; i t is therefore essential that we know how these coefficients may be determined. To unify our notation we shall represent the rectangular coordinates z, y, z of a point P by the letters xl, xz,x 3 respectively. Then (40)
dsz
=
+
( d ~ l ) ~(dx')'
+( d ~ ~ ) ~ ,
I n this most elementary of all systems the metrical coefficients are (41)
gil
= 6tjl
(&
=
1,
6,j
=
0 when i # j).
From the orthogonality of the coordinate planes and the definition (9)) it is evident that the unitary and the reciprocal unitary vcctors are identical, are of unit length, and are the base vectors customarily represented by the letters i, j, k. Suppose now that the rectangular coordinates are related functionally t o a set of general coordinates as in (2) by the equations (42) z1= z l ( u l ,u2,u31,
zz =
z2(u1,u2,u 3 ) ,
x3
=
x3(u1,u2,u3).
44
T H E FIELD EQUATIONS
[CHAP.I
The differentials of the rectangular coordinates are linear functions of the differentials of the general coordinates, as we see upon differentiating Eqs. (42).
(43)
According to (26) and (40) 3
(44)
ds2 =
3
3
22
gii
d d dui =
2 (dxk)2,
k=l
i = l j=1
whence on squaring the differentials in (43) and equating coefficients of like terms we obtain (45)
1.16. The Differential Operators.The gradient of a scalar function +(ul,u2, u3)is a fixed vector defined in direction and magnitude as the with respect to the cordinates. The maximum rate of change of variation in + incurred during a displacement dr is, therefore,
+
Now the dui are the contravariant components of the displacement vector dr, and hence by (20), (47)

dui = ai dr.
This value for dui introduced into (46) leads to
and, since the displacement dr is arbitrary, we find for the gradient of a scalar function in any system of ciirvilinear coordinates : (49)
In this expression the reciprocal unitary vectors constitute the base
SEC.1.151
45
T H E DIFFERENTIAL OPERATORS
system, but these may be replaced by the unitary vectors through the transformation 3
a t u2, the right at u2
(55)
+ du2.
The area
l
0
J
46
THE FIELD EQUATIONS
[CHAP.
1
The curl of the vector F is found in the same manner b y calculating the line int>cgral of F around an infinitesimal closed path. According to Eq. (21), page 7, the component of the curl in a direction defined by a unit normal n is

(V x F) n
=
lim cio

s
(F a3d
0
4
~

~  )(F ~a3d ~~ ~ +) , ~~;
from th e bottom and top parallel t o u2
a?\,
(57) This quantity must now be divided by the area of the rectangle, or d ( a z x as) (a2 x a3) du2du3. As for the unit normal n, we note th a t the reciprocal vector a', not the unitary vector al, is always normal to the ulsurface. Its magnitude must be unity; hence

These values introduced into (56) now lead to (59)
a1
(v x F)  dG* 
1
By (9) and f37) (60)

a2 x a3 = [al (az x a3>]al= d i a l ;
~
~
47
0RT HOGON A L S YST E M S
SEC.1.161
hence (59) reduces t o
The two remaining components of V X F are obtained from (61) by permutation of indices. Then by (21) 3
vx
F
(v x
= i=l

F a“>;.
Remembering that F * a; is the covariant component fi, we have for the curl of a vector with respect t o a set of general coordinates
Finally, we consider the operation V24, by which we must understand We need only let F = V + in ( 5 5 ) . The contravariant components of the gradient are
v  v+.
3
3
f ” = ~ . a i = C a i . .a84 J==gii~. (64)
dUi
j=1
j=l
dUj
Then
1.16. Orthogonal Systems.Thus far no restriction has been imposed on the base vectors other than that they shall be noncoplanar. Now it, happens that in almost all cases only the orthogonal systems can be usefully applied, and these allow a considerable simplification of the: formulas derived above. Oblique systems might well be of the greatest practical importancc; but they lead, unfortunately, t o partial differential equations which cannot be mastered by presentday analysis. The unitary vectors al, az, a3of a n orthogonal system are by definition mutually perpendicular, whence i t follows that ai is parallel t o ai and is its reciprocal in magnitude.
1 1 a$ = ___ ai =  a(. ai ai gii

(66) Furthermore (67)

al a,
= dz
 a3 = a 3 . al = 0;
48
[CHAP.I
THE FIELD EQUATIONS
hence gij = 0, when i # j. It is customary in this orthogonal case t c introduce the abbreviations
The hi may be calculated from the formula
although their value is usually obvious from the geometry of the system. The elementary cell bounded by coordinate surfaces is now a rectangular box whose edgcs are
(71)
dsl
=
hi du',
dsz
=
hz dU2,
d ~ 3 =
h3 du3,
and whose volume is
(72)
dv
=
hlhZh3 du' dU2du3.
All offdiagonal terms of the determinant for g vanish and hence
(73)
& = hihzh3.
The distinction between the contravariant and covariant components of a vector with respect to a unitary or reciproca1,unitary base system is essential to an understanding of the invariant properties of the differential operators and of scalar and vector products. However, in a fixed reference system this distinction may usually be ignored. It is then convenient to express the vector F in terms of its components, or projections, F1,FZ,P3 on an orthogonal base system of unit vectors il, i2, i3. By (22) and (66) . 1. a.,  h.i. aE =  1 ; . (74) h, %,
In terms of the components Fi the contravariant and covariant components are 1 fi = Fi, fz = h z ; . (75) h, Also
The gradient, divergence, curl, and Laplacian in an orthogonal system of curvilinear coordinates can now be written down directly from the results of the previous section.
SEC.1.161
49
ORTHOGONAL SYSTEMS
From (49) we have for the gradient
According to (55) the divergence of a vector F is
For the curl of F we have by (63)
It may be remarked that (80) is t’he expansion of the determinant
Finally, the Laplacian of an invariant scalar 4 is
By an invariant scalar is meant a quantity such as temperature or energy which is invariant to a rotation of the coordinate system. The components, or measure numbers, Fi of a vector F are scalars, but they transform with a transformation of the base vectors. Now in the analysis of the field we encounter frequently the operation (83)
V X V
X
F
=
V V  F V  V F .

No meaning has been attributed as yet to V VF. In a rectangular, Cartesian system of coordinates xl, x2,z3,it is clear that this operation is equivalent to
L e . , the Laplacian acting on the rectangular components of F.
I n gener
50
T H E FIELD EQUATIONS
[CHAP.I
alized coordinates V x V x F is represented by the determinant
la
(85) V X V X F = s I

The vector V VF may now be obtained by subtraction of (85) from the expansion of V V F, and the result differs from that which follows a direct application of the Laplacian operator to the curvilinear components of F. 1.17. The Field Equations in General Orthogonal Coordinates.In any orthogonal system of curvilinear coordinates characterized by the coefficients hl, hz, h3, the Maxwell equations can be resolved into a set of eight partial differential equations relating the scalar components of the field vectors.

SEC.1.181 PROPERTIES OF SOME ELEMENTARY SYSTEMS
51
It is not feasible to solve this system simultaneously in such a manner as to separate the components of the field vectors and to obtain equations satisfied by each individually. .In any given problem one must make thc most of whatever advantages and peculiaritics the various coordinate systems have to offer. 1.18. Properties of Some Elementary Systems.An orthogonal coordinate system has been shown to be completely characterized by the three metrical coefficients, h l , hz, h3. These parameters will now be determined for certain elementary systems and in a few cases the diffcrential operators set down for convenient 23 reference. 1. Cylindrical Coordinates.Let P' be the projection of a point P ( z , y, z ) on the zplane and r, 0 be the polar coordinates of P' in this plane (Fig. 7 ) . The variables
(87)
x
=
r cos 8,
y
=
r sin 8,
z
=
z.
The coordinate surfaces are coaxial cylinders of circular cross section intersected orthogonally by the planes B = constant and z = constant. The infinitesimal line element is ds2 = dr2
(88)
+ r2de2 + dz2,
whence it is apparent that the metrical coefficients are (89)
If
hl
=
1,
ha = r,
hS = 1.
+ is any scalar and F a vector function we find: VG
=
a+ it 4 189.  1.
a+ i,. +
52
THE FIELD EQUATIONS
2. Spherical Coordinates.The (91)
u1
=
variables u2
l1
[CHAP.1
=
0,
u3
=
41
related to the rectangular coordinates by the transformation (92)
= r sin
x
e cos 4,
y
=
r sin 0 sin 4,
z =
r cos 0,
are called the spherical coordinates of the point P. The coordinate surfaces, r = constant, are concentric spheres intersected by meridian planes, 4 = constant, and a family of cones, 0 = constant. The unit vectors i,, i, i, are drawn in the direction of increasing r, 8, and 4 such as to constitute a righthand base system, as indicated in Fig. 8. The line element is (93) 2
Js2 = dr2
+ r2de2 + r2 sin2 0 d@,
whence for the metrical coefficients we obtain
(94)
hl
=
h2 = r,
1,
h3 = r sin FIG.8.Spherical
coordinates.
These values lead to
a* la*. 1 a+. ~#=i~+i~+dr r a0 r sin e a+ la* V
F
i a
= r 2 ar (r2F1)
8.

i a 1 8FS + r sin e a e (sin OFz) + r sin e a+'
3. Elliptic Coordinates.Let two fixed points PI and PZbe located a t x = c on the xaxis and let rl and r2 be the distances of a variable point P in the xplane from P1 and P2. Then the variables 1
= c and
(96) defined by equations
(97)
u1 =
E,
u2
= 7,
u3
=
2,
SEC.1.181
PROPERTIES OF SOME ELEMENTARY S Y S T E M S
From these relations i t is evident that
are called elliptic coordinates.
5 I 1,
(98)
53
 1 5 9 5 1.
The coordinate surface, 4 = constant, is a cylinder of elliptic cross section, whose foci are P1 and 1’2. The semimajor and semiminor axes of an ellipse E are given by
(99)
a = c&
b
=
c d F i ,
and the eccentricity is
The surfaces, q = constant, represent a family of confocal hyperbolic X
u= T
FIG.g.Coordinates
of the elliptic cylinder. Ambiguity of sign is avoided by placing E = cosh u , 7 = COB a.
cylinders of two sheets as illust,rated in Fig. 9. two confocal systems are
Thc equations of these
from which we deduce the transformation
(102)
z = c[q,
y =c
d(p 1)(1
 +),
2
=
2.
The variable 9 corresponds to the cosine of an angle measured from the xaxis and the unit vectors il, iz of a righthand base system are therefore
54
[CHAP.I
THE FIELD EQUATIONS
drawn as indicated in Fig. 9, with i3 normal to the page and directed from the reader. The metrical coefficients are calculated from (102) and (70)) giving
4. Parabolic Coordinates.If r , 0 are polar coordinates of a variable point in the zplane, one may define two mutually orthogonal families of parabolas by the equations
5
=
4%sin ,2e
The surfaces, t = constant and q
q = =
constant, are intersecting parabolic
I
5=5
4%cos .2e
FIG.10.Parabolic
7/=.5 coordinates.
cylinders whose elements are parallel to the zaxis as shown in Fig. 10. The parameters u' = {,
(105)
u2
= 7,
u3=  2
are called parabolic coordinates. Upon replacing r and 0 in (104) by rectangular coordinates we find (106)
p
= 1/22
+  2) y2
12
= 1/22
+ + y2
2)
SEC.1.181 PROPERTIES OF SOME ELEMENTARY SYSTEMS
55
whence for the transformation from rectangular to parabolic coordinates we have (107)
z=
g.112
 p),
y = 5.11,
2
=
2.
The unit vectors il and i, are directed as shown in Fig. 10, with i, normal to the page and away from the reader. The calculation of the metrical coefficients from (107) and (70) leads to (108)
hl = hz = dp
+
72,
h3
=
1.
5. Bipolar Coordinates.Let, Pl and P z be two fixed points in any zplane with the coordinates (a, 0), (  u , 0) respectively. If 5 is a parameter, the equation
(109)
(X 
u coth [)'
+ y2 = u2 CsCh2 4,
describes two families of circles whose centers lie on the zaxis. These two families are symmetrical with respcct to the yaxis as shown in
I
FIG.11.Bipolar coordinates.
+
Fig. 11. The point P1 at (u, 0) corresponds to 5 = QI , whereas its image P , at (a, 0 ) is approached when 5 =  to. The locus of (log), when $, = 0, coincides with the yaxis. The orthogonal set is likewise a family of circles whose centers all lie on the yaxis and all of which pass through the fixed points P1and Pa. They are defined by the equation,
(110)
5 2
+ (y  a cot
r])2
= a 2 csc2 7,
56
[CHAP.I
THE FIELD EQUATIONS
wherein the parameter q is confined t o the range 0 5 q S ZT. I n order that the coordinates of a point P in a given quadrant shall be singlevalued, each circle of this family is separated into two segments by the points P I and Pz. A value less than ?r is assigned to the arc above the xaxis, while the lower arc is denoted by a value of q equal to T plus the value of q assigned to the upper segment of the same circle. The variables (111)
?J1 = f ,
u2 = q,
u3
= 2,
are called bipolar coordinates. From (109) and (110) the transformation to rectangular coordinates is found t o be (112)
5 =
a sinh [ cod1  cos q ’
=
a sin q cash c;  cos q’
= ”
The unit vectors il and i2 are in the direction of increasing f and q as indicated in Fig. 11, while is is directed away from the reader along the zaxis. The calculation of the metrical coefficients yields
6. Spheroidal Coordinates.The coordinates of the elliptic cylinder were generated by translating a system of confocal ellipses along the zaxis. The spheroidal coordinates are obtained by rotation of the ellipses about an axis of symmetry. Two cases are to be distinguished, according to whether the rotation takes place about the major or about the minor axis. In Fig. 9 the major axes are oriented along the zaxis of a rectangular system. If the figure is rotated about this axis, a set of confocal prolate spheroids is generated whose orthogonal surfaces are hyperboloids of two sheets. If 4 measures the angle of rotation from the yaxis in the xplane and r the perpendicular distance of a point from the xaxis, so that (114)
y
=
r cos 4,
z
=
r sin 4,
then the variables (115)
u1
= f,
u2 = q,
263
=
4,
defined by (97) and (114) are called prolate spheroidal coordinates. In place of (101) we have for the equations of the two confocal systems
SEC.1.181
57
PROPERTIES OF SOME ELEMENTARY S Y S T E M S
from which we deduce
(117)
2 =
c.&,
d r p  1)(1  q z ) cos 4 ,
y =c
z = c d ( ”  1)(1  72) sin
t L 1,
(118)
1STJS1,
4,
OSCpS27r.
A calculation of the metrical coefficients gives (119)
hl
=
c
JF, Jq, d(P hz
2  1
=c
ha
1
 1)(1  7”.
=c
When the ellipses of Fig. 9 are rotated about the yaxis, the spheroids are oblate and the focal points P I , P , describe a circle in the plane y = 0. Let r, 4, y, be cylindrical coordinates about the yaxis,
r eos 6,
z =
(120)
z =
T
sin 4.
If by P1 and P , we now understand the points where the focal ring of radius c intercepts the plane 4 constant, t h e variables 4 and TJ are still defined by (97) ; but for the equations of the coordinate surfaces we have
from which we deduce the transformation from oblatc spheroidal coordinates (122)
u1
t,
u2
y =c
d(t2  1)(1  $),
=
= q,
u3
=
4,
to rectangular coordinates (123)
5 = c.$q
sin 4,
z
= ~ $ cos 7
+.
The surfaces, E = constant, are oblate spheroids, whereas the orthogonal family, TJ = constant, are hyperboloids of one sheet. The metrical coefficients are
The practical utility of spheroidal coordinatrs may be surmised from the fact that as the eccentricity approaches unity the prolate spheroids berome rodshaped, whereas the oblate spheroids degenerate into flat, elliptic disks. I n the limit, as the focal distance 2c and the eccentricity approach zero, the spheroidal coordinates go over into spherical coordinates, with 5 + r , 7 + cos 0. 7. Paraboloidal Coordinates.Another set of rotational coordinates may be obtained b y rotating the parabolas of Fig. 10 about their axis
58
[CHAP.I
THE Y I E L D EQUATIONS
of symmetry.
The variables
are called paraboloidal coordinates. The surfaces, 5 = constant, q = constant, are paraboloids of revolution about an axis of symmetry which in this casc has been taken coincident with the zaxis. The plane, y = 0, is cut by these surfaces along the curves
which are evidently parabolas whose foci are located at the origin and whose parameters are t2and q 2 . The metrical coefficients are
hl
=
h,
=
d m ,
8. Ellipsoidal Coordinates.The 2 2
y2
h3
= 511.
equation 22
+y+=l, a2 b c2
(a
> b > c),
is that of a n ellipsoid whose semiprincipal axes are of length a, b, c.
Then
are the equations respectively of an ellipsoid, a hyperboloid of one sheet, and a hyperboloid of two sheets, all confocal with the ellipsoid (129). Through cach point of space there will pass just one surface of each kind, and to each point there will correspond a unique set of values for f , 7,{. The variables (131)
u' = '$,
uz = 7,
u3
=
(,
are called ellipsoidal coordinates. The surface, 5 = constant, is a hyperboloid of one shect and 9 = constant, a hyperboloid of two sheets. The transformation to rectangular coordinates is obtained b y solving
SEC.1.191
59
ORTHOGONAL T R A N S F O R M A T I O N S
(130) simultaneously for 2, y, z.
This gives a 2 )(17
+ 4 (.r + a?
u”(c2
 a2)
The mutual orthogonality of the three families of surfaces may be verified by calculating the c,oefficients g7. from (132) by means of (45). They are zero when i # j ; for the diagonal terms we find
It is convenient to introduce the abbreviation
46’7U ’ ) ( S + b 2 ) ( s + c2), For the Laplacian of a scalar + we then have (134)
R,
=
(s =
t, 17, r).
THE FIELD TENSORS
1.19. Orthogonal Transformations and Their Invariants.In the theory of relativity one undertakes the formulation of the laws of physics, and in particular the equations of the clectromagnetic field, such t h a t they are invariant t o transformations of the system of referenw. Although in the present volume we shall have no occasion to examine the foundations of the relativity theory, it will nevertheless prove occasionally advantageous t o employ the symmetrical, fourdimensional notation introduced b y Minkowski and Sommerfeld and t o deduce the Lorcntz transformation with respect to which the field equations are invariant. To discover quantities which are invariant to a transformation from one system of general curvilinear coordinates t o another, it is essential t h a t
60
[CHAP.I
THE FIELD EQUATIONS
one distinguish between the covariant and contravariant components of vectors and between unitary and reciprocal unitary base systems. For our present purposes i t will be sufficient, however, t,o confine t h e discussion to systems of rectangular, Cartesiai coordinates in which, as we have seen, covariant and contravariant components are identica1.l Let il, iz,i3be three orthogonal, unit base vectors defining a rectangular coordinate system X whose origin is located a t the fixed point 0, and let r be the position vector of any point P with respect to 0.
(1)
r
=
zlil
+ xziz+ z&,
and since
ii ik
(2)
= 8jk,
the coordinates of P in the system X are

xk = r ik.
(3)
Suppose now that i:, i;, i: are the base vectors of a second rectangular system X' whosc origin coincides with 0 and which, therefore, differs from X only by a rotation of the coordinate axes. Since
r
(4)
=
x;ii
+ xkil + zLih,
the coordinates of P with respect to X' are
(5)
z! = r . i!
=
.
.
xlil . ii. + xziz ii. + z3i3 il;
each coordinate of P in X' is a linear function of its coordinates in X , whereby the coefficients
(6)
ajk
=
il
.
ik
of the linear form are clearly the direction cosines of the coordinate axes of X' with respect t o the axes of X. A rotation of a rectangular coordinate system effects a changc in the coordinates of a point which may be represented by the linear transformation
(7) The coefficients a j k arc subject to certain conditions which are a consequence of the fact that the distance from 0 to P , t h a t is to say, the 1 This section is based essentially on the following papers: MINKOWSKI, Ann. Physik, 47, 927, 1915; SOMMEREELD, Ann. Physik, 32, 749, 1910 and 33, 649, 1910; MIE, Ann. Physik, 37, 511, 1912; PAULI, Relativitatstheorie, in the Encyklopadie der
mathematischen Wissenschaften, Vol. V, part 2 , p. 539, 1920.
ORTHOGONAL T R A N S F O R M A T I O N S
Ssc. 1.191
61
magnitude of r, is independent of the orientation of the coordinate system.
whence it follows that when i = k , when i f k. Equation (10) expresses in fa& the relations which must prevail among the cosines of the angles b c t m c n coordinate axes in order that they be rectangular and which are, therefore, known as conditions of orthogonality. The system (7), when subject to (lo), is likewise called an orthogonal transformation. As a direct consequence of (lo), i t may be shown that the square of the determinant jajkl is equal to unity and hence la,k[ = A1. Any set of coefficients a j k which satisfy (10) define a n orthogonal transformation in the sense that the relation (8) is preserved. Geometrically the transformat ion (7) represents a rotation only when the determinant Iajkl = +l. The orthogonal transformation whose determinant is equal to 1 corresponds t o a n inversion followed by a rotation. Since the determinant of an orthogonal transformation docs not ianish, the x k may be expressed as linear functions of the x:. These relations are obtained most simply by writing as in ( 5 ) :
(11)
xk
=



r ie = xiii ik + zkii ik + xkii ie,
(L = 1, 2, q, whence i t follows from (8) that,
Let A be any fixed vector in space, so that 3
A
(14)
=
2 .qkik
k1
=
3
ALiL.
k=l
The component A; of this vector with respect t o the system X’ is given by
(15)
A!
=
.
A i!
3
3
A k i k .if.
= k=l
2 aikAk;
:
k= 1
62
[CHAP.I
T H E FIELD EQUATIONS
thus the rectangular components of a fixed vector upon rotation of the coordinate system transform like the coordinates of a point. Now while evcry vector has in gencral three scalar components, it does not follow that any three scalar quantities constitute the components of a vector. I n order that three scalars A1, Az, AS m a y be interpreted as the components of a vector, i t i s necessary that they transform like the coordinates of a point. Among scalar quantitics onc must distinguish the variant from the invariant. Quantities such as temperature, pressure, work, and the like are independent of the orientation of the coordinate system and are, thcrefore, called invariant scalars. On the other hand the coordinates of a point, and the measure numbers, or components, of a vector have only magnitude, but they transform with the coordinate system itself. We know that the product A B of two vectors A and B is a scalar, but a scalar of what kind? In virtue of (12) and (13) we have
the scalar product of two vectors i s invariant to a n orthogonal transformation of the coordinate system. Let cp be an invariant, scalar and consider the triplet of quantities
(i = 1) 2) 3). Now by (12), (18) and hence
the Bi transform like the components of a vector and therefore the gradient of 4,
calculated at a point P , i s a jixed vector associated with that point. Let Ai be a rectangular component of a vector A, and (21)
Then by (18) and (15),
SEC.1.191
whence, from
63
ORTHOGONAL TRANSFORMATIONS
(lo), i t follows th at
the divergence of a vector i s invariant to a n orthogonal transformation of the coordinate system. Lastly, since the gradient of an invariant scalar is a vector and since the divergence of a vector is invariant, it folIows that the Laplacian (24)
is invariant to an orthogonal tra.nsformation. The transformation properties of vectors may be extended t o manifolds of more than three dimensions. Let xl, x2,x 3 , 5 4 be the rectangular coordinates of a point P with respect to a reference system X in a fourdimensional continuum. The location of P with respect to a fixed origin 0 is determined by the vector
2 xjij, . . . = sik. 4
(25)
r
=
ii i k
j=1
The linear transformation A
will be called orthogonal if the coefficients satisfy the conditions
c 4
ajzajk = 6a.
j=1
The characteristic property of an orthogonal transformation is that it leaves the sum of the squares of the coordinates invariant:
2 4
2 (xi)Z. 4
=
JZ.(
j= 1
j=1
The square of the determinant formed from the a j k is readily shown to be positive and equal to unity, and hence the determinant, itself may equal 1. However, if (26) is to includc the identical transformation (29)
2'. = 2. 1
31
(j
=
1, 2, 3, 41,
it is obvious that the determinant must be positive. Henceforth we shall confine ourselves to the subgroup of orthogonal transformations
64
T H E FIELD EQUATIONS
[CHAP.I
characterized by (27) and the condition
(30)
lUlkl =
+l.
The transformation then corresponds geometrically to a rotation of the coordinate axes. A fourvector is now defined as any set of four variant scalars Ai (i = 1, 2, 3, 4) which transform with a rotation of the coordinate system like the coordinates of a point. 4
4
It is then easy to show, as above, that the scalar product of two fourvectors and the fourdimensional divergence of a fourvector are invariant to a rotation of the coordinate system. 4
k=l
4
~j=1
(33)
Furthermore the derivatives of a scalar, (34)
(i = 1, 2, 3) 4)
transform like the components of a fourvector and hence the fourdimensional Laplacian of an invariant scalar, (35) is also invariant to an orthogonal transformation. 1.20. Elements of Tensor Analysis.Although most physical quantities may be classified either as scalars, having only magnitude, or as vectors, characterized by magnitude and direction, there are certain entities which cannot be properly represented by either of thesc terms. The displacement of the center of gravity of a metal rod, for example, may be defined by a vector; but the rod may also be stretched along the axis by application of a tension at the two ends without displacing the center a t all. The quantity employed to represent this stretching must thus indicate a double direction. The inadequacy of the vector concept becomes all the more apparent when one attempts the description of a volume deformation, taking into account the lateral contraction of the
SEC.
1.201
ELEMENTS OF TENSOR ANALYSIS
65
rod. I n the present section we shall deal only with the simpler aspects of tensor calculus, which is the appropriate tool for the treatment of such problems. I n a threedimensional continuum let each rectangular component of a vector B be a linear function of the components of a vector A.
In order t ha t this association of the components of B with the components of A in the system X be preserved as the coordinates are rotated, it is necessary that the coefficients T2ktransform in a specific manner. The T j k are therefore variant scalars. A tensoror more properly, a tensor of rank twowill now be defined as a linear transformation of the components of a vector A int,o the components of a vector B which is invariant to rotations of the coordinate system. The nine coefficients Tik of the linear transformation are called the tensor components. To determine the manner in which a tensor component must transform we write first (36) in the abbreviated form
(37) If (37) is to be invariant to the transformation defined by
then the Tjk must transform to Til such that 3
(i = 1, 2, 3).
(39) Multiply (37) by a;j and sum over the index j. 3
66
T H E FIELD EQUATIONS
[CHAP.
I
The components of a tensor of rank two transform according to the law 3
(43)
Ti', =
2
3
UijaikTjk,
(i, z
=
1, 2, 3);
j=1 k=l
inversely, any set of nine quantities which transform according to (43) constitutes a tensor. By an analogous procedure one can show that the reciprocal transformation is
(44)
If the order of the indices in all the components of a tensor may be changed with no resulting change in the tensor itself, so that Tjk = Tki, the tensor is said to be completely symmetric. A tensor is completely antisymmetric if an interchange of the indices in each component results in a change in sign of the tensor. The diagonal terms Tij of an antisymmetric tensor evidently vanish, while for the offdiagonal terms, = Tkj, then also Ti', = Tii. Tjk =  Tkj. It is clear from (43) that if Likewise if Ti, = Tkj, it follows that Ti, = Tii. The symmetric or antisymmetric character of a tensor is invariant to a rotation of the coordinate system. The sum or difference of two tensors is constructed from the sums or differences of their corresponding components. If 2Ris the sum of the tensors 2S and 2T,1 its components are by definition
I n virtue of the linear character of (43) the quantities Rjk transform like the S j k and Tjk and, therefore, constitute the components of a tensor 2R. From this rule it follows that any asymmetric tensor may be represented as the sum of a symmetric and an antisymmetric tensor. Assuming 2R to be the given asymmetric tensor, we construct a symmetric tensor 2S from the components
and an antisymmetric tensor
2Tfrom the components
Then by (45) the sum of 2S and 2Tso constructed is equal to 2R. In a threedimensional manifold an antisymmetric tensor reduces to three independent components and in this sense resembles a vector. The Tensors of second rank will be indicated by a superscript as shown.
SEC. 1.201
67
ELEMENTS OF TENSOR ANALYSIS
tensor (36), for example, reduces in this case to
(48)
B1
=
Bz B3
= =
+
0  T ~ I A Z T13A3, TziA1 0  T32A3, Ti3Ai T32A2 0.
+
+
+
These, however, are the components of a vector, B=TxA,
(49)
wherein the vector T has the components
Now it will be recalled that in vector analysis it is customary to distinguish polar vectors, such as are employed to represent translations and mechanical forces, from axial vectors with which there are associated directions of rotation. Geometrically, a polar vector is represented by a displacement or line, whereas an axial vector corresponds to an area. A typical axial vector is that which results from the vector or cross product of two polar vectors, and we must conclude from the above that an axial vector is in fact an antisymmetric tensor and its components should properly be denoted by two indices rather than one. Thus for the components of T = A X B we write (51)
Tik = AiBk  AkBi
=
Tk. 11
( j , k = 1, 2, 3).
If the coordinate system is rotated, the components of A and B are transformed according t o 3
3
Upon introducing these values into (51) we find 3
(53)
AiBk  AkBi
=
3
2 2 aljaik(AiBi  A:B’,),
2=1 i=l
a relation which is identical with (44) and which demonstrates that the components of a “vector product” of two vectors transform like the components of a tensor. The essential differences in the properties of polar vectors and the properties of those axial vectors by means of which one represents angular velocities, moments, and the like, are now clear: axial vectors are vectors only in their manner of composition, not in their law of transformation. It is important to add that an antisymmetric tensor can be represented by an axial or pseudovector only in a threedimensional space, and then only in rectangular components.
68
[CHAP.I
T H E FIELD EQUATIONS
Since the cross product of two vectors is in fact an antisymmetric tensor, one should anticipate that the same is true of the curl of a vector. That the quantity aAj/axk, wherc Ai is a component of a vector A, is the component of a tensor is at once evident from Eq. (22).
(54) The components of
vx
A,
(55) therefore, transform like the components of an antisymmetric tensor. The divergence of a tensor is defincd as the operation 3
( j = 1, 2, 3). k=l
The quantities Bi are easily shown t o transform like the components of a vector. (57)
or, on summing over 1 and applying the conditions of orthogonality,
The divergence of a tensor of second rank is a vector, or tensor of first rank. The divergence of a vector i s a n invariant scalar, or tensor of zero rank. These are examples of a process known in tensor analysis as contraction. As in the case of vectors, the tensor concept may he extended t o manifolds of four dimensions. Any set of 16 quantities which transform according to the law, (59)
Ti, =
52 j=l
k=l
4
4
i=l
1=1
aijarkTjk,
(i, I
=
1, 2, 3, 4),
aijazkT,!l,
(j,k
=
1, 2, 3, 41,
or its reciprocal, (60)
Tik
=
]c 2
will be called a tensor of second rank in a fourdimensional manifold. As in the threedimensional case, the tensor is said to be completely
SEC. 1.211
69
THE SPACETIME SYMMETRY
symmetric if Tjk = Tkj, and completely antisymmetric if Tik =  T k j , with Tji = 0. I n virtue of its definition i t is evident that an antisymmetric fourtensor contains only six independent components. Upon expanding (59) and replacing Tkj by  Tik, then recollecting terms, we obtain as the transformation formula of an antisymmetric tensor the relation
Any six quantities that transform according to this rule constitute a n antisymmetric fourtensor or, as it is frequently called, a sixvector. I n threespace the vector product is represented geometrically by the area of a parallelogram whose sides are defined by two vectors drawn from a common origin. The components of this product are then the projections of the area on the three coordinate planes. By analogy, the vector product in fourspace is defned as the “area” of a parallelogram formed by two fourvectors, A and B , drawn from a common origin. The components of this extended product are now the projections of the parallelogram on the six coordinate planes, whose areas are
(62)
Tjk
=
A,Bk  AkBj
=
Tkj,
(j,k
=
1, 2, 3, 4).
The vector product of two fourvectors is therefore a n antisymmetric fourtensor, or sixvector. If again A is a fourvector, the quantities
can be shown as in (54) to transform like the components of a n antisymmetric tensor. The l l l k may be interpreted as the components of the curl of a fourvector. As in the threedimensional case the divergence of a fourtensor is defined by
a set of quantities which are evidently the components of a fourvector. 1.21. The Spacetime Symmetry of the Field Equations.A remarkable symmetry of form is apparent in the equations of the electromagnetic field when one introduces as independent variables the four lengths (65)
x1 = 2,
22
== y,
5 3
= 2,
where c is the velocity of light in free space.
2 4
=
ict,
When expanded in rec
70
T H E FIELD EQUATIONS
[CHAP.I
tangular coordinates, the equations
aD (11) V x H   = J,
(IV) V * D
at
= p,
are represented by the system 0
aHz . aD1 = + aHo __  __  ZCax2 ax3 ax4
J1,
We shall treat the righthand members of this system as the components of a “fourcurrent ” density, (67)
J1
=
J,,
Jp
=
J,,
Js
= Jz,
i ~ p ,
J q =
and introduce in the lefthand members a set of dependent variables defined by Gi2
=
613
Gzz
=
Gz3 =
Hz Hi
GI4
Ha Hz iCD1
G3z
= =
0 icDa
G3q
GI1 = 0 Gzi
(68) Gal G4l
= = =
G42
Ha 0 = Hi = i~Dz
G33 G43
=
G24
Gqq
= icDl = $Dp = =
wD~ 0.
Then in the reference system X , Eqs. (11) and (IV) reduce to
(j
=
1, 2, 3, 4).
Only six of the G 3 k are independent, and the resemblance of this set of quantities to the components of an antisymmetric fourtensor is obvious. Since the divergence of a fourtensor is a fourvector, it follows from (69) that if the G l k constitute a tensor, thcn the J k are the components of a fourvector; inversely, if we can show that J is indeed a fourvector, we may then infer the tensor character of 2G. However, we have as yet offered no evidence to justify such an assumption. I n the preceding sections it was shown that the vector or tensor properties of sets of scalar quantities are determined b y the manner in which they transform on passing from one reference system to another. Evidently an orthogonal transformation of th e coordinates 2 3 corresponds to a simultaneous change in both the space coordinates x, y, z and the time t , and only recourse t o experiment will tell us how the field intensities may
71
T H E SPACETIME S Y M M E T R Y
SEC. 1.211
be expected t o transform under such circumstances. I n Sec. 1.22 we shall set forth briefly the experimental facts which lead one to conclude that the J k are components of a fourvector, and the Gjk the components of a field tensor; in the interim we regard (69) and the deductions that follow below merely as concise and symmetrical expressions of the field equations in a fixed system of coordinates. The two homogeneous equations
aB
(I) V x E + 27
=
(111) V * B
0,
=
0,
are represented by the system
After division of the first three of these equations by ic, an antisymmetrical array of Components is defiiicd as follows:
Fzi
=
Ba
F,, = 0
F61
=
BZ
Fa2
Bi
F23
=
F33
=0
F24
a
= 
C
(71)
i
F ~ =I  El C
F42
= B1 =
i 
C
i
Ez
F43
= C
F34
i
= 
C
E3
F44
=
Ez E3
0.
Then all the equations of (70) are contained in the system
where i, j , k are any three of the four numbers 1, 2, 3, 4. The arrays (68) and (71) are congruent in the sense th a t in each the real components pertain to the magnetic field, while the imaginary components are associated with the electric field. To indicate this partition it is convenient t o represent the sets of components by the symbols
(73)
,
'G = (H, icD).
72
T H E FIELD EQUATIONS
[CHAP.
1
Now the field equations may be defined equally well in terms of the “ dual” systems : (74)
2F* =
(fE, B),
2G* = (icD, H),
or
FYI = 0 Ffl =
i E3
FFl
=
 E z
F4:
=
B1
(75)
C
i
C
i
Fz
= 
F&
=
C
F&
E3
=
5
Ez
 E 1
p,”z = 2.C El
F&
0
F&
F23 = B3
Bz
=
=
B1
FF4
=
B,
FF4
=
B3
i
Ff. =
0
F&
C
F4“4 = 0,
and a corresponding system for the components of 2G*. Upon introducing these values into (I) and (111),and (11) and (IV), respectively, we obtain 4
k=l
(77)
It has been pointed out by various writers that this last representation is artificial, in that (74) implies th at E is an axial vector in threespace and B a polar vector, whereas the contrary is known to be true. The representation
9%
= Ji,
k=l
must in this sense be considered the I‘ natural” form of the field equations. To these we add the equation of continuity,
vJ+
=
0,
which in fourdimensional notation becomes (78)
9% =
k=
“
1
axk
0.
If the components F j k are defined in terms of the components of a fourpotential ” Q, by
(79)
73
THE SPACETIME SYMMETRY
SEC. 1.211
one may readily verify that Eq. (72) is satisfied identically. Now in threespace the vectors E and B are derived from a vcctor and scalar potential. aA E=V+, B=vxA; (80) at or, in component form,
where the indices i, j , k are l3obe taken in cyclical order. Clearly all these equations are comprised in the system (79) if we define the components of the fourpotential by =
+I
(82)
A,,
=
A,,
ips =
A,,
a* = i 4. C
As in three dimensions, the fourpotential is useful only if we can i E). determine from it the field 2G(H, ZcD) as well as the field 2F(B,
c
Some supplementary condition must, therefore, be imposed upon @ in order that it satisfy (69) as well as (72); thus i t is necessary that the components G i k be related functionally to the F j k . We shall confine the discussion here to the usual case of a homogeneous, isotropic medium and assume the relations to be linear. To preserve symmetry of notation i t will be convenient to write the proportionality factor which characterizes the medium as y j k , so that
(83)
Gik = YikFjk;
but i t is clear from (68) and (71) that' (84)
yjk
1
=  when j , P
k
=
1, 2, 3,
yjk
= ccz
when j or k
=
4.
These coefficients are in fact components of a symmetrical tensor, and with a view t o subsequent necds the diagonal terms are given the values
Equation (69) is now to be replaced by ( j = 1, 2, 3, 4). 1 A medium which is anisotropic in either its electrical properties or its magnetic properties may be represented as in (83) provided the coordinate system is chosen to coincide with the principal axes. This also is the case if its principal axes of electrical anisotropy coincide with those of magnetic anisotropy.
74
[CHAP.I
T H E FIELD EQUATIONS
Upon introducing (79) we find that (86) is satisfied, provided solution of
is a
(87) subject to the condition
(j= 1, 2. 3, 4). k=l
This last relation is evidently equivalent to (89) and (87) comprises the two equations
In free space ~ o e o= c', Y j k = PO', for all values of the indices. Equations (87) and (88) then reduce to the simple form:
3%
k=l
4 a@k
= p0Jj,
= 0, ( j = 1, 2, 3, 4).
k=l
1.22. The Lorentz Transformation.The physical significance of these results is of vastly greater importance than their purely formal elegance. A series of experiments, the most decisive being the celebrated investigation of Michelson and Morley,' have led to the establishment of two fundamental postulates as highly probable, if not absolutely certain. According to the first of these, called the relativity postulate, it is impossible to detect by means of physical measurements made within a reference system X a uniform translation relative to a second system X'. That the earth is moving in an orbit about the sun we know from observations on distant stars; but if the earth were enveloped in clouds, no measurement on its surface would disclose a uniform translational motion in space. The course of natural phenomena must therefore be unaffected by a nonaccelerated motion of the coordinate systems to which they are referred, and all reference systems moving linearly and uniformly relative to each other are equivalent. For our present needs we shall state the relativity postulate as follows : When properly formulated, the laws of 1 MICHELSON and MORLEY,Am. J . Sci., 3, 34, 1887.
SEC.
1.221
T H E LORENTZ TRANSFORMATION
75
physics are invariant to a transformation f r o m one reference system to another moving with a linear, u n i f o r m relative velocity. A direct consequence of this postulate is that the components of all vectors or tensors entering into an equation must transform in the same way, or covariantly. The existence of such a principle restricted to uniform translations was established for classical mechanics by Newton, but we are indebted t o Einstein for its extension to electrodynamics. The second postulate of Einstein is more remarkable: The velocity of propagation of a n electromagnetic disturbance in free space i s a universal constant c which i s independent of the reference system. This proposition is evidently quite contrary to our experience with mechanical or acoustical waves in a material medium, where the velocity is known t o depend on the relative motion of medium and observer. Many attempts have been made to interpret the experimental evidence without recourse to this radical assumption, the most noteworthy being the electrodynamic theory of Rita.l Thc results of all these labors indicate that although a constant velocity of light is not necessary to account for the negative results of the MichelsonMorley experiment, this postulate alone is consistent with that experiment and other optical phenomena.2 Let us suppose, then, that a source of light is fixed a t the origin 0 of a system of coordinates X(z,y , z). At the instant t = 0, a spherical wave is emitted from this source. An observer located at the point x, y , z in X will first note the passage of the wave a t the instant ct, and the equation of a point on the wave front is therefore (92)
52
+ y2 +
22
 c2t2 = 0.
The observer, however, is free to measure position and time with respect to a second reference frame X’(x’, y’, z’) which is moving along a straight line with a uniform velocity relative t o 0. For simplicity we shall assume the origin 0‘ to coincide with 0 a t the instant t = 0. According to the second postulate the light wave is propagated in X’ with the same velocity as in X , and the equation of the wave front in X’ is (93)
+ / + z’z  c2tJZ= 0.
By t’ we must understand the time as measured by an observer in X’ with instruments located in that system. Here, then, is the key to the transformation that connects the coordinates x, y , z, t of an observation or event in X with the coordinates x’, y’, z’,t’ of the same event in X’: it must be linear and must leave the quadratic form (92) invariant. The linearity follows from the requirement that a uniform, linear motion of a particle in X should also be linear in X‘. 2
RITZ,Ann. chim. p h y s . , 13, 145, 1908. An account of these investigations will be found in Pauli’s article, 20c. cit., p. 549.
76
[CIIAP.I
T H E FIELD EQUATIONS
Let (94)
5 1
= 5,
5 2
= y,
5 3
= 2,
54
=
ict,
be the components of a vector R in a fourdimensional manifold X(51, 5 2 , 5 3 , 5 4 ) .
R 2 = X!
(95)
+ X: + + X!
5".
The postulate on the constancy of the velocity c will be satisfied by the group of transformations which leaves this length invariant. But in Sec. 1.19 i t was shown that (05) is invariant to the group of rotations in fourspace and we conclude, therefore, th at the transformations which take one from the coordinates of an event in X to the coordinates of that event in X' are of the form
(26)
XI =
5
(j = 1, 2, 3, 41,
ajkxk,
k=l
where 4
2
(27)
ajiajk
=
(i, k
6ikl
= 1, 2, 3,
4),
j= 1
the determinant lajk[ being equal t,o unity. We have now to find t,hese coefficients. T h e calculation will be simplified if we assume th at the rotation involves only the axes x 3 and 2 4 , and the resultant lack of generality is inconsequential. We take, therefore, xi = 51, xk = 2 2 , and write down the coefficient matrix as follows:
The conditions of orthogonality rcduce to
(97)
a23
+ a& = 1,
4 4
+
a"4
= 1,
ckL3a34
+
a43a44
= 0.
If we put a33 = a, a34= iaD, we find from (97) th a t a44 = +a, ~4~ = f i a p , a dl  p 2 = & 1. Only the upper sign is consistent with the requirement that the determinant of the coefficients be positive unity, and this in turn is the necessary condition th a t the group shall contain the identical transformation. I n terms of the single parameter /3 the coefficients are 1 iP a33= a44= a 3 4 = a43 = (98)
dFj9'
di+
SEC.
77
T H E LORENTZ T R A N S F O R M A T I O N
1.221
for the transformation itself, we have
.:=
(99)
21,
z: = z.2,
Reverting t o the original spacetime manifold this is equivalent to z’ = 2,
(100)
z‘ =
Y’ = Y,
1
diq5
(2 
t’ =
pet),
d~1
(1

z).
The parameter p may be det,ermined by considering z’, y’, z’ to be the coordinates of a fixed point in X’. The coordinates of this point with respect to X are z, y, z. Since dz’ = 0, it follows th a t
_  v dt dz
= pc,
p
= ?, C
and hence the rotation defined in (96) and (97) is equivalent to a translation of the system X’ along the zaxis with the constant velocity v relative to the unprimed system X . The transformation
obtained from (100) by substitution of the value for p, or its inversion,
has been named for Lorentz, who was the first t o show that Maxwell’s equations are invariant with respect to the change of variables defined by (l02), but not invariant under the “Galilean transformation:” (104)
2’ = z 
vt,
t’
=
t.
All known electromagnetic phenomena may be properly accounted for if the position and t i m e coordinates of an event in a moving system X’ be related t o the coordinates of that event in an arbitrarily fixed system X by a Lorentz transformation. The Galilean transformation of classical mechanics represents the limit approached by (102) when v =
1:
1
.(p> dp,
1 7vp, VP = 
where po is the pressure a t a point in the fluid where E = 0. If ds is an element of length along any path connecting two points whose pressures are respectively p and PO, then V P . ds = dP, and (64) is satisfied by
ADAMS,Phil. Mag., (6) 22, 889, 1911. KEMBLE, Phys.Rev., (2) 7,614, 1916. SACERDOTE, Jour. physique, (3) 8,457, 1899; 10, 196, 1901. 4 International Critical Tables, Vol. VI, p. 207, 1929.
SEC.2.261
FORCE ON A BODY I M M E R S E D I N A F L U I D
151
d7 In liquids d p = XI T, Eq. (77), page 96, or PPO
(67)
7
= roe
x1
.
The parameter X,I the reciprocal of the compressibility, is a very large quantity and hence r cv ro. For liquids, therefore,
or, upon applying the ClausiusMossotti law,
RT where R is the M
In gases (67) is replaced by the relation p = 7,
gas constant, T the absolute temperature, and M the molecular weight.’ 2.26. Force on a Body Immersed in a Fluid.The results of our analysis may be summed up profitably by a consideration of the following
..
/
Fluid
Fluid
E
FIQ.24.A solid body immersed in a fluid dielectric.
problem. In Fig. 24, a dielectric or conducting body is shown immersed in a fluid. An electrostatic field is applicd and we desire an expression for the resultant force o n the entire body. Let us draw a surface S1 enclosing the body and located in the fluid just outside the boundary X. On every volume element of the solid there is a force of density
+
f = pE  $E2Ve f”, (70) where f” is given by Eq. (40), and on each element of the surface of discontinuity there is a traction given by (53). Rather than evaluate the integrals off and t over the volume and bounding surface X, we need only 1 ABRAHAM, BECKER, and DOUGALL, “The Classical Theory of Electricity and Magnetism,” p. 98, Blackie, 1932.
152
[CHAP.I1
STRESS A N D ENERGY
calculate the integral of t from Eq. (51) over the surface 3€ lies in the fluid, a1 = a2 = T  and (51) reduces t o
81.
Since
S 1
aT

t = eE(E n)  5 E2n 2
a€ E2n, + 2 di7
where n is t h e unit normal directed from the solid into the fluid and E and T apply t o the fluid. Now the integral of this traction over XI givrs the force exerted directly b y the field on the volume and surface of thc body. B u t the field, as we have just seen, also generates a pressure (68) in t h e fluid which a t every point on SIacts normally inward, i.~., toward the solid. The resultant force per unit area transmitted across S1 is, therefore, equal t o (71) diminished by (68), and the net force exerted b y the field on the solidthe force which must be cornpensated b y exterior supportsis
Thus t o obtain the resultant force on a n entire body we need not know the constants a1 and a2 within either the solid or the fluid, for the forces with which they are associated are compensated locally by elastic stresses. Furthermore, since the fluid is in equilibrium, i t is not essential t h a t S1 lie in the immediate neighborhood of the body surface S. The net forcc on the liquid contained between XI and a n arbitrary enclosing surface Sz is zero and consequently
if we adhere t o t h e convention that n is directed outward from the enclosed region. The surface Sz may, therefore, be chosen in any manner that will facilitate integration, provided only t h a t no foreign bodies are intersected or enclosed. I n case the solid is a conductor, the field E is normal a t the boundary. The resultant force on a n isolated conductor is then (74)
where w is t h e charge density on S. This last is true, of course, only for a surface S1just outside the conductor. It should be evident now t h a t the force on one solid embedded in another can be calculated only when the nature of the contact over their common surface is specified; the problem is complicated by the fact t h a t
153
NONFERROMAGNETIC M A T E R I A L S
SEC. 2.271
tangential shearing stresses as well as normal pressures must be compensated by elastic stresses across the boundary. FORCES IN THE MAGNETOSTATIC FIELD
2.27. Nonferromagnetic Materials.The analysis of Sec. 2.22 may be applied directly t o the magnetostatic field i n all cases where the relation between B and H is of the form 3
with the components pik of the permeability tensor functions possibly of position but independent of field intensity. The change in magnetic energy of an isotropic body resulting from a variation in p is given by Eq. (52), Sec. 2.17.
‘I’
6T = 2
6pH2dv,
and the generalization of this expression t o a n anisotropic body occupying the volume V1 is
(3)
6T
=
aJvl
+ ~ P z J+G
(8~1lH21
+ 26~12H1Hz + 26Pz3HzH3 + 26p31H3H1) dv.
8~33H2
This variation may be expressed in terms of the components of strain, assuming a linear relation between the components of the permeability and strain tensors. (4)
If we assume further that in the unstrained state the medium i s isotropic, the coefficients @% reduce t o two, (5) so that the variation in magnetic energy due t o a pure strain is
(6)
6T
sv, + + + + +
=
[(blHq
(bzHI
bzH;
bzH;
b ~ H 3Jell
blH2,) 6e33
+ (bzHq + b ~ H +i b2W3
+ ( b l  bd(H1Hz
6czz
6e12
4 HZf13 6 e 2 3 4H3H1 6c3d1 dv.
Equation (6) differs formally from (31), page 143, only in algebraic sign. Now the electrostatic energy U represents the work done in
154
STRESS AND ENERGY
[CHAP.I1
building up the field against the mutual forces between elements of charge. The magnetic energy T , on the other hand, represents work done against the mutual forces exerted by elements of current, plus the work done against induced electromotive forces. The work required to bring about a small displacement or deformation of a body in a magnetic field is done partly against the mechanical forces acting on the body, partly on the current sources to maintain them constant.' When the work done against the induced electromotive forces is subtracted from the total magnetic energy, there remains the potential energy of the mutual mechanical forces, and this we found in See. 2.14 to be equal and opposite to T .
SU
(7)
=
ST.
The forces exerted by the jield on the body within V1 are to be calculated from ( 6 ) with sign reversed. I n complete analogy with the electrostatic case we now find that the body force exerted by a magnetic field on a medium which in the unstrained state is isotropic, whose permeability is independent of field intensity, which is free of residual magnetism, but which may carry a current of density J, is
If the medium is homogeneous and carries no current, (8) reduces to f =
(9)
:(bi
I n a gas or liquid bl = b, =
7
+ b2)VH2. aP , a7
and (8) can be written in the form
As in the electrostatic case, the body force whose components are given by (8) can be expressed as the divergence of a tensor 2S, (11)
f
=
v * 2s,
whose components are
l This remark does not apply if the source is a permanent magnet; in that case the energy of the field is not given by T = +JB. H dv.
SEC.2.281
FERROMAGNETIC MATERIALS
155
(12)
The force exerted by the magnet.ic field on the matter within a closed surface S1is, therefore, the same as would result from the application over S1 of a force of surface density
(13) Likewise the force per unit area exerted on a surface of discontinuity can be obtained by calculating (13) on either side of the surface. The result is the magnetic equivalent of (53). A case of particular importance is that of a magnetic body immersed in a fluid whose permeability to a sufficient approximation is independent of density and equal to PO, the value in free space. Let the body be represented by the region (1) in Fig. 23 and the fluid by the region (2). The force per unit area on the bounding surface is then found to be
where the constants bl, bz apply to the magnetic body and H1 is measured just inside its surface. Not very much is known about the parameters b l and bz, although there are indications that they may be very large. They must, of course, be determined if the elastic deformation of a body is to be calculated. I n most practical problems, fortunately, one is interested only in the net force acting on the body as a whole. The forces arising from deformations are compensated locally by induced elastic stresses and consequently terms involving bl and b, will drop out. As in the electric case, the resultant force exerted by a magnetostatic field on a nonferromagnetic body immersed in any fluid is obtained by evaluating the integral
pH(H.n) 
1
H2n da
over any surface enclosing the body. 2.28. Ferromagnetic Materials.The preceding formulas apply to ferromagnetic substances in sufficiently weak fields. If, however, the permeability p depends markedly on the intensity of the field, the energy of a magnetic body can no longer be represented in the form of Eq. (2) or (3) and we must use in their place the integral derived in Sec. 2.17.
156
STRESS A N D ENERGY
[CHAP.I1
An analysis of the volume and surface forces for such a case has been made by Pockels,' who also treats briefly the problem of magnetostriction. The phenomena of magnetostriction are governed, however, b y several important factors other than the simple elastic deformation considered here. A specimen of iron, for example, which in the large appears isotropic, exhibits under the microscope a finegrained structure. The properties of the individual grains or microcrystals are strongly anisotropic. Of the same order of magnitude as these grains but not necessarily identified with them are also groups or domains of atoms, each domain acting as a permanent magnet. I n the unmagnetized state the orientation of the magnetic domains is random and the net magnetic moment is zero. A weak applied field disturbs these little magnets only slightly from their initial positions of equilibrium. A small resultant moment is induced and under these circumstances the behavior of the iron may be wholly analogous to a polarizable dielectric in an electric field. As the intensity of the field is increased, however, the domains begin to flip over suddenly t o new positions of equilibrium in line with the applied field, with a consequent change in the elastic properties of the specimen. A dilatation in weak fields may be followed by a contraction as the field becomes more intense, quite contrary to what would be predicted by the magnetostriction theory when applied to strictly isotropic solids. We are confronted here with a problem in which the macroscopic behavior of matter cannot be treated apart from its microscopic structure. FORCES IN THE ELECTROMAGNETIC FIELD
2.29. Force on a Body Immersed in a Fluid.The
expressions
for the energy of a body in a stationary electric or magnetic field have been based in Secs. 2.10 and 2.17 on the irrotational character of the vectors E and H  HI as well as upon their proper behavior a t infinity. In a variable field these conditions are not satisfied and therefore (1) cannot be applied t o the determination of the force on a body or an element of a body without a thorough revision of the proof. At best the analysis will contain some element of hypothesis, for our assumption that dD dB the quantities E  and H  represent the snergy densities in an

at
 dt
electromagnetic field is, after all, only a plausible interpretation of Poynting's theorem. POCKELS, loc.
cit., p. 369.
SEC.2.291
157
FORCE ON A BODY I M M E R S E D I N A F L U I D
In Sec. 2.5 it was shown that the total force transmitted by an electromagnetic field across any closed surface Z in free space is expressed by the integral
F
(2)
=
z
[eo(E n)E + 1 (B .n)B  1(eoE2+ 1 B z ) n ] da. 

PO
PO
Subsequently we were able to demonstrate for stationary fields that if the surface Z: is drawn entirely within a fluid which supports no shearing stress, the net force is given by (2) if only we replace e0 and po by their appropriate values in the fluid.
This is the resultant force on the charge, current, and matter within 2. The matter within B need not be fluid and there may be sharp surfaces of discontinuity in its physical properties; the relations between E and D, and B and H, however, are assumed to be linear. Now since (2) is valid in the dynamic as well as in the stationary regimes, there is reason to suppose that (3) may be applied also to variable fields. It is not difficult to marshal support for such a hypothesis, particularly from the theory of relativity. However, the righthand side of (3) must now be interpreted in the sense of See. 2.6 not as the force exerted by the field on the matter within B but as the inward flow of momentum per unit time through 2 . We denote the total mechanical momentum of the matter within 2 , including the ponderable charges, by G,,,, and the electromagnetic momentum of the field by G,. Then (4)
+ p(H  n)H  1 (eE2+ pW)n
+
d  (Gmech G,) = e(E n)E dt Sz
I
da.
The surface integral (4) may be transformed into a volume integral. For in the first place:
eEZnda
(5) Then
(6)
(7)
*V(eE2) = +E2Ve (D V)E =
a
(DS)
=
:L

+ (D
+ dY
*
v(eEZ)do.
+ D X V X E. + a (D,E)  E V
V)E
d  (DUE)
*
D.
and this obviously cancels the 2component of the vector e(E n)E in
158
STRESS A N D ENERGY
[CHAP.
I1
(3). Proceeding similarly with the magnetic terms and then making aD use of the relations V X E = aB/at, V x H =  J, V  D = p, we at
+
are led t o
(9)
+
dt (Gmech G,)
=
V
[pE
+ J X B  21 E2Ve  21 H 2 V p + ata (D x B)] dv.
The increase in the mechanical momentum is the result of the forces exerted by the field on charges and neutral matter.
F
(10)
d
=  Gmeoh.
dt
If, therefore, the righthand side of (9) can be split into two parts identified with Gmechand G,, the force can be determined. Just how this resolution is to be made is by no means obvious and various hypotheses have been suggested.' According to Poynting's theorem the flow of energy, even within ponderable matter, is determined by the vector (11)
S = E X H
joules/sec.meter2.
Abraham and von Laue take for the density of electromagnetic momentum 1 g, = /.Lo& =s kg. /sec.meter2. (12) c2 Then according to this hypothesis, the resultant force on the charges, currents, and polarized matter within 2 is
(13) F =
or
(14) F =
s s[ V
(pE
+ J x B  21 E2Vc  21 H 2 V p + ___ 

c2

E(E n)E

dt
+ p(H  n)H  1 (eE2 + pH2)n] da 2.s c2dt
p
E x Hdv.
Practically, the exact form of the electromagnetic momentum term is of no great importance, for the factor 1/c2 makes it far too small to be easily detected. 1 PAULI,Encyklopadie der mathematischen Wissenschaften, Vol. V, Part 2, pp. 662667, 1906.
SEC.2.291
FORCE ON A BODY I M M E R S E D I N A FLUID
159
On the grounds of (13) it is sometimes stated t h a t the force exerted by an electromagnetic field on a unit volume of isotropic matter is (15)
f = pE
+J x B 
Such a conclusion is manifestly incorrect, for (15) does not include the forces associated with the deformation. As previously noted, these strictive forces are compensated locally by elastic stresses and do not enter into the integrals (13) and (14) for the resultant force necessary to maintain the body as a whole in equilibrium.
CHAPTER I11 THE ELECTROSTATIC FIELD
From fundamental equations and general theorems we turn our attention in the following chapters to the structure and properties of specific fields. The simplest of these are the fields associated with stationary distributions of charge. Of all branches of our subject, however, the properties of electrostatic fields have received by far the most adequate and abundant treatment. In the present chapter we shall touch only upon the more outstanding of these properties and of the methods which have been developed for their analysis. GENERAL. PROPERTIES OF AN ELECTROSTATIC FIELD
3.1. Equations of Field and Potential.The equations satisfied by the field of a stationary charge distribution follow directly from Maxwell's equations when all derivatives with respect to time are placed equal to zero. We have, then, at all regular points of an electrostatic field : (I) V X E = 0, (11) V * D = p . According to (I) the line integral of the field intensity E around any closed path is zero and the field is conservative. The conservative nature of the field is a necessary and sufficient condition for the existence of a scalar potential whose gradient is E. (1)
E = V4.
The algebraic sign is arbitrary but has been chosen negative to conform with the convention which directs the vector E outward from a positive charge. Equation (1) does not define the potential uniquely, for there might be added to 4 any constant Cb0 without invalidating the condition
+
v x v(4 40) = 0. (2) I n Chap. I1 it was shown that the scalar potential of an electrostatic field might be interpreted as the work required t o bring a unit positive charge from infinity to a point (z, y, z ) within the field: (3) We shall show below that the field of a system of charges confined to a finite region of space vanishes at infinity. The condition that $ shall vanish at infinity, therefore, fixes the otherwise arbitrary constant $ 0 . 160
SEC.3.11
E QUAT I ON S OF FIELD A N D P O T E N T I A L
161
+
Those surfaces on which is constant are called equipotential surfaces, or simply equipotentials. At every point on an equipotential the field intensity E is normal to the surfact:. For let
+(x, y , z )
(4)
=
constant
be an equipotential, and take the first differential.
The differentials d z , d y , d z , are the components of a vector displacement dr along which we wish to determine the change in +, arid since d+ = 0, this vector must lie in the surface, = constant. The partials a+/ax, a+/ay, a+/&, on the other hand, are rates of change along the x, y, zaxes respectively and as such have been shown to be the components of another vector, namely the gradient
+
d,$ is manifestly the scalar product of these two and, since this product vanishes, the vectors must be orthogonal. An exception occurs a t those points in which the three partial dcrivativcs vanish simultaneously. The field intensity is zero and the points are said t o be points of equilibrium. The orthogonal trajectories of the equipotential surfaces constitute a family of lines which a t every point of the field are tangent to the vector E. They are the lines of force. It is frequently convenient t o represent graphically the field of a given system of charges by sketching the projection of these lines on some plane through the field. Let d s represent a small displacement along a line of force, where
+
+
d s = i dx’ j dy‘ k dz’, (7) the primes being introduced to avoid confusion with a variable point (2, y, z ) on an equipotential. Then, since by definition the lines of force are everywhere tangent to the fieldintensity vector, the rectangular components of ds and E(x’, y’, 2 ’ ) must be proportional.
(8)
E, = X dx‘,
E,
= X
dy‘,
E,
=
X dz‘.
The differential equations of the lines of force are, therefore, (9)
The relations between the components of D and those of E are almost invariably linear. If the medium is also isotropic, one may put (10)
D
=
EE= cV+,
162
T H E ELECTROSTATIC FIELD
[CHAP.111
whence, by (11),4 must satisfy
+
v * (ev4) = €V2+ V €* v4 = p. (11) In case the medium is homogeneous, 4 must be a solution of Poisson’s equation, 1 (12) v24 =   p . At points of the field which are free of charge (12) reduces to Laplace’s equation,
v24
(13)
=
0.
The fundamental problem of electrostatics is to determine a scalar function +(x, y, z ) that satisfies at every point in space the Poisson equation, and on prescribed surfaces fulfills the necessary boundary conditions. A much simpler inverse problem is occasionally encountered : Given the potential as an empirical function of the coordinates reprcsenting experimental data, to find a system of charges that would produce such a potential. The density of the necessary continuous distribution is immediately determined by carrying out the differentiation indicated by Eq. (12). There will in general, however, be supplementary point charges, whose presence and nature are not disclosed by Poisson’s equation. At such points the potential becomes infinite and, inversely, one may expect to find point charges or systems of point charges located a t the singularities of the potential function. The nature of these systems, or multipoles as they are called, is the subject of a later section, but a simple example may serve to illustrate the situation. Let the assumed potential be
where r is the radial distance from the origin to the point of observation and a is a constant. By virtue of the spherical symmetry of this function, Poissons’s equation, when written in spherical coordinates, reduces to
on differentiation one finds for the density of the required continuous charge distribution
The charge contained within a sphere of radius r is obtained by integrating p over the volume, or
(17)
f
p
dv = e=r(ar
+ 1)  I .
SEC.3.21
163
B O U N D A R Y CONDITIONS
If the radius is made infinite, the total charge of the continuous distribution is seen to be (18)
L m p d v = 1.
But this is not the total charge required to establish the potential (14). For a t T = 0 the potential becomes infinite and we must look for a point charge located a t the singularity. To verify this we nced only apply (11) in its integral form. If q is a point charge a t the origin,
the surface integral extending over a sphcrc S bounding the volume V .
D, da
(21) and, hence,
q
(22)
=
(ar
=
+ 1)e*',
+l.
The potential defincd by (14) arises, therrfore, from a positive unit charge located a t the origin and arq {equal negative charge distributed about it with a density p , the system as a whole being neutral. 3.2. Boundary Conditions.Thc transition of the field veciors across a surface of discontinuity in the medium was investigated in See. 1.13 and the results of that section apply directly to the electrostatic case. The two media may be supposed to meet a t a surface S and the unit normal n is drawn from medium (1) into medium (2), so that (1) lies on the negative side of S, medium (2) on the positive side. Then (23) where
n w
x (E2  El)
=
n . (Dz  D1)
0,
=
W,
is the density of any surface charge distributed over S.
It will be convenient to introduce the unit vector t tangent to the surface S. The derivativcs &$/an and represent respectively the rates of change of q5 in the normal and in a tangential direction. Then the boundary conditions (23) can be expressed in terms of the potential by (24)
($)2

(2)
 el($)
e.($),
= 0,
= w.
From the conservative nature of the field it follows also that the potential itself must be continuous across S, for the work required t o carry a small charge from infinity to either of two adjacent points located on opposite sides of S must be the same Hence (25)
41
=
42.
164
THE ELECTROSTATIC FIELD
[CHAP.
111
The two conditions
are independent. Conductors play an especially prominent role in electrostatics. For the purposes of a purely macroscopic theory it is sufficient to consider a conductor as a closed domain within which charge moves freely. If the conductor is a metal or electrolyte, the flow of charge is directly proportional t o the intensity E of the electric field: J = uE. Charge is free to move on the surface of a conductor but can leave it only under the influence of very intense external fields or at high temperatures (thermal emission). If the conductor is in electrostatic equilibrium all flow of charge has ceased, whence i t is evident t h a t at every interior point 07 a conductor in an electrostatic field the resultant field intensity E i s zero, and at every point on its surface thP tangential component of E i s zero. Furthermore the electrostatic potential 4 within a conductor i s constant and the surface of every conductor i s an equipotential. Let us suppose that an unchargcd conductor is introduced into a fixed external field Eo. I n the first instant there occurs a transient current. According to Sec. 1.7, no charge can accumulate a t an interior point, but a’redistribution will occur over the surface such t h a t the surface density a t any point is w , subject t o the condition
Ss
w
da
= 0.
This surface distribution gives rise t o an induced or secondary field of intensity El. Equilibrium is attained when the distribution is such t h a t at every interior point
Eo
(28)
+ El = 0.
Likewise, if a charge q is placed on an isolated conductor, the charge will distribute itself over the surface with a density w subject t o (28) and the condition
Ss
(29)
w
da
= q.
We shall denote the interior of a conductor in electrostatic equilibrium by the index (1) and the exterior dielectric by (2). Then a t the surface S
(30)
El
=
D1 = 0,
n x Ez = 0,
n.Dz=
or in terms of the potential, (31)
C$
=
constant,
a4 dn
e2=
w.
O,
SEC.3.31
165
GREEN’S THEOREM
If a solution of Laplace’s equation can be found which is constant over the given conductors, the surface density of charge may be determined by calculating the normal derivative of the potential. CALCULATION OF THE FIELD FROM THE CHARGE DISTRIBUTION
3.3. Green’s TheoremLet V be a closed region of space bounded by a regular surface S , and let 4 and $ be two scalar functions of position which together with their first and second derivatives’ are continuous throughout V and on the surface 8. Then the divergence theorem applied to the vector $V+ gives
J,v  ( W 4 >dv
(1)
=
($v4> n da. +
Upon expanding the divergence to (2)
v  (*v4) = v+. vlp + 1c.v * vlp
= V1c.e vlp
+ 1c.V2&,
and noting that
(3) where &$/an is the derivative in the direction of the positive normal, we obtain what is known as Green’s first identity:
(4)
If in particular we place $= 4 and let 4 be a solution of Laplace’s equation, Eq. (4) reduces to
Next let us interchange the roles of the functions 4 and the divergence theorem to the vector 4v$.
1c.; ie., apply
Upon subtracting (6) from (4)a relation between a volume integral and a surface integral is obtained of the form
(7) known as Green’s second identity or also frequently as Green’s theorem. This condition is more stringent than is necessary. The second derivative of one function Ic. need not be continuous.
166
T H E ELECTROSTATIC FIELD
[CHAP.111
3.4. Integration of Poisson’s Equation.By means of Green’s theorem the potential a t a fixed point (d, y’, z’) within the volume V can be expressed in terms of a volume integral plus a surface integral over X. Let us suppose that charge is distributed with a volume density p(x, y, z). We shall assume that p(x, y, z ) is bounded but is an otherwise arbitrary function of position. An arbitrary, regular surface S is now drawn enclosing a volume V , Fig. 25. It is not necessary that S enclose all the charge, or even any of it. Let 0 be an arbitrary origin and x = x’, y = y‘, z = z’, a fixed point of observation within V . The potential a t this point due to the entire charge distribution is c$(x’, y’, 2’). For
FIQ.26.Application of Green’s theorem t o a region V bounded externally by the surface S and internally by the sphere 81.
the function I) we shall choose a spherically symmetrical solution of Laplace’s equation, $(z, y, 2; z’, y’, 2’)
(8)
=
1 r
7
where r is the distance from a variable point (z, y, z ) within V to the fixed point (x’,y’, 2’). (9)
r
=
d ( x ’  x)2
+ (y’ 
y)2
+ (z’

z)2.
This function $, however, fails to satisfy the necessary conditions of continuity a t r = 0. To exclude the singularity, a small sphere of radius r1 is circumscribed about (z’, y’, z’) as a center. The volume V is then bounded externally by S and internally by the sphere S1. Within B both 4 and I) now satisfy the rcquirernents of Green’s theorem and furthermore V21) = 0. Thus (7) reduces to
the surface integral to be extended over both S and S1. Over the sphere S1 the positive normal is directed radially toward the center (d, y’, z’),
SEC.
167
BEHAVICIR AT I N F I N I T Y
3.51
since this is out of the volume V . Over 81,therefore,
Since r1 is constant, the contribution of the sphere to the righthand side of (10) is
If is
6 and s / a r denote mean values of
+ and a+/ar on S1,this contribution
which in the limit rl+ 0 reduces to  4 ~ + ( z ‘ , y’, 2 ’ ) . Upon introducing this value into (10) the potential a t any interior point (d,y‘, 2’) is
or in terms of the charge density when the medium is homogeneous, 1 v2+ =  p €
(13)
’
+(d,y’,
2’) =
1
47r€
.(1 f dv + I 47r
(‘)I
[Ar a29n  4 % 7
da.
I n case the region V bounded by S contains no charge, (13) reduces t o +(2’, y’, 2 ’ ) =

47r
It is apparent that the surface integrals in (13) and (14) represent the contribution to the potential a t (d, y’, 2 ’ ) of all charges which are exterior t o S. If the values of + and its normal derivative over S are known, the potential a t any interior point can be determined by integration. Equation (14) may be interpreted therefore as a solution of Laplace’s equation within V satisfying specified conditions over the boundary. The integral c#l(X’,
y’,
1 4T€ Jv
2’) = 
f dv
is a particular solution of Poisson’s equation valid a t (z’, y’, 2 ’ ) ; the general solution is obtained by adding the integral (14) of the homogeneous equation v2+= 0. If there are no charges exterior t o s, the surface integral must vanish. 3.6. Behavior at InfinityLet us suppose that every element of a charge distribution is located within a finite distance of some arbitrary
168
[CHAP.I11
T H E ELECTROSTATIC FIELD
origin 0. We imagine this distribution to be confined within the interior of a surface Sl which also contains the origin 0. The distances r, r1 and R are indicated in Fig. 26.
r
(16)
=
( R 2f r:  2rlR cos 0)h.
The potential a t any point P(z’, y’,
2’)
outside S1 is then
As P recedes to infinity, the terms r; and 2rlR cos 6 become negligible
/
/
’,Z‘
/
I I I
I
\
\
\ \ \
‘\

‘
/
/ ,
___*
FIU.%Figure t o accornpany Sec. 3.5.
with respect to R2, and in the limit as R + co we find
where q is the total charge of the system. A potential function is said to be regular at injinity if R4 is bounded as R ;r 0 0 . The field intensity E = V+ at great distances is directed radially from 0 and the function R21El is bounded. All real charge systems are contained within domains of finite extent and their fields are, therefore, regular a t infinity. Frequently, however, the analysis of a problem is simplified by assuming the external field to be parallel. Such a field does not vanish a t infinity and can only arise from sources located a t an infinite distance from the origin. It is important to note that a closed surface S divides all space into two volumes, an interior V 1and an exterior V 2 ,and that if the functions 4 and $ are regular a t infinity Green’s theorem applies to the external region V2 as well as to V1. For certainly the theorem applies to a closed region bounded internally by S and externally by another surface S2.
a4 and 4 all. vanish If now S2recedes towards infinity, the quantities $ an an
169
COULOMB F I E L D
SEC. 3.61
as l/r3 and the integral over S z approaches zero.
Consequently
but the positive normal a t points on S is now directed out of V z and therefore into V1. Let us suppose, for example, that a chargesystem is confined entirely to the region V1 within S. The potential a t any point (z’, y’, x ’ ) in V z outside S may be calculated from (17), or equally well from a knowledge of 4 and &$/an on S , applying (14) as indicated in Fig. 27.
FIG.27.Application
of Green’s theorem to the exterior Vz of a closed surface S.
3.6. Coulomb Field.According to (18) thc potential of a charge q in a homogeneous medium a t distances very great relative to the dimensions of the charge itself approaches the value +(z’, y’, 2’) =
1 . q

47re r
If q is located a t (z, y, x ) , the distance from p to the point of observation (d,g’, z’) is (21)
T
= d(s’ 
.)Z
+ (y’

y)2
+ (d 
2)2.
Let now ro represent a unit vector directed from the source; Le., from (2,y, z ) towards (z‘, y’, 2’). Thcn
where the prime above the gradient operator denotes differentiation with respect to the variables (d, y’, 2’) a t the point of observation. The field intensity a t this point is
The field of a point charge i s inversely proportional to the square of the distance and i s directed radially outward when q i s positive. This is the law
170
T H E ELECTROSTATIC FIELD
[CHAP.
111
established experimentally by Coulomb and Cavendish which is usually taken as a point of departure for the theory of electrostatics. The term “point charge” is employed here in the sense of a charge whose dimensions are negligible with respect to r. Mathematically one may imagine the dimensions of q to grow vanishingly small while the density p is increased such that q is maintained constant. I n this fashion a point singularity is generated and (23) is then valid a t all points except r = 0. There is no reason to believe that such singularities exist in nature, but it is convenient to interprct a field at sufficient distances as that which might be generated by systems of mathematical point charges. We shall have occasion to develop this concept in the subsequent section. The potential a t (z’, y’, 2’) is obtained by integrating the contributions of charge elements dq = p dv over all space. (24)
The field intensity due to a complete charge distribution in a homogeneous, isotropic medium is, therefore,
E(d, y’,
2’)
1
=
4ae
p(z, y, z)V
(:)
dv.
3.7. Convergence of Integrals.The proof of convergence of the integrals for potential and field intensity is implicit in the method by which they have been derived, but an alternative treatment will be described which may serve as a model for proofs of this kind. Since the element of integration at (2,y, z ) can coincide with (z’, y r , z’), with the result that the integrand becomes infinite at this point, it is not obvious that the integral has a meaning. It will be shown that, although an improper integral of this type cannot be defined in the ordinary manner as the limit of a sum, it can by suitable definition be made to converge absolutely to a finite value. Let the point &(z’, y’, z r ) at which the field is to be determined be surrounded by a closed surface S of arbitrary shape, thus dividing the total volume V occupied by charge into two parts: a portion Vl representing the volume within S and a portion Vz external to S. Throughout V z the integral of Eq. (26) is bounded and consequently the charge outside X contributes a finite amount to the resultant field intensity at Q , a contribution E2,
SEC. 3.71
C'ONVKRGENCE OF' INTEGRALS
171
The contribution of the charge within X to the field intensity at Q may be called El. If now the field intensity E = E l Ez at Q is t o have a definite significance, it is necessary that El vanish in the limit as S shrinks about Q. A moment's reflection will indicate t h a t this is indeed the case for, although the denominator of the integrand vanishes as the square of the distance, the element of charge in the numerator is proportional to the volume V1 which vanishes as the cube of a linear dimension. It has been tacitly assumed that thc charge density p is finite a t every point in the volume V . There exists, therefore, a number m such t h a t IpI < m, and Ip/rl < m / r , a t every point in V . It follows, furthermore, for this upper bound m that
+
The surface S bounding the volume V , is of arbitrary form, and so, to avoid the awkwardness of evaluating the integral over this region, a sphere of radius a concentric with Q is circumscribed about 81. The volume element of integration is positive; consequently the integral extended throughout V1 must be less than, or a t the most equal to, the integral ex tended throughout the volume enclosed by the circumscribed sphere.
which evidently vanishes wit,h a. The contribution of the charge within S to the field a t Q becomes vanishingly small as S shrinks about Q and hence (26) converges for interior as well as exterior points of a charge distribution. The potential integral
is also an improper integral when the point of observation is taken within the charge but, since the denominator of the integrand vanishes only 5ts the first, power of r , the proof above holds here a fortiori. If p is a bounded, integrable function of position, (31) is a cont,inuous function of the coordinates x', y', z', has continuous first derivatives, and satisfies the condition E =  V ' + everywhere. It can be shown, furthermore,
172
T H E ELECTROSTATIC FIELD
[CHAP.I11
that i f p and all its derivatives of order less than n are continuous, the potential 4 has continuous derivatives of all orders less than n 1.l
+
EXPANSION OF THE POTENTIAL I R SPHERICAL HARMONICS
3.8. Axial Distributions of Charge.We shall suppose first t h a t an element of charge q is located a t the point z = 1 on the zaxisof a rectangular coordinate system whose origin is at 0. We wish t o express the potential of q a t any other point P in terms of the coordinates of P with respect to the origin 0. The rectangular coordinates of P are 2, y, z, but since the field is symmetric about the zaxis i t will be sufficient t o locate P in terms of the two polar coordiz nates r and 6, Fig. 28. T h e distance from q to P is r2and the potential at P is, therefore, +(r, 6 )
(1)
Q > 1 = ~4ac r 2
the medium being assumed homogeneous and isotropic. r2 = ( r 2
(2)
+ l 2  2r{ cos e)*.
28.Figure to Sec. 3.8.
There are now two cases to be considered. The first and perhaps less common is t h a t in which P lies within a sphere drawn from 0 as a center through Then T < { and we shall write
r.
(3)
7.2
=
Ai[]+
(;y 
2TCOS i0
I’
.
The bracket may be expanded by the binomial theorem if
2);(I
r
 2 ? cos
01
< 1.
< 1, the rcsultant series converges absolutely and consequently the various powers may be multiplied out and the terms rearranged a t will. If the terms of the series are now ordered in ascending powers of r / { , we find
See for example KELLOGO,“Foundations of Potential Theory,” Chap. VI, Springer, 1929, or PHILLIPS, “Vector Analysis,” pp. 122 f.,Wiley, 1933.
SIX. 3.81
173
A X I A L DISTRIBUTIONS OF CHARGE
which shall be written in the abbreviated form
(5) The coeficients of r /p are polynomials in cos 0 and are known as the Legendre polynomials.
(6)
PO(cose) = 1, P~(COS e) = COS
e,
+
o  1) = +(3 cos 28 11, P3(cos e) = + ( 5 c0s3 e  3 COY e) = + ( 5 cos 38 3 cos ...................................................
p Z ( c O s e) = +(3 cos2
+
e).
The absolute value of the coefficients P, is never greater than unity; hence the expansion convergcs absolutely provided r < /(I. I n the second case P lies outside the sphere of radius {. The corresponding expansion is obtained by interchanging r and { in ( 3 ) and (5).
This last result may be obtained in a slightly different manner. Consider the inverse distance from the point z = [ t o P as a function of { and expand in a Taylor series about the origin, { = 0.
f(s) = 1 = (r2 + c2  Zr{ cos e)*,
(9)
r2
Now in rectangular coordinates, r2 is
r2
(1 1) and
=
[x2
(12) Hence.
and, sincef(0)
=
l / r , we have
+ y2 + (z  ~7~1+,
174
T H E ELECTROSTATIC FIELD
[CHAP.111
The pwkntial a t a point P outside the sphere through p can be written in either of the forms
from which it is apparent that
Finally, let us suppose that charge is distributed continuously along a length 1 of the zaxis with a density p = p @ ) . The potential a t a sufficiently great distance from the origin is
this expansion,
where p is now the total charge on the line, is evidently the Coulomb potential of a point charge q located at the origin. However, the density may conceivably assume negative as well as positive values such that the net charge (19)
is zero. The dominant term approached by the potential when r >> 1 is then
is called the dipole moment of the distribution. write
a s an axial rnultipole of nth order
I n general, we shall
SEC.3.91
175
T H E DIPOLE
3.9. The Dipole.In order that the potential of a linear charge distribution may be represented by a dipole i t is neccssary that the net charge be zerothe system as a whole is neutraland that the distance to the point of observation be very great relative t o the length of the line. We have seen that the potential dois that which would be generated by a mathematical point charge located a t the origin. do has a singularity a t r = 0, for a true point charge implies an infinite density. We now ask whet,her a configuration of point charges can be constructed which will give rise to the dipole potential 41. Let us place a point charge + q a t a point z = 1 on the zaxis and an equal negative charge  q a t the origin. According to Eq. (8) thc potential a t a distant point is
The product p = ql is evidently the dipole moment of the configuration Suppose now that 13 0, but a t the same time q is increascd in magnitudt: in such a manner that the product p remains constant. Then in the limit a doublepoint singularity is generated whose potential is
(25) everywhere but a t the origin. A direction has been associated with a point. The dipole moment is in fact a vector p directed, in this case, along the zaxis. The unit vector directed along r from the dipole towards the point of observation is again ro, and the potential is, theref ore, 1 1 pro 4(x, y, 2) =  __ = 4ae r2 4ae The field of a dipole is cylindrically symmetrical about the axis; hence in any meridian plane the radial and transverse components of field intensity are
(27)
+
1 I n (26) r = 4 z 2 $ y2 z2 and V ( l / r ) implies differentiation a t the point of observation. If the dipole wcre located at (z, y, z ) and the potential measured at
(d,y’,
z ’ ) , we should have
Eq. (22), p. 169.
+(d,y’,
2’)
=
 p * V’ 47rs
1
176
THE ELECTROSTATIC FIELD
[CHAP.I11
The potential energy of a dipole in an external field is most easily determined from the potential snergies of its two point charges. Let a charge + q be located a t a point a, and a charge  q at b displaced from t,he first by an amount 1, in an external field whose potential is +(x, y, 2). The potential energy of the system is then
i Y
(28) or as b
= Y+(U> 
Y+(b),
3 a,
(29)
li
= qd+ = plV+ = p.E
=
pE cos 0,
where 0 is the angle made by the dipole with the external field E.
J?IQ. 29.Lines
of force in a meridian plane passing through the axis of a dipole p .
The force exerted on the dipole by the external field is equal to the negative gradient of U when the orientation is fixed. (30)
F
= v(p
E>B oomtant.
On the other hand a change in orientation at a fixed point of the field also leads to a variation in potential energy. The torque exerted on a dipole by an external field is, therefore,
or vectorially, (32)
T=pXE.
3.10. Axial Multipo1es.Let us refer again to Eq. (17) for the potential of a linear distribution of charge. This expansion is valid at all points outside a sphere whose diameter is the charged line. Now the first term +o of the series is just the potential that would be produced by a point charge y located at the center of the sphere. The second term represents the potential of a dipole p located at the same origin. We shall show that the remaining terms may likewise be interpreted as the potentials of higher order charge singularities clustered at the center.
+,,
SEC.
3.101
A X I A L MULTIPOLES
177
The dipole, whose moment we now denote as p ( l ) , was constructed by placing a negative charge  q a t the origin and a positive charge +q at a point z = ZO along the axis. The two are then allowed t o coalesce such that p ( l ) = qZo remains constant. The potential of the resultant singularity is
(33) The singularity of next higher order is constructed by locating a dipole of negative moment  p ( l ) a t the origin and displacing from it another equal positive moment by the small amount 11. For the present we confine ourselves to the special case wherein the axes of both dipoles as well as the displacement 11 are directed along the zaxis. The potential of the resultant configuration is
(35)
An axial quadrupole moment is drfined as the product
(36)
p(2) = Z(p(ljZ1) = Z(qZ011).
The mathematical quadrupole is generated by letting Z0 f 0, Zl  0 , q 3 co such that tJhe product (36) remains finite. The potential of this configuration is then strictly
(37) a t all points excluding r = 0. By induction one constructs charge singularities, or multipoles, of yet higher order. I n each case a multipole of order n  1 and negative moment P("~) is located at the origin and a n equal positive multipole p(n1) displaced from it by 1,. I n the gcnrral case t o he dealt with below, the displacements are arbitrary in direction; i f , as in the present special case, all are along t,he same straight line, the multipole is said t o be axial. The potential is given approximately by the first term of a Taylor series, which again by (12) may be written
The npole moment is defined as the limit of the product p(n) = np (n1)l n1 (39)
178
when finite.
[CHAP.I11
T H E ELECTROSTATIC FIELD
I, + 0 and
~ ( n  ~+ )
in such a manner that p("')ln remains
m
The potential of a point charge of nth order is then
&=
4ae
P (
a).
SEC.3.151
SINGLELAYER CHARGE DISTRIBUTIONS
187
I n Fig. 32, C#I and its first derivative E are plotted as functions of r. Note that although E passes continuously through the surface r = a, its slopethe second derivative of 9suffers an abrupt change because of the failure in continuity of the density p . Consider next a volume distribution of dipole moment of density P. The potential a t any exterior point is
+(dly’,
2’) = 4nco 
.I 
P V(i)dv.
This can be resolved into three integrals of the type
(7)
+ l =
4nto
P,
&);1(
dv
=  a
4~ a d
pdv. r
From what has just been proved regarding (1) it follows that (6) converges within the dipole distribution as well as at points extprior to it, and i s a continuous function of position provided only P(x, y , z ) i s bounded and piecewise continuous. Since a dielectric body is equivalent to a region of dipole moment, we have here proof of the continuity of the potential across a dielectric surface without recourse to an energy principle. Across a surface of discontinuity in P, the normal derivative of (6), and consequently the normal component of E, is discontinuous. The magnitude of this discontinuity can be determined most readily by reverting from the volume distribut>ion of moment to the equivalent volume and surfacecharge distribution dcfined in Sec. 3.13. The vector E passes continuously into the volume charge, but we shall now show t h a t its normal component suffers an abrupt change in passing through a layer of surface charge. 3.16. Singlelayer Charge Distributions.Let charge be distributed over a surface S with a density w which we shall assume to be a bounded, piecewise continuous function of position on S. The potential a t any point not on S is
where T as usual is drawn from the charge element w(z, y, z) da t o the point of observation. If now (d,y’, 2’) lies o n the surface, the integral (8) is improper and its finiteness and continuity must be examined. About the point (d,y’, 2’) on S let us circumscribe a circle of radius a. If the radius is sufficiently small, the circular disk thus defined may be assumed plane. Now the potential a t (z’, y’, z’) due t o surface charges outside the disk is a bounded and continuous function of position in the vicinity of (z’, y’, 2’). Call this portion of the potential 42. There remains a contribution 41 due t o the charge on the disk itself. The
188
[CHAP.I11
T H E ELECTROSTATIC FIELD
surface density w is bounded, and we proceed as in Sec. 3.7. There exists a number m such that at every point on the disk Iw[ < m, Iw/rl < m/r.
(9) where S1 indicates that the surface int>egralis to be extended over the y’, z’) on the disk. The resultant potential at an arbitrary point (d, surface is, hence,
4
(10)
=
41
+ 42.
Both 41 and 42have been shown to be bounded, and +2 is also continuous. But 41 vanishes with the area of the disk and consequently, since 4 differs as little as desired from a continuous function, it is itself continuous. The specialization of the disk t o circular form is no restriction on the generality of the proof, since the circle may be considered to be circumscribed about a disk of arbitrary shape. T h e potential due to a surface distribution of charge i s a bounded, continuous function of position at all points, both o n and o$ the surface. The function defined by (S), therefore, passes continuously through the surface. The integral expression of the field intensity,
is continuous and has continuous derivatives of all orders a t points not on the surface, but suffers an abrupt change as the point ( d , y’, 2 ’ ) passes through S. The nature of the discontinuity may be determined directly from ( l l ) , I but we shall content oursolves here with the simple method employed in Sec. 1.13 based on the divergence and rotational properties of E. The transition of the vector E through the layer S is subject to a discontinuity defined by
E+  E
(12)
=
1 wn, €0
where n is the unit normal drawn outward from the positive face of the surface. If now by w we understand the true plus the bound charge, w  n (P+  P), then (12) is equivalent t o
+
.
+
.

(13) (eOE+ P+) n  (e0E P) n = (D,  D) n = w. This specifies at the same time the transition of E at a surface of discontinuity in a dipole distribution. 3.16. Doublelayer Distributions.I t frequently happens that the potential of a charge distribution is identical with that which might be Cf. KELLOGG, Zoc. cit.; PHILLIPS, Zoc. cit., Chap VI.
189
DOUBLELAYER DISTRIBUTIONS
Snc. 3.161
produced by a layer of dipoles distributed over a surface. We imagine such a surface distribution to be generated by spreading positive charge of density +w over the positive sitlr of a regular surface S , and an identical distribution of opposite sign on its negative side. The result is a double layer of charge separated by thc infinitesimal distance 1. The dipole moment per unit area, or surface density 7 , is a vector directed along the positive normal t o S and is dcfined as the limit
(14)
T =
n lim (wZ) as Z + 0,
w
+ cc.
The dipole moment corresponding t o an clcmcn 1 of area d a on the double layer is T da, and its contribution l o the potcntial at a fixed point (x’,y’, 2’) not on the surface is
Now
cos
e
 da measures exactly the solid angle r2 d n a t the point of observation (s‘,y’, z‘) subtendcd by the element of area da. This angle FIG. 33.The element of is positirc if the radius vector r drawn from area da subtends a positive (s’,y’, z ’ ) to the clement du makes an acute solid angle a t P I and a negative angle a t 1’2. anglc with the positive normal n. Thus in Fig. 33 thc clement du subtends a positive angle a t P1 arid a negative one a t Pz. The potential due to the entire distribution may be written
(16)

+(x’, y’,
1 47rq
2’) = 

~ ( x y,, z ) V
(:)
da =
4aeo
where Q is the solid angle subtended a t (2’)y’, z’) by the surface S. T h a t the second integral must be precrded by a negative sign is evident, if one notes that do is positive when the laycr is viewed from the lower or negative side, where the potential is certainly negative. The potential $.I has distinct v:zlues on opposite sides of a double layer. Suppose first t h a t the surface S is closed and that the distribution is of constant density, so t h a t r may be taken outside the integral sign. The positivc layer lies on the outer side of S so that z has the same direction as the positive normal. There is a wellknown theorem on solid angles which applies to this case: “If S forms the complete boundary of a threedimensional region, the total solid angle subtended by S a t P is zero, i f P lies outside the region, and 47r if P lies inside.”’ The potential 1 a t any interior point of S is, therefore, 4 =  r ; a t any exterior point €0
On the arialytic properties of solid angles see Phillips, Zoc. cit., p. 112.
190
++ = 0.
T H E ELECTROSTATIC FIELD
[CHAP.I11
The difference in potential on either side of the double layer is
++
(17)

+
1
=  7. €0
Next we observe that (17) represents correctly the discontinuity in + as one traverses the double layer also when S is an open surface. For S may be closed by adding an arbitrary surface S’. At every point within or without this closed surface the resultant potential can be resolved into two parts, a fraction due t o the distribution on S , and a fraction +’, due to that on S’. Inside the closed surface the resultant potential is again 1  7; outside, it vanishes. Thus, as we traverse the surface S, +’
+
++
€0
1
+’,due to the layer S‘, is certainly continuous across S and hence the entire discontinuity is in +, changes discontinuously by an amount
2 T ; but €0
as specified by (17). It remains to show that (17) is also valid when T is a function of position on S. About an arbitrary point P on the surface S draw a circle of radius a. Let the radius a be so small that, over the area enclosed by the circle, 7 may be assumed to have a constant value 7 0 . The potential C$ in the neighborhood of P may again be resolved into two parts, a fraction 4’ due to the infinitesimal circular disk and a fraction 4‘’ due t o that portion of the dipole layer lying outside the circle. 4‘’ is con1 tinuous at P. 4‘ on the other hand suffers a discontinuous jump of  T,, €0
+
on crossing the circular disk. The resultant potential + = +‘ 4’’ must, therefore, exhibit a discontinuity of the same amount and, in the limit as a +0, we find that (17) holds for variable distributions if for T we take the value at the point of transition through S. These results might be interpreted to mean that the work done in moving a unit positive charge around a closed path which passes once through a surface bearing a uniform dipole distribution of density r0 is 1
f  ro, depending on the direction of circuitation.
The potential in
€0
presence of such a double layer is a multivalued function; for to its value 1 at (x’, y’, 2’) one may add  mr, where m is any positive or negative €0
+
one need only return to the integer. To obtain a different value of initial point after traversing the surface S. All this appears to be in contradiction to the conservative nature of the electrostatic field. As a matter of fact, the mathematical double layer constitutes a singular surface which has no true counterpart in nature. We shall show below by means of Green’s theorem that the potential within a closed domain
SEC.
191
DOUBLELA YER D I S T R I B U T I O N S
3.161
bounded by a surface S due to external charges is identical with that which might be produced by a certain dipole distribution over S. This equivalent double layer, however, does not lead to correct values of 4 outside S. 1 The application of the integral E n d a =  q to a small right
§
€0
cylinder, the ends of which lie on cither side of the double layer, after the manner of Sec. 1.13, indicates at, once that there is no discontinuity in the normal component of the field vector E across the surface since the total charge wit.hin the cylinder is now zero.
(E+  E). n
(18)
=
0.
The line integral of E around a closed path, however, is no longer necessarily zero; consequently we anticipate a possible discontinuity in the tangential components. Let t,he ./at+ contour start at 1, Fig. 34, at which point the dipole density is r. The difference in potential between R E t p0int.s 1 and 2 is, according to (17), s FIG.34.Transition of the tangential com1 41  42 =  T. (Note th at t.he ponents of E across a double layer.
7= 
€0
intensity E is given b y the derivative of the potential, not the diference in potential across the discontinuity. The normal derivative is continuous.) If A1 is the length of the path tangential to t,he surface, t'he density at 3 is r V r . Al, and the potential difference between 4 and 3 is
+
$4

43
1
=:

(T
60
Now the quantity ( 4 2 identically zero; hence, (19)
1  7 €0
 41)+ ( 4 3
+ VT
 42)
*
Al).
+ (44  + (41 44) is 43)
+ (43  + 1 (r + Vr  Al) + (41  44) = 0.
Abbreviating 44 
42)
41
= &b+,
$3
 $2
= A$,
(19) reduces to
where t is a unit vector tangent, to S. The limit of A+/Al as 1 i 0 is the component of E in the direction of t, so that in virtue of the continuity of the normal component of E the transition of the field vector is specified by 1 E+  E = vT. (21) €0
192
[CHAP.I11
T H E ELECTROSTATIC FIELD
Clearly V r signifies here the gradient of 7 in the surface and is, therefore, a vector tangent t,o 8. The proof is subject to the assumption t h a t r and its first derivatives are continuous over S. 3.17. Interpretation of Green’s Theorem.In Sec. 3.4 it was shown that the potential a t any interior point of a region V bounded by a closed, regular surface S could be expressed in the form (22)
+(d,y‘,
2’)
da 
=
1 &.(I4 ana () da. r 
From the analysis of the preceding paragraph we are led t o interpret this result in the following way. The volume intcigral represents, of course, the contribution of the charge within S , and the surface integrals account for all charges exterior to it. However, the first of these surface integrals is also equivalent t o the potential of a single layer of charge distributed over S with a density (23)
a+ a = et
dn
and the second can rvidently be interpreted as the potential of a double layer on S whose density is
(24)
7
=
€4,
T h e charges outside S m a y be replaced by a n equivalent single and double layer, the densities of which are specified by (23) and (24), without modifying in a n y way the potential at an interior point. The potential outside S produced by these surface distributions corresponds in no way, however, t o that arising from the true distribution. On the contrary, it is most important t o note that these equivalent siirface layers, which give rise t o the proper value of the potential a t all interior points of S , are just those required to reduce both potential and field E to zero at every point outside. We observe first t h a t the single layer givcs rise t o a discontinuity in the normal derivative of 4 equal t o
The normal derivative in Eq. (23) must be calculated on the inner or negative side of S , since this alone belongs t o V. Replacing (a4/dn)by its value (23)) it follows that (a+/&)+ = 0. Furt,hermore, the discontinuity in potential due to the double layer is specified by (17), which together with (24) shows that 4+ = 0. T h a t and its derivatives vanish everywhere outside S is readily shown by applying (22) to the volume V 2 exterior to S. Since there are now no charges in V2, V24 = 0. At infinity the potential is regular, and consequently the potential within C#I
Next Page
SEC.3.181
IMAGES
193
V z is dctermined solely by the values of 4+ and (a+/&)+ on S. But, these have just been shown to vanish. The function 4(d, y’, 2’) of (22) is continuous and has continuous derivatives. The field intensity E outside X is, thtrcforc, also zero. From the foregoing discussion it is clear t h a t one may always close off any portion of a n elwtro5taf ic ficld by a surface, reducing the field and potential outside to zero, and taking account of the effect of external charges on the field within by proper single and doublclayer distributions on the bounding surface. 1t is instructive to consider these results from the standpoint of the ficld intensity E. The single layer introduces the proper discontinuity in the normal component E,, but does not affect the transition of the tangential component. The double layer, on the othrr hand, in no wise affects the transition of the normal component, but may be adjusted to introduce the proper discontinuity in Et, or according t o (211, Et+
=
0,
where 1 is any direction tangent to 8. If in particular the surface X is an equipotcntial, n X E = 0, and no dipole distribution is neccssary; the field inside S due to external chargvs can then be accounted for by a single a4 on the layer of density (J = E dn equipotential. 3.18. Images.An important application of these principles is t o be found in the theory of images. The equipotential surfaces of a pair of equal point charges, one positive and FIG. BB.Application of Green’s theorem to the theory of images. the other negative in a homogeneous dielectric of inductive capacity E , form a family of spheres whose centers lie along the line joining t h e charges. Let t h e surface S be represented by any equipotential about  q , Fig. 35. This surface thus divides all space into two distinct regions, and in view of the regularity of the potential a t infinity, Green’s theorem applies t o both, S being t h e bounding surface in either case; i~., either region may be taken as t h e “interior” of S. If, therefore, the charge  q be removed, the field in the region occupied b y +q is unmodified if a charge of density w is spread over S as specified b y (23). Inversely, if the charge + q be located as
Previous Page
194
THE ELECTROSTATIC FIELD
[CHAP.111
shown with respect to a conducting sphere, a surface charge w will be induced upon it such that it becomes an equipotential. The contribution of this induced charge to the field outside the Conductor S is now determined most simply by replacing the surface distribution by the equivalent point charge q. The charge  q is said to be the image of + q with respect to the given sphere. In case the equipotential surface is the median plane x’ = 0, the calculation is extremely simple. The potential at any point (x’,y’, 2’) to the right of x’ = 0 is (27)
+(XI,
y’,
2’)
=
1
9
The normal derivative a t x’ = 0 must be taken in the direction of the negative x’axis, since the region on the right of x‘ = 0 is to be considered the interior of the equipotential.
hence by (23) the charge density is w =
where r2 = a2
+ yf2 + z ‘ ~ . ~
1 aq, 29 r3
BOUNDARYVALUE PROBLEMS
3.19. Formulation of Electrostatic Problems.The analysis of the preceding sections enables one to calculate the potential at any point in an electrostatic field when the distribution of charge and polarization is completely specified. In practice, however, the problem is rarely so elementary. Ordinarily only certain external sources, or an applied field, are given from which the polarization of dielectrics and the surface charge distribution on conductors must be determined such as to satisfy the boundary conditions over surfaces of discontinuity. Among electrostatic problems of this type are to be recognized two classes : the homogeneous boundaryvalue problem and the inhomogeneous problem. To illustrate the first, consider an isolated conductor embedded in a dielectric. A charge is placed on the conductor and we wish to know its distribution over the surface and the potential of the conductor with respect to earth or to infinity. At all points outside the conductor the “Mathematical Theory of Electricity and * On the method of images, see JEANS, Magnetism,” 5th ed., Chap. VIII, Cambridge University Press; or MASONand WEAVER,“The Electromagnetic Field,” pp. log#., University of Chicago Press, 1929.
SEC.3.191
FORMULATION OF ELECTROSTATIC PROBLEMS
195
potential must satisfy Laplace’s equation. At infinity it must vanish (regularity), and over the surface of the conductor it must assume a constant value. We shall show that these conditions are sufficient t o determine uniquely. The density of surface charge can then be determined from the normal derivative of +, subject to the condition t h a t JW da over the surface of the conductor must equal the total charge. An inhomogeneous problem is represented by the case of a dielectric or conducting body introduced into the $xed field of external sources. A charge is then induced on the surface of a conductor which will distribute itself in such a manner that the resultant potential is constant over its surface. The integral J w d u is now zero. Likewise there will be induced in the dielectrics a polarization whose field is superposed on the primary field t o give a resultant field satisfying the boundary conditions. A schedule can be drawn up of the conditions which must be satisfied in every boundaryvalue problem. To simplify matters we shall assume henceforth that the dielectrics are isotropic and homogeneous except across a finite number of surfaces of discontinuity.
+
V2+ = 0 at all points not o n a boundary surface or within external sources; $I i s continuous everywhere, including boundaries of dielectrics or of conductors, but excluding surfaces bearing a double layer; + i s finite everywhere, except at external point charges introduced as primary sources; e2
fg)2
el
= 0
across a surface bounding two dielectrics;
a+ at the interface of a conductor and dielectric; an O n the surface of a conductor either ( a ) + i s a known constant +i, or (b) + i s a n unknown constant and e  = w
+ i s regular at infinity provided all sources are within a finite distance of the origin. In (4) it is assumed t h a t the interface of the dielectrics bears no charge, as is almost invariably the case. The normal is directed from (1) to (2), and in ( 5 ) from conductor into dielectric. An electrostatic problem consists in finding among all possible solutions of Laplace’s equation the particular one that will satisfy the conditions of the above schedule over the surfaces of specified conductors and dielectrics.
196
THE ELECTROSTATIC FIELD
[CHAP.
111
3.20. Uniqueness of Solution.Let 4 bc a function which is harmonic (satisfies 1,aplacc’s equation) and which has continuous first and second derivatives throughout a region V and ovcr its bounding surface S. According t o Green’s first identity, (5), page 165, 9 satisfies
Suppose now that 9
=
0 on the surface 8. Then
sv
dv
(v+)2
= 0.
The
integrand is an essentially positive quantity and consequently V 4 must vanish throughout 8. This is possible only if 4 is constant. Over the boundary 4 = 0 and, since by hypothesis 4 is continuous throughout V , it follows that 6 = 0 over the ent,ire region. Now let 41 and 4 2 be two functions that arc harmonic throughout the closed region V and let
Then if and 42are equal over the boundary S, their difference vanishes identically throughout V . A function which i s harmonic and which possesses continuous first and secondorder derivatives in a closed, regular region T7 i s uniquely determined by its values o n the boundary X. Consider next a system of conductors embedded in a homogeneous delectric whose potentials are specified. We wish t o show t h a t the potential a t every point in space is thereby uniquely determined. The reasoning of the preceding paragraph is applied t,o a volume V which is bounded interiorly by the surfaces of the conductors, and on the exterior by a sphere of very large radius R. Lct us suppose that there are two solutions, 41 and &, that satisfy the prescribed boundary conditions. Then on the surfaces of the conductors, 4 = $1  $2 = 0. Since $1 and $2 are assunied t o be solut,ions of the problem, they must bot,h satisfy the conditions of See. 3.19; hence their differcnce 4 is harmonic, has the value zero over the conductors, and is rcgular a t infinity. Thc surface integral on the righthand sidc of (1) must now be extended over both the interior and exterior boundaries. Over the interior boundary 9 = 0 ; hence the integral vanishes. On the outer sphere 8 4 / 8 n = &$/ah!. If €2 approaches infinity, 4 vanishes as 1/R and a+/aR as 1/R2. The integrand 9 89  thus vanishes as 1 / R 3 , whereas the area of the sphere an becomes infinite as R2. The surface integral over the exterior boundary is, therefore, zero in the limit R+ 00. It follows too t h a t the volume integral in (1) must vanish when extended over the entire space exterior to the conductors, and we conclude as before t h a t if the two functions $1
SEC.
3.211
SOLUTION OF LAPLACE’S EQUATION
197
and 4 2 are identical on the boundaries they are identical everywhere: there is only one potential function that assumes the specified constant values over a given set of conductors. The lefthand side of (1) m a y also be made to vanish b y specifying that d4ldn shall be zero over th(1 enclosing boundary S. Then throughout V we h a w again v+ = 0, whence i t follows that is constant everywhere, although not necessarily zero since the condition &plan = 0 does not imply the vanishing of 4 on S . One concludes, as above, that if the normal derivatives d & / a n and d & / d n of two solutions are identical on the boundaries, the solutions themselves can differ only by a constant. I n other words, the potential i s uniquely determined except for a n additive constant by thp values of the normal derivative o n the boundaries. But the normal derivative of the potential is in turn proportional to the surface charge density and, consequently, there is only one solution corresponding to a given set of charges on the conductors. In case there are dielectric bodies present in the field the requirement t h a t the first derivatives of +, as well as itself, shall be continuous is no longer satisfied and (1) cannot be applied directly. However, the region outside the conductors may be rcsolved into partial volumes V , bounded by the surfaces S , within which the dielectric is homogeneous. To each of these regions in turn, (1) is then applied. The potvntial is continuous across any surface S , and the derivatives on one side of S, are fixed in terms of the derivatives on the other. It is easy to see that also in this more general case the electrostatic problem i s completely determined by the values either of the potentials or of the charges specified on the conductors of the system. 3.21. Solution of Laplace’s Equation.It should be apparent t h a t the fundamental task in solving an electrostatic problem is the determination of a solution of Laplace’s equation in a form that will enable onc to satisfy the boundary conditions by adjusting arbitrary constants. There are a certain number of spccial methods, sucli as t h r method of images, which can sometimes be applied for this purpose. Apart from the theory of integral equations, the only procedure that is bolh practical and general in character is the method known as “separation of the variables.” Let us suppose that the surface S bounding a conductor or dielectric body satisfies the equation
+
+
We now introduce a set of orthogonal, curvilinear coordinates ul, u2,u3, as in See. 1.14, such that one coordinate surface, say u1 = C, coincides with the prescribed boundary ( 3 ) . If then a harmonic function +(ul, u2,u3)can be found in this coordinate system, it is evident that the normal derivative a t any point on the boundary is proportional to the derivative
198
THE ELECTROXTATIC FIELD
[CHAP.
111
of 4 with respect to u l , and th at the derivatives with respect to u2and u3 are tangential. According to (82), page 49, Laplace's equation in curvilinear coordinates is
(4) Let us suppose th at the scale factors hi satisfy the condition
9
 = Mif,(u1)f,(u2)f3(u3).
(5)
hi
Each of the product functions fi(ui) depends on the variable ui alone, but the factor M i does not contain ui. It may, however, depend on the other two. We next assume that likewise may be expressed as a product of three functions of one variable each.
+
+ = F~(U')F,(U~)P~(U~).
(6)
Then (4) can be written in the form
(7)
If finally the M ; are rational functions, Eq. (7) can be resolved into three ordinary differential equations. The method is best described by example. 1. Cylindrical Coordinates.From (I), page 51, we have
whence by inspection
(9)
fl
=
r,
f2
= f3 =
1,
MI
=
M3
=
1,
M2
=
1 7.
Equation (7) becomes
The first two terms of (10) do not contain z ; the last term is independent of
r and 4. A change in x cannot affect the first two terms and therefore the last term must be constant if (10) is to be satisfied identically for any range of z. 1 d2F,  c,. Fa dz2
SOLUTION OF LAPLACE'S EQUATION
SEC. 3.211
199
The arbitrary constant CI, called the separation constant, has been chosen negative purely as a matter of convenience and the partial derivative has been changed to a total derivative since FS is a function of z alone. Upon replacing the third term of (10) by C1 and multiplying by r2 one obtains
It is again apparent that the second term of (12) is constant, leading to two ordinary equations,
d2F,
(13)
+ CzFz
=
0,
The equations for Fz and F3 are satisfied by exponential functions of real or imaginary argument depending on the algebraic sign attributed to the separation constants; F1 is a Besselfunction. In the not uncommon case of a potential 9 which is independent of z, we find F3 constant, C1 = 0, and in place of (14)
2. Spherical Coordinates.From
whence (17)
fl
=
r2,
.f2
(2), page 52, we have
= sin 8,
f3
=
1,
where $, has in this case been employed in place of 9 to represent the azimuthal angle. Separation of the variables leads to the three ordinary equations 1 d (rz?) = cl, F 1 dr sin 6 d ___  (sin 0 = cz  c1sin2 0, F z d0
s)
E3 + C2F3 = 0. dlL2 Of these three, only the Leqendre equation (19b) is of any complexity.
200
T H E ELECTROSTATIC FIELD
3. Elliptic Coordinates.According
[CHAP.I11
t,o (3), page 52, we have
which by inspcction lead t o
The Laplace equation for t,hc elliptic cylinder is
which upon separation gives
1 d2F3
(23a)

c1,
Both F1 and F , are Mathieu junctions, but simplify notably when C1 = 0 as is the case when 4 is uniform along thc lcngth of the cylinder. 4. Spheroidal Coordinates.According t o (6), page 56, we have
di c (_ P_ _ VZ)_ _ (24) T 4 c(tZ l),  __ hl fl = .p  1, fz = 1  7 2 , f 3 = 1, (25) C C C(t2  ~ M , =   . ~ ,M 3 == ______ . Mi = (2  1 1  172, [(t2 1)(1 s"1' Laplace's equation rcduces to ~
~
ded in a homogeneous, isotropic dielectric of inductive capacity € 2 . At z = .( > r1 on the zaxis, there is located a point
 32=t
..
;?(r,
e)
1931. 3 WATSON, “Treatise on the Theory of Hessel Functions,” Cambridge Ilniversity Press, 1922. WHITTAKER and WATSON,“ Modern Analysis,” Cambridge University Press, 1922. 6 JEANS, loc. cil. 6 SMYTHE, “Static and Dynamic Electricity,” McGrawHill, 1939.
202
THE ELECTROSTATIC FIELD
[CHAP.
111
+
~ ( C OO),S imposing on C1 the value C1 = n(n l), n = 0, 1, 2, . . . . Under these circumstances (19u) is satisfied by either rn or r%l. The condition that the potential be singlevalued is fulfilled by the function
where an,, b,, are arbitrary constants. But 41 must also be regular a t infinity, which necessitates our placing unm= 0. Furthermore, the primary potential +o is symmetric about the zaxis; consequently m = 0 in this case. The potential of the induced distribution is, therefore, represented by the series m
The expansion of the primary potential 40 in spherical coordinates was carried out in Sec. 3.8. When r < {, (3)
and the resultant potential on the surface r =
r1
is
Now is a constant, and since (4) must hold for all values of 8, it follows that the coefficients of P,(cos 0 ) must vanish for all values of n greater than zero. The coefficients b, are thus determined from the set of relations
(5) At any point outside the sphere
To determine the charge density, we compute the normal derivative on the surface.
(7)
SEC.3.221
203
CONDUCTING SPHERE I N FIELD
and the induced charge density is
The total charge on the sphere is q1 =
(9)
I"d""
ur: sin 0 d0 d#.
Now a wellknown property of the Legendre functions is their orthogonality.
6"
(10)
~ , ( c o se)P,(cos
e) sin e ae
= 0,
n # rn.
We may take m = 0, Po(cos 0) = 1, and learn that P,(cos 0) vanishes when integrated from 0 to T if n > 0. q1
(11)
=
q
rl
l
+4 ~ ~ ~ ~ 1 4 . .
The potential of the sphere is, therefore,
q1 representing an excess charge that has been placed on the isolated ~ be put equal to zero. sphere. If the sphere is grounded, $ Jmay It is interesting to observe that the potential 91 of the induced distribution at any point outside the sphere is that which would be produced by a charge 4ae2bo = q1, a dipole moment 4n& =  q r: 51 etc., all located
l
a t the origin and oriented along the zaxis (Sec. 3.8). There is, however, another simple interpretation. The point x = l', Fig. 36, where ll' = r;, is said to be the inverse of z = { with respect to the sphere. The reciprocal distance from this inverse point to the point of observation is, by (8), Page 173,
Thus the resultant potential ( 6 ) may be written
Outside the sphere the potential is that of a charge q at z = l, an image r1
charge q' = q  located a t the inverse point z =
l
b', a charge q1 (which
204
T H E ELECTROSTATIC FIELD
[CHAP.
111
is zero when the sphere is uncharged) located a t the origin, and a charge q
r
at the origin which raises the potential of th e floating sphere to the
proper value in th e external field. 3.23. Dielectric Sphere in Field of a Point Charge.At any point outside the sphere, whose conductivity is zero and whose inductive capacity is €1, the potential is (15)
The notation ++ mill be employed to denote a potential or field a t points outside, or on the positive side, of a closed surface. The expansion of +1 in inverse powers of r does not hold within the sphere, for the potential must be finite everywhere. We resort, therefore, t o a n alternative solution of Laplace's equation obtaincd from (1) by putting the coefficients b,, equal to zero. At any interior point the resultant potential is m
4 includes the contribution of the charge q as well as that of the induced polarization, for the singularity occasioned by this point charge lies outside thc region to which (16) is confined. I n the neighborhood of the surface, r < jcj, so that +o can be expanded as in (3). Just outside the sphere,
Across the surface (r
=
r1).
A calculation of the coefficients from these boundary conditions leads to 2n+1 , 47rp+1nE1 (n i)Z (19) q r?+l _ €1 n _ b, = . L _ 47r {"+I €2 nE1 (n 1)ez The potential a t any point outside the sphere is
a,
4= 
+ +
+ +
while a t a n interior point
205
SPHERE I N A PARALLEL FIELD
SEC.3.241
It is important to observe that a region of infinite inductive capacity behaves like an uncharged conductor. As e l becomes very large, it will be noted that the first term in the series (20) corresponding to n = 0 vanishes, so that in the limit one obtains (6) for the case q 1 = 0. 3.24. Sphere in a Parallel Field.As the point source q recedes from the origin, the field in the proximity of the sphere becomes homogeneous and parallel. We shall consider the case of a sphere embedded in a dielectric of inductive capacity e2 under the influence of a uniform, parallel, external field Eodirected along the positive zaxis. The primary potential is then (22)
(Po =
Eox = or
cos 0 = EorPl(cos 8).
Note that 4 0 is no longer regular a t infinity, for the sourcc itself is infinitely remote. The potential outside the sphere due to either induced surface charge or polarization is again P,(cos 8) n=O
If the sphere is conducting, the resultant potential on its surface and throughout its interior is a conshnt &.
is independent of 0; whence it follows that bo
=
bl
Eor cos 8
=
r!Eo, b, cos 8
= 0
whenn
> 1.
+ Ear: __+ 4*.: r2
The charge density and the total charge are respectively
(27)
w =
3e2E0cos 8 f
€24 )'
r1
q1 = 4 ~ r ~ e ~ 4 ~ .
The potential of the induced surface charge is, therefore, that of a dipole of moment p = 4mzEor:; a moment, in other words, which is proportional to the volume of the sphere. To this is added the potential of q1 in case the sphere is charged. If the sphere is a dielectric of inductive capacity e l , the potential at an interior point is of the form (16). To satisfy the boundary conditions (18), the coefficients must now be
206
THE ELECTROSTATIC FIELD
[CHAP.I11
The resultant potential is then
Within the sphere the field i s parallel and uniform.
The dielectric constant ~1 of the sphere may be either larger or smaller than K ~ . Thus the field within a spherical cavity excised from a homogeneous dielectric K Z is
Next we note that the induced field outside is that of a dipole oriented along the zaxis whose moment is
Apparently even a spherical cavity behaves like a dipole. This effect may be readily accounted for by recalling that thc walls of the cavity bear a bound charge of density w’ = n Pz, wher3 P2 is the polarization of the external medium. In the case of a dielectric sphere in air, € 2 = €0. The polarization of the sphere is then K1  1 Pi = E O ( K ~1)E = 3 ___ (33) K1 2 roEo’ and its dipole moment

+
The energy of this polarized sphere in the external field is
The dielectric polarization modifies the field within the sphere. It will be convenient to express this modification directly in terms of P1. A depolarixing factor L is defined by

E = Eo LP1. (36) From (30) and the relation PI= C O ( K I  l)E, one calculates for a sphere in a parallel external field
SEG.3.251
FREE CHARGE O N A CONDUCTING ELLIPSOID
207
(37) I n the case of a sphere immersed in air K~ = 1, L
FIG.37a.Conducting sphere in a parallel field. The external medium is air.
=
.1
3Eo
FIG.37b.Dielectric sphere in a parallel field. The external medium is air.
PROBLEM OF T H E ELLIPSOID
3.26. Free Charge on a Conducting Ellipsoid.In ellipsoidal coordinates Laplace’s equation reduces by virtue of Eq. (135), page 59, to
The properties of the ellipsoidal harmonics that satisfy this equation have been extensively studied, but we shall construct here only certain clementary solutions that will prove sufficient for the problems in view. Consider first a conducting ellipsoid embedded in a homogeneous dielectric Q. The semiprincipal axes of the ellipsoid are a, b, c. It carries a total charge q, and we assume initially that there is no external field. We wish t o know the potential and the distribution of charge over the conducting surface. T o solve this problem a potential function must be found which satisfies (I), which is regular a t infinity, and which is constant over the given ellipsoid. Now 4 is the parameter of a family of ellipsoids all confocal with the standard surface .$ = 0 whose axes have the specified values a, b, c. The variables 9 and { are the parameters of confocal hyperboloids and as such serve to measure position on any ellipsoid 4 = constant. On the surface ( = 0, therefore, q5 must be independent of q and {. If we can find a function depending only on E which satisfies (1) and behaves properly a t infinity, it can be adjusted to represent the potential correctly a t any point outside the ellipsoid C; = 0.
208
T H E ELECT ROST A TIC FIELD
Let us assume, then, that
+ = +(E).
[CHAP.111
liaplace’s equation reduces to
which on integration leads to
(3) where C1 is a n arbitrary constant. The choice of the upper limit is such as to ensure the proper behavior a t infinity. When t becomes very large, REapproaches t $and 2c1 (4)
77
On the other hand, the equation of an ellipsoid can be written in the form
+ +
If r2 = x 2 y2 z 2 is the distance from the origin to any point on the ellipsoid t , it is apparent that as ( becomes very large 4 + r2 and hence a t great distances from the origin
The solution (3) is, therefore, regular a t infinity. Moreover (6) enables us to determine a t once the value of C1; for it has been shown that, whatever the distribution, the dominant term of the expansion a t remote points is the potential of a point charge at the origin equal to the total charge of the distributionin this case p. Hence C1 = ,9 and the 8aez
potential at any point is
(7) The equipotential surfaces are the ellipsoids t = constant. Equation (7) is an elliptic integral arid it,s values have been tabulated.‘ To obtain the normal derivative we must remember that distance along a ciirvilincar coordinate u’is measured not by du’ but by hl du’ (See. 1.16). In ellipsoidal coordinates
a _ +   l_a +  
(9)
dn 1
hldt
4
~
~ T C Z
_

1 ~
_
~
~ {)~

.
~
)
(
See for example JahnkeEmde, “Tables of Functions,” 2d ed., Teubner, 1933
~
SEC.3.261
209
CONDUCTING ELLIPSOID I N A PARALLEL FIELD
The density of charge over the surface 5 = 0 is
If now in the three equations (132), page 59, defining x, y, z in terms of (, 7, we put E = 0, it may be easily verified that
r,
Consequently, the charge density in rectangular coordinates is
Several special cases are of interest. If two of the axes are equal, the body is a spheroid and (7) can be integrated in terms of elementary functions. Thus, if a = b > c, the spheroid is oblate and
When c = 0, the spheroid degenerates into a circular disk. On the other hand, if a > b = c the spheroid is prolate and we find for the potential
The eccentricity of a prolate spheroid is e =
J1 
c>’ 
.
&e+1,
the spheroid degenerates into a long thin rod. 3.26. Conducting Ellipsoid in a Parallel Field.We assume first that a uniform, parallel field Eo is directed along the xaxis, and consequently along the major axis of the ellipsoid. The potential of the applied field is
the value of x in ellipsoidal coordinates being substituted from Eqs. (132), page 59. This primary potential is clearly a solution of Laplace’s equation in the form of a product of three functions,
It is not, however, regular a t infinity.
210
THE ELECTROSTATIC FIh’LD
[CHAP.
111
Now if the boundary conditions are to be satisfied, the potential 61 of the induced distribution must vary functionally over every surface of the family f = constant in exactly the same manner as $0. It differs from b0 in its regularity at infinity. We presume, therefore, that 41 is a function of the form bl = CZGI(f)F2(7)F3({),
(17) where (18)
Fz(7) = d 7
+ a2,
Z73G1
=
d m .
To find the equation satisfied by Gl(f) we need only substitute (17) and (18) into (l),obtaining as a result
(19) Equation (19) is an ordinary equation of the second order and as such possesses two independent solutions. One of these we know already to be F1 = z / E u2. There is a theorem’ which states that if one solution of a secondorder linear equation is known, an independent solution can be determined from it by integration. If y l is a solution of
+
then an independent solution yz is given by
In the present instance
The limits of integration are arbitrary, but G1(f)is easily shown to vanish properly at infinity if the upper limit is made infinite. The potential of the induced charge is, therefore, (24)
* See 1927.
for example Ince, “Ordinary Differential Equations,” p. 122, Longmans,
SBC.3.271
DIELECTRIC ELLIPSOID I N PARALLEL FIELD
21 1
The constant Cz is determined finally from the condition that on the ellipsoid [ = 0 the potential is a constant
At any external point the potential is
As in the analogous problem of the sphere, the constant +a can be calculated in terms of the total charge on the ellipsoid. The integrals occurring in (26) are elliptic and of the second kind.’ In case the field is parallel to either of the two minor axes it is only necessary to replace the parameter a2 above by b2 or c2. Thus the potential about a conducting ellipsoid oriented arbitrarily with respect to the axis of a uniform, parallel field Eo can be found by resolving Eo into three components parallel to the principal axes of the ellipsoid and then superposing the resulting three solutions of the type (26). 3.27. Dielectric Ellipsoid in Parallel Field.It is now a simple matter to calculate the perturbation of a uniform, parallel field due to a dielectric ellipsoid. We shall assume that the inductive capacity of the ellipsoid is ~ 1 ,and that it is embedded in a homogeneous medium whose inductive capacity is again €2 The applied Eo is directed arbitrarily with respect t o the reference system and has the components Eo,, Em, E o along ~ the axes of the ellipsoid. Consider first the component field E g Z . Outside the ellipsoid the resultant potential must exhibit the same general functional behavior as in the preceding example, and will differ from it only in the value of the constant Cz. In this region, therefore,
The variable s has replaced [ under the integral to avoid confusion with the lower limit. The interior of the ellipsoid corresponds to the range c2 5 I0 if a 2 b 1 c. In this region 3 must vary with 77 and < as determined by the function Fz(v)F3( b = c.
(45) where e
=
=
lm +
JT 1
(s
1 (2, + + a2)a  ~ a3e3
ds b2)(s
is the eccentricity.
As e + 1, the spheroid degen
erates into a long, needleshaped cavity and the product ab2A1 approaches zero. A t any point inside this cavity E = Eo: the field intensity E i s exactly the same as that prevailing initially in the dielectric. These cavity definitions of the vectors E and D in a ponderable medium were introduced by Lord Kelvin. Obviously a direct measurement of E or D in terms of the force or torque exerted on a small test body is not
SEC. 3.291
215
TORQUE EXERTED ON A N ELLIPSOID
feasible within a solid dielectric, and if a cavity is excised the field within it will depend on the shape of the hole. Our calculations have shown, however, that the electric field intensity within the dielectric is exactly that which might be measured within the needleshaped opening of Fig. 38b, whereas the field measured within the flat slit of Fig. 3% differs from D within the dielectric only by a constant factor. 3.29. Torque Exerted on an Ellipsoid.The intensity of an electrostatic field may be measured by introducing a small test body of known shape and inductive capacity suspended by a torsion fiber and observing the torque. Inversely, if the intensity of the field is known, such an experiment may be employed to determine the inductive capacity or susceptibility of a sample of dielectric matter. In general the field and polarization throughout the interior of the probe are nonuniform and an accurate computation is difficult, or impossible. On the other hand, the advantages of an ellipsoidal test body for these purposes are obvious. The polarization of the entire probe is constant and the applied torque depends essentially only on its volume and its inductive capacity. According to Eq. (49), page 113, the energy of a dielectric body in an external field is
where as above E denotes the resultant field inside the body and Eo the initial field. If the body is ellipsoidal and Eo is homogeneous, we have by (33) and (37):
+
E,"z abc
1+ 2 and, since the volume of an ellipsoid is +rabc,
U
(48)
=
~1

K~
(a) A3
.
&abc(E2  el)E Eo.
This energy depends not only upon the intensity of the initial field but also upon the orientation of the principal axes with respect to the field. Let the vector 60 represent a virtual angular displacement of the ellipsoid about its center and T the resultant torque exerted by the field. Both T and 60 are axial vectors (page 67), and the components SW,, SW,, SW, are the angles of rotation about the axes x, y, z, respectively. The work done in the course of such a rotation is (49)
SW
=

T 60
=
T , SW,
+ T,SW, + T,6
~ .
216
[CHAP.I11
THE ELECTROSTATIC FIELD
This work must be compensated by a decrease in the potential energy U . I n virtue of the homogeneous, quadratic character of (47), we may write
6U = $ T U ~ C ( ~Z ~1)6(E* Eo) = + T u ~ c ( E ~EI)E * 6Eo. (50) The reference axes have been chosen to coincide with the principal axes of the ellipsoid. Relative t o this system fixed in the body the variation of Eo corresponding to a virtual rotation 60 is 6Eo = Eo X 6 ~ 3 (51) whence for the energy balance we obtain (52)
6br = + T u ~ c (c ~e1)E
 (Eo X 60) = + T U ~ C ( E~
ei)(E X Eo) * 60
=
6W,
and from the arbitrariness of the components of rotation i t follows that the torque exerted by the field is
T = gTabc(E1  EZ)E X ED. (53) Thc components of this torque are T,
=
+ r ( ~ b c (E* )~
E2I2
€2
EoyEoz
A3  A ,
l+XabCKi  K 2 A 3 ) ] 7
1+
K2
(54)
T,
€2 = % ( u ~ c ) ’___ EoJoz (€1

€2
A1
[(
I + abc r  KI K2
T,
=
+ r ( ~ b c (€1 )~
€212
€2
K2
 A3
(
abc K i  Kz A 3 ) 1’T K 2
EoSoU
r
A,  A,
1
T o investigate the stability of the ellipsoid we must determine the relative magnitudes of the constants A1, Az, AB. I n the first place, it is clear from their definition, (32) and (36), that all three are positive, whatever the values of a, 6, c. It is easy t o show, furthermore, that the order of their relative magnitudes is thc inverse of the order of the three parameters. That is, if a > b > c , A1 < A 2 < A S . Next, one finds that the sum of the three integrals can be reduced to a simple integral when u = R: is introduced as a new variable, giving
(55)
A1
+ + A2
A3
=
2 . Sc,
217
PROBLEMS
whence from the essentially positive character of the constants it follows that 2 0 I A  < 9 ( j = 1, 2, 31, (56) ’  abc a b c ~ 1 K~ (j = 1, 2, 3). 1 f  Z  P Ai > 0, (57) KZ The denominators of (54) are, therefore, positive both when € 1 > ez and when e l < €2, and the direction of the components of torque i s independent of the relative magnitudes of € 1 and e z . If the applied field is parallel to any of the three principal axes, all components of torque arc zero, so that these constitute three directions of equilibrium. The stability of equilibrium depends on the direction of the torque and this, we see, depends solely on the sign of A i  A k , which in turn is determined by the relative magnitudes of the axes a, b, c. Thus A t  A z is positive if b > c, negative when c > b. The components of torque are such as t o rotate the longest axis into the direction of the field by the shortest route. An ellipsoid whose major axis i s oriented along the applied jield i s in stable equilibrium; the equilibrium positions of the minor axes are unstable. Problems 1. The coordinates 6,7,< are obtained from the rectangular coordinates 5, y, z by the transformation
5
+ i?
= f(z
+ iy),
r
= 2,
where f is any analytic function of the complex variable z following properties of this transformation:
+
+ iy. +
Demonstrate the
a. The differential line element is ds2 = h2(,dt2 dq2) dr2, where h = l/lj’l, f’ denoting differentiation with respect to the variable z iy; b. The system 5, 7, is orthogonal; c. The transformation is conformal, so that every infinitesimal figure in the zyplane is mapped as a geometrically similar figure on the .$?plane. Show that in these coordinates the Laplacian of a scalar function 4 assumes the form
+
and find expressions for the divergence and curl of a vector. 2. With reference t o Problem I, discuss the coordinates defined by the following transformations:
218
[CHAP.111
THE ELECTROSTATIC FIELD
+
4
(5)
~ T J=
In
3. A set of ring or toroidal coordinates A,
x
= r cos
+,
y = r sin
+ iY + a ,
5
x+iyu
p,
+ are defined by the relations
+,
z =
sin p cosh h cos
+
9
p
where r =
cosh
sinh X cos
+
1.1
Show that the surfaces, $ = constant, are meridian planes through the zaxis, that the surfaces, h = constant, are the toruses whose meridian sections are the circles
+
22
and t h a t the surfaces, circles
 22 coth X
22
+ 1 = 0,
constant, are spheres whose incridian sections are the
p =
x2
+ z2 + 22 t a n p  1 = 0.
Show that the system is orthogonal apart from ccrtain exceptional points, and find these points. Find the expression for the differential element of length and show that
Is Laplace's equation separable in these coordinates? 4. Let F ( x , y, z ) = h represent a family of surfaces such that F has continuous partial derivatives of first and second orders. Show that a neccssary and sufiicient condition that these surfaces may be equipotentials is
Show that if this condition is fulfilled the potential
wherej(h) is a function of X only.
is
where c1 and c2 are constants. 6. Show that $ = F ( z iz cos u
+
an+ ax2
+ i y sin u ) is a solution of Laplace's equation
+
2 %* = +
ap
a22
in three dimensions for all values of the parameter u and for any analytic function F . Show further that anv linear combination of 2n 1 independent particular solutions can be expressed by the integral
+
$ =
J:
(z T
+ iz cos u + iy sin u ) " f , ( u ) du,
219
PROBLEMS
where fn(u) is a rational function of e”, and finally that every solution of Laplace’s equation which is analytic within some spherical domain can be expressed as an integral of the form J. =
s”
*
f(z
4ix cos u + iy sin u,u) du.
(See Whittaker and Watson, “ Modern Analysis,” Chap. XVIII.) 6. Charge is distributed along an infinite straight line with a constant density of q coulombs/meter. Show that the field intensity a t any point, whose distance from the line is r, is
and that this field is the negative gradient of a potential function 1 ro +(z, y) =  q In 1 2se r
+
where ro is an arbitrary constant representing thc radius of a cylinder on which = 0. From these results show that if charge is distributed in a twodimensional space with a density w(x, y) the potential a t any point in the zyplane is
where r = d ( z ’
x
+
) ~ (y’  y)’ and that +(x, y) satisfies a26 a’+ $.  =
ax2
ay2
1 4x9 Y).
Show furthermore t h a t this potential function is not in general regular at i n h i t y , m. but that if the total charge is zero, so that J w du = 0, then r+ is bounded as r 7. Charge is distributed over a n area of finite extent in a twodimensional space with a dcrisity w(x, y). Express the potcntinl a t an external point z‘, y‘ as a power serics in r, the distance from a n artiitrary origin to the fixed point x‘, y‘. Show t h a t the successive terms are the potentials of a series of twodimensional multipoles. 8. Let C be any closed contour in the xyplane bounding a n area S and let + b e a twodimensional scalar potential function. By Green’s theorem show that f
where u formula
=
4(x’, y’) if z‘, y‘ is an i n h i o r point, and u = 0 if d,y’ is exterior.
a*+ + a’+ ax2 ag
V Z =~
9
r = 2/(xf
 2)’ + (y’  y
I n this
) ~
and 7~ is the normal drawn outward from the contour. 9. A circle of radius a is drawn in a twodimensional space. Position on the circle is specified by an angle 8‘. The potential on the circle is a given function +(u, 6 ’ ) .
220
T H E ELECTROSTATIC FIELD
[CHAP.TI1
Show that a t any point outside the circle whose polar coordinates are r, e the potential is
This integral is due to Poisson. 10. An infinite line charge of density q per unit length runs parallel to a conducting cylinder. Show that the induced field is that of a properly located image. Calculate the charge distribution on the cylinder. 11. The radii of two infinitely long conducting cylinders of circular cross section b. The are respectively a and b and the distance between centers is c, with c > a external medium is a fluid whose inductive capacity is B . Thc potential difference between the cylinders is V volts. Obtain expressions for the charge density and the mechanical force exerted on one cylinder by the other per unit length. Solve by introducing bipolar coordinates into Laplace's equation and again by application of the method of images. 12. An infinite dielectric cylinder is placed in a parallel, uniform field Eo which is normal to the axis. Calculate the induced dipole moment per unit length and the depolarixing factor L (p. 206). 13. Two point charges are immersed in an infinite, homogeneous, dielectric fluid. Show that the coulomb force exerted by one charge on the other can be found by evaluating the integral
+
F
=
[

BE(E n) 
2
[Eq. (72), p. 1521 over any infinite plane intersecting the line which joins the two charges. 14. Two fluid dielectrics whose inductive capacities are € 1 and c 2 meet a t an infinite plane surface. Two charges p l and qz are located a distance u from the plane on opposite sides and the line joining them is normal to the plane. Calculate the forces acting on q1 and qz and account for the fact that they are unequal. 16. A conducting sphere of radius T I , carrying a charge Q, is in the field of a point charge q of the same sign. Assume q Q. Plot the force exerted by the sphere on q as a function of distance from the center and calculate the point at which the direction reverses. 16. A charge q is located within a spherical, conducting shell of radius b a t a point whose distance from the center is a, with a < b. Calculate the potential of the spherr, the charge density on its internal surface, and the force exerted on q. Assume the sphere to be insulated and uncharged. 17. Find the distribution of charge giving rise to a twodimensional field whose potential a t any point in the zyplane is
a),
(T
< a).
P2 T
2Pl Bor sin 8, +. P1+ P2
By means of (11) the field may be calculated.
H o sin 8,
The nature of the field inside the cylinder is made somewhat clearer by a transformation to rectangular coordinates. (17)
H,
=
H , cos 8  H o sin 8,
H , = H , sin B
+ H s cos 8.
The magnetization of the cylinder is given by the relation
M
 P1  P o H. PO
Evidently the magnetization induced by the applied field Bo is in the direction of the positive xaxis. Upon this is superposed the magnetization induced by the current I. 4.21. Force Exerted on the Cylinder.If the medium supporting the wire is fluid, or a nonmagnetic solid, the force per unit length exerted by the field can be computed from (15), page 155. Resolved into rectangular components, this becomes
(20)
The rectangular components of the field outside the cylinder are first calculated from (15) and (17).
262
T H E MAGNETOSTATIC FIELD
H$ (21)
=
H+ 
Ho P1 P1
[CHAP.IV
I sin 9,  ma2 + ~+ Ho cos 29  2nr I cos 8, + H o sin 29 + 27rr PI
PZT2
P1
PPa2
P2T2
while H 2 is equal to the sum of the squares of these two components.
Upon int,roducing (21) and (22) into (20) and evaluating the integrals, one obtains the components of the total force exerted by the field upon the wire and the current it carries.
Fz
(23)
FU
= 0,
=
p,HoI.
That the induced magnetic moment of the wire contributes nothing to the resultant force follows from the uniformity of the applied field. If this field, on the other hand, were generated by the current in a neighboring conductor, the distribution of induced magnetization would no longer be symmetrical; a force would be exerted on the cylinder even in the absence of a current I . Problems
1. An infinitely long, straight conductor is bounded externally by a circular cylinder of radius a and internally by a circular cylinder of radius b. The distance between centers is c, with a > b c. The internal cylinder is hollow. The conductor carries a steady current I uniformly distributed over the cross section. Show that the field within the cylindrical hole is
+
H =
CI
2a(a2  b2)
and is directed transverse to the diameter joining the two centers. 2. Two straight, parallel wires of infinite length carry a direct current I in opposite directions. The conductivity u of the wires is finite. The radius of each wire is a and the distance between centers is b. a . Using a bipolar coordinate system find expressions for the electrostatic potential and the transverse and longitudinal components of electric field intensity a t points inside and outside the conductors. b. Find expressions for the corresponding components of magnetic field intensity. c. Discuss the flow of energy in the field.
3. A circular loop of wire of radius a carrying a steady current I lies in the xyplane with its center a t the origin. A point in space is located by t h e cylindrical coordinates
PROBLEMS
263
4, z, where x = r cos 4, y = r sin 4, Show that the vector potential at any point in the field is T,
= 7rk !!, I
(:)'

[(I
kz) K  E ] ,
+ + + >>
where k2 = 4 a r / [ ( a r ) 2 91, and K and E are completc elliptic integrals of the first and second kinds. Show that when (r2 9)): a, this reduces to Eq. (37), p. 237, for the vector potential of a magnetic dipole. 4. From the expression for the vector potential of a circular loop in Problem 3 show that the components of field intensity are
T
r
6. The field of two coaxial Helmholtz coils of radius a and sepnrated by a distance
a between centers is approximately uniform in a region near the axis and halfway between them. Assume that each coil has n turns, that the cross section of a coil is small relative to a, and t h a t each coil carries the same current. From the results of Problem 4 write down expressions for the longitudinal and radial components of H at points on the axis and a t any point in a plane normal to the axis and located midway between the coils. What is the field intensity a t the midpoint on the axis? Find a n expansion for the longitudinal component H , in powers of z / a valid on the axis in the neighborhood of the midpoint, and corresponding expressions for H , and H , in powers of r / a valid over the transverse plane through the midpoint. 6. Two linear circuits CI and CZ carry steady currents I I and ZZ respectively. Show that the magnetic energy of the system is
where dsl and dsz are vector elements of length along the contours and r12 the distance between these elements. The permeability p is constant. The coeficient of mutual inductance is defined by the relation
7. Show that the mutual inductance of two circular, coaxial loops in a medium of constant permeability is
264
THE MAGNETOSTATIC FIELD
[CHAP.I V
where a and b are the radii of the loops, c the distance between centers, K and E the complete elliptic integrals of the first and second kinds, and k is defined by k2 =
4ab ____.
(a
+ + c2 b)2
When the separation c is very small compared to the radii and a formula reduces to
b show that this
+
where d = [ ( a  bI2 czl*. 8. The coeficient of selfinductance LI1 of a circuit carrying a steady current 11is defined by the relation TI = &lz; where T 1 is the magnetic energy of thc circuit. either of the expressions
Hence L I I can be calculated from
provided B is a linear function of H. The volume integral in the first case is extended over the region occupied by current, in the second over the entire field. That portion of Lll associated with t,he energy of the field inside the conductor is called the internal selfinductance Lil. Show that the internal selfinductance of a long, straight conductor of constant perheability p1 is
L;,
=
PI
8u
henrys/meter.
9. Show that the selfinductance of a circular loop of wire of radius R and crosssectional radius r is
where p1 is the permeability of the wire and p2 t h a t of the cxternal medium, both being assumed constant. 10. A toroidal coil is wound uniformly with a single layer of n turns on a surface generated by the revolution of a circle T meters in radius about an axis IZ meters from the center of the circle. Show that the selfinductance of the coil is
L = pn2(R  d R 2  r2)
henrys.
11. A circular loop of wire is placed with its plane parallel to the plane face of a semiinfinite medium of constant permeability. Find the increase in the selfinductance of the loop due to the presence of the magnctic material. Show that if the plane of the loop coincides with the surface of the magnetic material, the selfinductance is increased b y a factor which is independent of the form of the loop. 12. Find an expression for the change in selfinductance of a circular loop of wire due to the presence of a second circular loop coaxial with the first.
PROBLEMS
265
13. An infinitely long, hollow cylinder of constant permeability p1 is placed in a fixed and uniform magnetic field whose direction is perpendicular to the generators nf the cylinder. The potential of this initial field is 4; = HOT cos 8, where r and e are cylindrical coordinates in a transverse plane with the axis of the cylinder as an origin. The cross section of the cylinder is bounded by concentric circles of outer radius a and inner radius b. The permeability of the external and internal medium has the constant value p2. Show that the potential a t any interior point is
L
\PI  P Z /
\a
14. A hollow sphere of outer radius a and inner radius b and of constant permeabill ity PI is placed in a fixed and uniform magnetic field Hn.The external and internal medium has a constant permeability p z . Show that the internal field is uniform and is given by
1 Discuss the relative effectiveness of a hollow sphere and a long hollow cylinder for magnetic shielding. 16. A magnetic field such as is observed on the earth’s surface might be generated either by currents or magnetic matter within the earth, or by circulating currents above its surface. Actual measurement indicates that the field of external sources amounts to not more than a few per cent of the total field. Show how the contributions of external sources may be distinguished from those of internal sources by measurements of the horizontal and vertical components of magnetic field intensity on the earth’s surface. 16. Assume that the earth’s magnetic field is due to stationary currents or magnetic matter within the earth. The scalar potential a t external points can then be represented as an expansion in spherical harmonies. Show that the coefficients of the first four harmonics can be determined by measurement of the components of magnetic field intensity a t eight points on the surface. 17. A winding of fine wire is to be placed on a spheroidal surface so that the field of the coil will be identical with that resulting from a uniform magnetization of the spheroid in the direction of the major axis. How shall the winding be distributed? 18. A copper sphere of radius a carries a uniform charge distribution on its surface. The sphere is rotated about a diameter with constant angular velocity. Calculate the vector potential arid magnetic field a t points outside and inside the sphere. 19. A solid, uncharged, conducting sphere is rotated with constant angular velocity w in a uniform magnetic field B, the axis of rotation coinciding with the direction of the field. Find the volume and surface densities of charge and the electrostatic potential a t points both inside and outside the sphere. Assume that the magnetic field of the rotating charges can be neglected. 20. The polarization P of a stationary, isotropic dielectric in a n electrostatic field E is expressed by the formula
P
=x
(C
 Q)E.
266
T H E MAGNETOSTATIC FIELD
[CHAP.
Iv
If each element of the dielectric is displaced with a velocity v in a magnetic field the polarization is P = ( E eo)(E v X B),
+


at least t o terms of the first order in l / c = Z / e o p o . A dielectric cylinder of radius a rotates about its axis with a constant angular velocity 0 in a uniform magnetostatic field. The field is parallel to the axis of the cylinder. Find the polarization of the cylinder, the bound charge density appearing on the surface, and the electrostatic potential at points both inside and outside the cylinder. 21. A dielectric sphere of radius a in a uniform magnetostatic field is rotated with a constant angular velocity w about an axis parallel to the direction of the field. Using the expression for polarization given in Problem 20, find the densities of bound volume and surface charge and the potential at points inside and outside the sphere. 22. Show t h a t the force between two linear current elements is
dF
=
P
 IlIZ 4r
p _ I l l = _
2
4r r2
ds2 X (dsl X ro) r2


[(dsz ro) dsl  (dsl ds2)ro],
where ro is a unit vector directed along the line T joining dsl and dsz from dsl toward dsz. 23. A long, straight wire carrying a steady current is embedded in a semiinfinite mass of soft iron of permeability p a t a distance d from the plane face. The wire is separated from the iron by a n insulating layer of negligible thickness. Find expressions for the field a t points inside and outside the iron. 24. A long, straight wire carrying a steady current is located in the air gap between two plane parallel walls of infinitely permeable iron. The wire is parallel to the walls and is assumed t o be of infinitesimal cross section. Find the field intensity a t any point in the air gap. Plot the force exerted on the conductor per unit length for a current of one ampere as a function of its distance from one wall. 26. A long, straight conductor of infinitesimal cross section lies in a n infinitely deep, parallelsided slot in a mass of infinitely permeable iron. The thickness of the slot is I and the conductor is located by the Air parameters s a n d h as in the figure. Calculate the field intensity at points within the slot and the force on unit length of conductor per ampere. (Cj. Hague, “Electromagnetic Problems in Electrical Engineering,” andLinder, J.Am. Znst. Elec. Engrs., 46,614,1927.) 26. An infinite elliptic cylinder of soft iron defined by the equation
b2x2
+ a2y2 =
a2b2
is placed in a fined and uniform magnetic field whose direction is perpendicular to the generators of the cylinder. Find expressions for the magnetic scalar potential a t internal and external points and calculate the force and torque exerted on the cylinder per unit length. Discuss the case of unit eccentricity in which the cylinder reduces t o a thin slab. 27. A particle of mass m and charge q is projected into the field of a magnetic dipole.
PROBLEMS
267
a. Write the differential equations for the motion of the particle in a system of spherical coordinates whose origin coincides with the center of the dipole. b. Show that the component of the angular momentum vector in the direction of the dipole axis is constant. c. Discuss the trajectory in the case of a particle initially projected in the equatorial plane. 28. A small bar magnet is located near the plane face of a very large mass of soft iron whose permeability is p. The magnet is located by its distance from the plane and the angle made by its axis with the perpendicular passing through its center. Considering only the dipole moment of the magnet, find the force and torque exerted on it by the induced magnetization in the iron. 29. Two small bar magnets whose dipole moments are respectively ml and m 2are placed on a perfectly smooth table. The distance between their centers is large relative to the length of the magnets. A t any instant the dipole axes make angles el and e2 with the line joining their centers. a. Calculate the force exerted by ml on m 2and the torque on mzabout a vertical axis through its center. b. Calculate the force exerted by m2 on ml and the torque on ml about a vertical axis through its center. c. Calculate the total angular momentum of the system about a fixed point in the plane. Is it constant?
30. Show that the force between two small bar magnets varies as the inverse fourth power of the distance between centers, whatever their orientation in space. 31. A magnetostatic field is produced by a distribution of magnetized matter. There are no currents a t any point. Show that the integral BHdu=O
2
if extended over the entire field. 32. If J ( T , y, z ) is the current density at any point in a region V bounded by a closed surface S, show that the magnetic field a t any interior point d , y’, z’ is
+
where r = 1 / ( z ’  E ) + ~ (y’  y)2 (2’  z ) ~ and , that a t exterior points the value of the surface integral is zero. 33. Determine the field of a magnetic quadrupole from the expansion of the vector potential given by Eq. (23), p. 234. Give a geometrical interpretation of the quadrupole moment in terms of infinitesimal linear currents.
CHAPTER V
PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA Every solution of Maxwell’s equations which is finite, continuous, and singlevalued at all points of a homogeneous, isotropic domain represents a possible electromagnetic field. Apart from the stationary fields investigated in the preceding chapters, the simplest solutions of the field equations are those that depend upon the time and a single space coordinate, and the factors characterizing the propagation of these elementary onedimensional fields determine also in large part the propagation of the complex fields met with in practical problems. We shall study the properties of plane waves in unbounded, isotropic media without troubling ourselves for the moment as to the exact nature of the charge and current distribution that would be necessary to establish them. PROPAGATION OF PLANE WAVES
6.1. Equations of a Onedimensional Field.It will be assumed for the present that the medium is homogeneous as well as isotropic, and of unlimited extent. We shall suppose, furthermore, that the relations (1)
D
=
EE,
B
= pH,
J
=
uE,
are linear so that the medium can be characterized electromagnetically by the three constants E , p , and u. If the conductivity is other than zero any initial freecharge distribution in the medium must vanish spontaneously (Sec. 1.7). I n the following, p will be put equal to zero in dielectrics as well as conductors. The Maxwell equations satisfied by the field vectors are then
(I) V X E
+
dH
p~
= 0,
l3E (11) V X H  e   UE = 0, at
(111) V * H = 0, (IV) V * E
=
0.
We now look for solutions of this system which depend upon the time and upon distance measured along a single axis in space. This preferred direction need not coincide with a coordinate axis of the reference system. Let us suppose, therefore, that the field is a function of a coordinate ( measured along a line whose direction is defined by the unit vector n. The rectangular components nz,%, n, of this unit vector are obviously 268
E Q U A T I O N S OF A O N E  D I M E N S I O N A L FIELD
SEC.5.11
269
the direction cosines of the new coordinate axis {. Our assumption implies that a t each instant the vectors E and H are constant in direction and magnitude over planes normal to n. These planes are defined by the equation

r n = constant, (2) where r is the radius vector drawn from the origin to any point in the plane as indicated by Fig. 44.
'Y FIG.44.Homogeneous
plane waves are propagated in a direction fixed by the unit vector n.
Since the ficlds are of the form (3 ) E = E(C, 0, H = W, 0, the partial derivatives with respect to a set of rectangular coordinates may be expressed by (4)
From these we construct the operator sions for the curl and divergence. (5)
v
=
v and so obtain simplified expres
a a a a + j  + k  = (in, + jn, + kn,)  = n , as ay az ar: ar: a
i
vx
(6)
E
=
v.E
(7)
n
=
aE 8 x= ar: ar: (n x
n.

dE
ai
=
E),
a  (nE).
ar:
In virtue of these relations, the field equations assume the form
aH (111) n .  = 0, (11) n
aH
aE
x aE = 0, ar:  E at
ar:
aE (IV) n . = 0.
ar:
270
PLANE W A V E S I N UNBOUNDED, ISOTROPIC M E D I A
[CHAP.V
This system is now solved simultaneously by differentiating first (I) with respect to { and then multiplying vectorially b y n. (11) is next differentiated with respect to t. We have
upon elimination of the terms in H, the vector E is found to satisfy the equation a2E aE _a2E _ p pta = 0. (9) at2 at
A similar elimination of E leads to an identical equation for H. a2H a2H aH ~
Since n . (n
n  aH at
=
x
a!?  p E  at2
g)
 p(T
at
=
0.
is identically zero, it follows from (I) th a t
0. Taken together with (111),this gives
We are forced to conclude that a variation of the {component of H with respect to either t or t is incompatible with the assumption of a field which is constant over planes normal to the laxis a t each instant. Equation (11) does admit the possibility of a static field in the fdirection but, since at present we are concerned only with variable fields, we shall put Hr = 0. Likewise, i t follows from (11) th at
(;
n . e++E
) =O;
this together with (IV) leads to (13)
n
($dt + f E dt + a< d { )
n  (dE
=
+ f E dl)
=
0.
The component of E normal to the family of planes, therefore,. satisfies the condition
which upon integration gives
(15)
_ t
El = Eore
T
SEC.5.11
EQUATIONS OF A ONEDIMENSIONAL FIELD
271
where Eof is the longitudinal component of E at t = 0 and r = e/u is the relaxation time defined on page 15. If the conductivity is finite, the longitudinal component of E vanishes exponentially : a static electric field cannot be maintained in the interior of a conductor. According to (15) there may be a component El in a perfect dielectric medium, but this particular integral does not contain the time. Equations (11) and (13) prove the transversality of the field. The vectors E and H of every electromagnetic $eld subject to the condition (3) lie in planes normal to the axis of the coordinate {. Let us introduce a second system of rectangular coordinates E, 7, {, whose origin coincides with that of the fixed system x , y , z and whose .
where  k 2 is the separation constant. The general solution of the equation in f l is (18)
fl({)
=
Aeikf
+ Be+T,
where A and B are complex constants; for fz we shall take the particular solution (19)
j z ( t ) = CePt.
Then p must satisfy the determinantal equation p "   , pu+  = Ok2 . EIE
There exists a fixed relation between p and the separation constant k 2 ; the value of either may be specified, whereupon the other is determined. From (18) and (19) may be constructed a particular solution of the form (21)
E E = EIEeikfpt + E 2EeikfPt*
likewise, for the other rectangular component, (22)
E,, = El,,eikrPt
+ EZ,,eiktPf.
272
P L A N E W A V E S I N U N B O U N D E D , ISOTROPIC M E D I A
[CHAP.V
The four complex constants Elt . . . Ez, are the components of two complex vector amplitudes lying in the 4s plane. Combining (21) and (22), one obtains (23)
E = Elgiklpt
+ EZeiklPt.
This vector solution of (9) in turn is introduced into the field equations t o obtain the associated magnetic vector H. Since H must have the
FIG.45.Relative
directions of the electric and magnetic vectors in positive and negative waves.
same functional dependence on { and t , we can write
(24)
H = HleikSPt
and then determine the constants
+ H,eiklpt
HI, H2 in terms of El and E2.
These derivatives are introduced into (I) and lead to
(26) (ikn x El  ppHl)ei"fPt  (i k n X Ez
+ ppHz)eiklpt= 0.
The coefficients of the exponentials must vanish independently and, hence, (27)
ik H1 =  n x El, PEL
ik Hz =  n X Ez.
I n terms of the rectangular components,
PP
PLANE W A V E S H A R M O N I C I N T I M E
SEC.5.21
.
273
.
Since E n X E = 0, it follows that E H = 0. The electric and magnetic vectors of a$eld of the type defined by Eq. (3) are orthogonal to the direction n and to each other. These mutual relations are illustrated in Fig. 45. 6.2. Plane Waves Harmonic in Time.There are now two cast’s t o be considered, dependent upon the choice of p and k , which it will be well to treat separately. Let us assume first that the field is harmonic in time, and that therefore p is a pure imaginary. The separation constant is then determined by (20). (29)
p
=
iw,
k2
= p€w2
+ ipuw.
I n a conducting medium k 2 , and hence k itself, is complex. The sign o j the root will be so chosen that the imaginary part of k is always positive.
k
(30)
= a
+ ip.
The amplitudes E ~ .E. . Ex, arc also complex and will be written now in the form Ell = aleis’, E21 = a2e292, (31) El,, = het*’, E,, = b2eZ#2, where the new constants a1 . . . b2, 81 . . . $xare real. In virtue of these definitions we have for the (component of the vector E
Et = aleBf+i(.f’Wt+S1)
(32)
+ a2eBft(af+wt8z).
Since (32) is the solution of a linear equation with real coefficients, both its real and imaginary parts must also be solutions. The real part of (32) is (33)
E,
=
alear cos (wt  a{  0,)
+ azeaf cos (wt + a{  ox).
The phase angles B1 and Ox are arbitrary and consequently a choice of the imaginary part of (32) does not, lead to a solution independent of (33). I n the same way the 7component of E is obtaincd from (22).
(34)
Ev = ble@f cos
(wt  a[
 $1) 3 hear cos
(wt
+ a[  +J.
The components of the associated magnetic field are found from (28).
We now have
Thc complex tcomponcnt of H is
upon substitution of the appropriate values of IilE and H z t , we obtain for
274
P L A N E W A V E S I N UNBOUNDED, ISOTROPIC M E D I A
[CHAP.
Y
the real part (37) HE =
d 2 2
b1
a
+
cos (wt  a t  $1  y )
e81
w
Similarly for the 11component,
(38)
H,
= UI
67 e81 Pw
cos (wt  a t  81  y)
These solutions correspond obviously to plane waves propagated along the {axis in both positive and negative directions. Consider first the most elementary case in which all amplitudes except a1 are zero in a nonconduct,ng medium. Then Q and consequently p are zero and the field is represented by the equations (39)
E~ = al cOS (wt  at  el),
H,
=
a
 al COS (wt  at  el). Pw
This field is periodic in both space and time. The frequency is w / 2 ~= v and the period along the time axis is 2 ~ / w= T . The space period is called the wave length and is defined by the relat,ion Ly.
=
2T , x
=
2T 1
a
The argument $1 = wt  a[  B1 of the periodic function is called the phase and the angle el, which will be determined by initial conditions, is the phase angle. At each instant the vectors E and H are constant over the planes { = constant. Let us now choose a plane on which the phase has some given value at t = 0, and inquire how this plane must be displaced along the {axis in order that its phase shall be invariant to a change in t. Since on any such plane the phase is constant, we have
(41)
41 = w t  a[

el = constant,
d& = w dt
 adt
=
0.
The surfaces of constant phase in a field satisfying (39) are, therefore, planes which displace themselves in the direction of the positive {axis with a constant velocity
v is called the phase velocity of the wave. It represents simply the velocity of propagation of a phase or state and does not necessarily coincide with the velocity with which the energy of a wave or signal is
SEC.5.21
275
P L A N E W A V E S HARMONIC I N T I M E
propagated. In fact v may exceed the critical velocity c without violating in any way the relativity postulate. In a nonconducting medium 1 c Ly=wl&, v=___ (43)
fi€ = d KK , m’
where c is the velocity of the wave in free space and K ~ K,,,, are the electric and magnetic specific inductive capacities. In optics the ratio n = c/v = a c / w is called the index of refraction. Since in all but ferromagnetic materials K~ is very nearly unity, the index of refraction should be equal to the square root of K ~ . This result was first established by Maxwell and was the basis of his prediction that light is an electromagnetic phenomenon. Marked deviations from the expected values of n were observed by Maxwell, but these were later accounted for by the discovery that the inductive capacity does not necessarily maintain a t high frequencies the value measured under static or quasistatic conditions. A functional dependence of n on t,he frequency results in a corresponding dependence of the phase velocity and leads to the phenomena known as dispersion. At a given frequency the wave length is determined by the properties of the medium. (44) where Xo is the wave length at the same frequency in free space. I n all nonionized media n > 1, and consequently the phase velocity is decreased and the wave length is shortened. Referring again to (39), we see that the relation of the magnetic to the electric vector is such that the cross product E x H is in the direction of propagation. The magnetic vector is propagated in the same direction with the same velocity, and in a nonconducting medium is exactly in phase with the electric vector. Their amplitudes differ by the factor (u =
(45)
0).
If in Eqs. (33) to (38) all amplitudes except a2 are put equal to zero, another particular solution is found which in nonconducting media reduces to (46)
E~ = a2 cos (wt
+ a!:  e,),
H, = a2$
cos (wt
+
 e2).
This field differs from the preceding only in its direction of propagation. a(  82 and the surfaces of constant phase are The phase is 4 2 = wt
+
276
P L A N E WAVES I N UNBOUNDED, ISOTROPIC M E D I A
[CHAP.V
propagated with the velocity v =  w / a in the direction of the negative {axis. The two particular solutions whose amplitudes are bl and b2 represent a second pair of positive and negative waves, both characterized by electric vectors parallel to the 7axis. Let us remove now the restriction t o perfect dielectric media and examine the effect of a finite conductivity. It is apparent in the first place that both electric and magnetic vectors are attenuated exponentially in the direction of propagation. Waves traveling in the negative direction are multiplied by the factor e+@r,but since { decreases in the direction of propagation this too represents an attenuation. Not only does a conductivity of the medium lead to a damping of the wave, but i t affects also its velocity The constants a! and p can be calculated in terms of p, E, and u by squaring (30) and equating real and imaginary parts respectively to the real and imaginary terms in (29). (47) Upon solving these relations simultaneously, we obtain (48)
(49) Ambiguities of sign that arise on extracting the first root are resolved by noting that a and must be real. The planes of constant phase are propagated with a velocity
which increases with frequency so long as the constants K,, K ~ and , u are independent of frequency. Attenuation of amplitudes in a surface of constant phase is determined by the attenuation factor /3 which also increases with increasing frequency. The complex factor k will be referred to as the propagation constant, and its real part a! may be called the phase constant, although this term is applied also to the angles 0 and +. The effect of frequency and conductivity on the propagation of plane waves is most easily elucidated by consideration of two limiting cases. Examination of (48) and (49) shows that the behavior of the factors the Q and is essentially determined by the quantity U ~ / E ~ W Now ~ .
PLANE WAVES H A R M O N I C I N T I M E
SEC.5.21
27 7
total current density a t any point in the medium is
J
=
UE + E
aE at
=
(U
 iw~)E,
whence it is apparent that U / U is equal to the ratio of the densities of conduction current to displacement current.
Case I .
> 1. The conduction current greatly predominates E W
ovcr the displacement current. This is invariably the case in metals, where u is of t he order of lo7 mhos/meter. Not much is known about the inductive capacity of metals but there is no reason t o believe th a t it assumes large values. Since the magnitude of e is probably of the order of the displacement current could not possibly equal the conduction current a t frequencies less than 10’7, lying in the domain of atomic phenomena to which the present obviously does not apply. For CY and /3 we obtain the approximate formula (54)
a =
p
=
J
?! 2
=
1.987 X lO’.\/VaK,.
An increase of frequency, permeability, or conductivity contributes in the same way t o an increase in attenuation. The phase velocity also increases with frequency, but decreases with increasing u or K ~ . Thus the higher order harmonics of a complex periodic wave are constantly advancing with respect to those of lower order. The amplitudes of the electric and magnetic vectors of a plane wave are related by (5.5)
278
PLANE W A V E S I N UNBOUNDED, ISOTROPIC M E D I A
[CHAP.V
(56) I n poorly conducting media expansion of (56) gives
whereas in good conductors,
In a perfect dielectric the electric and magnetic vectors oscillate in phase; if the medium is conducting, the magnetic vector lags by an angle y.
If
L>> 1, this ratio reduces to unity; hence the magnetic vector 2W2
of a
plane wave penetrating a metal lags behind the electric vector by 45 deg. 6.3. Plane Waves Harmonic in Space.The assumption that p in Eq. (20) is a pure imaginary leads to complex values of k and to electromagnetic fields which are simple harmonic functions of time. A field which at any given point on the {axis is a known periodic function of t can be resolved by a Fourier analysis into harmonic components propagated along the {axis as just described. The time variation at any other fixed point can then be found by recombining the components at the point in question. I n place of the variation of the field with t at a certain point c, one may be given the distribution with respect to at a specified value of t and asked to determine the field a t any later instant. A harmonic analysis must then be made with respect to the variable 1. Let the separation constant k 2 be real. Then p is a complex quantity determined by
p =
a+ i
2E 
If for the moment we write q in (23) assumes the form
=
JF$.
,,/x2,
the electric vector defined
279
POLARIZATION
SEC.5.41
If now a2/42 < P / p , the quantity iq is a pure imaginary and the field may be interpreted as a plane wave propagated along the {axis with a phase velocity
The amplitude of oscillation a t any point { decreases exponentially with t, the rate of decrease being determined essentially by the relaxation time 7
= e/u.
When a2/4e2 > k 2 / p e , the quantity iq is real and there is no propagation in the sense considered heretofore. The field is periodic in { but decreases monotonically with the time. There is no displacement along the space axis of an initial wave form: the wave phenomenon has degenerated into d i f u s i o n . 6.4. Polarization.Inasmuch as the properties of the positive and negative waves differ only in the direction of propagation, we shalI confine our attention a t present to the positive wave alone. Furthermore, since the attenuating effect of a finite conductivity in a n isotropic, homogeneous medium enters as an exponential factor common to all field components, it plays no part in the polarization and will be neglected. Let us suppose then that the amplitudes and phase constants of the rectangular components of E have been specified and investigate the locus of IEl = d E t 2 EV2in the plane p = constant. To determine this locus one must eliminate from the equations
+
(63)
Ec
=
a cos (4
+ 0),
the variable component 4 is written in the form (64)
$
=
cos (4
+
E,
=
b cos (4
+ +),
 wt of the phase.
= cr{
7
Er = 0, To this end (63)
+ 0) cos 6  sin (4 + e) sin 6, Upon squaring, the periodic factor cos (4 + 0) can
+ O),
= cos
(4
where 6 =  0. be elirninat>edand it is found that the rectangular components must satisfy the relation
The discriminant of this quadratic form is negative, 4 cos2 6  4 = __ 4 , sin2 6 5 0, a2b2 a2b2 a2b2
and the locus of the vector whose components are Ec, E, is, therefore, an ellipse in the Eqplane.
280
P L A N E W A V E S I N UNBOUNDED, ISOTROPIC M E D I A
[CHAP.
V
I n this case the wave is said to be elliptically polarized. The points of contact made by the ellipse with the circumscribed rectangle are found from (65) to be ( + a , & b cos 6) and ( + a cos 6, Ab). I n general the principal axes of the ellipse fail to coincide with the coordinate axes [, 7 , but the two systems may be brought into coincidence by a rot,ation of the coordinate system about the {axis through a n angle 6 defined b y tan 26
(67)
46.Po1ari2ation
(68) Et.2
= __2ab
a2  b2
cos 6.
When the amplitudes of the rectangular components are equal and their phases differ by some odd integral multiple of ~ / 2 ,the polarization ellipse degenerates into a circle.
+ EV2= a2,( a = b,
6
m?r 2 j
=
The wave is now said to be circularly polarized. Two cases are recognized according to the positive or negative rotation of the electric vector about the {axis. It is customary to describe as righthanded circular polarization a clockwise rotation of E when viewed in a direction opposite to that of propagation, looking towards the source. By far the most important of the special cases is th at in which the polarization ellipse degenerates into a straight line. This occurs when 6 = I m n , where Righthanded Lefthanded FIG. 47.Circular polarization. m is any integer. The locus of E in the the circles indicating the locus of E. t7plane then reduces to a straight line The direction of propagation is normaking an angle 6 with the &axis de mal to the page and toward the observer. fined by Ev b t a n 6 =   (l)m
00
J%
Q
The wave is linearly polarized. The magnetic vector of a plane wave is a t right angles to the electric vector and H oscillates, therefore, parallel to a line whose slope is ( l ) % / b . It is customary to define the polarization in terms of E and to denote the line (69) as the axis of linear polarization. In optics, however, the orientation of the vectors is specified traditionally by the “plane of polarization,” by which is meant the plane normal to E containing both H and the axis of propagation.
SEC.5.51
28 1
ENERGY FLOW
6.6. Energy Flow.The rate a t which energy traverses a surface in an electromagnetic field is measured by the Poynting vector, S = E x H
ik
defined in Sec. 2.19. Since by (27) H = rt  n PP
x E, the sign depending
upon the direction of propagation, it is apparent that the energy flow is normal to the planes of constant phase and in the direction of propagation. To calculate the instantaneous value of the flow one must operate with the real parts of the complex wave functions. The mean value of the flow can be quickly determined, on the other hand, by constructing the complex Poynting vector S* = +E x fi as in Sec. 2.20. For simplicity lct us consider a plane wave linearly polarized along the xaxis and propagated in the direction of positive x . Then
The complex flow vector is, therefore,
S:
=
1 E,H, 

2
=
a2
e.@*iy
2uw
7
whose real part represents the energy crossing on the average unit area of the zyplane per second.
The factor cos y arising from the relativc phase displacement between E and H may be expressed in terms of a and 0. (73)
tan
y =
P >
a
cos y
=
~ CY .\/,yz+pz
.
(74) According to Poynting's theorem the divergence of the mean fiow vector measures the energy transformed per unit volume per second into heat. I n the present instance
(75) which in turn is obviously equal t o the conductivity times the mean square value of the electric field vector E.
282
P L A N E WAVES I N U N B O U N D E D , ISOTROPIC M E D I A
[CHAP.v
6.6. Impedance.In a world which outwardly is all variety, it is comforting t o discover occasional unity and tempting t o speculate upon its significance. T o the untutored mind a vibrating set of weights suspended from a network of springs appears t o have little in common with the currents oscillating in a system of coils and condensers. But the electrical circuit may be so designed t h a t its behavior, and the vibrations of the mechanical system, can be formulated b y the same set of differential equations. Between the two there is a onetoone correspondence. Current replaces velocity and voltage replaces force; and mass and the elastic property of the spring are represented by inductance and the capacity of a condenser. It would seem that the L‘absolute reality,” if one dare think of such a thing, is an i n w t i a property, of which mass and inductance are only representations or names. Whatever the philosophic significance of mechanical, electrical, and chemical equivalences may be, the physicist has made good use of them to facilitate his own investigations. The technique developed in the last thirty years for the analysis of electrical circuits has been applied with success to mechanical systems which not long ago appeared too difficult to handle, and mechanical problems of the most complicated nature are represented by electrical analogues which can be investigated with ease in the laboratory. Not only the methods but the concepts of electrical circuits have been extended t o other branches of physics. Certainly the most important among these is the concept of impedance relating voltage and current in both amplitude and phase. This idea has been applied in mechanics to express the ratio of force to velocity, and in hydrodynamics, notably in acoustics, to measure the ratio of pressure to flow. The extension of the impedance concept to electromagnetic fields is not altogether new, but it has recently been revived and developed in a very interesting paper by Sche1kunoff.l The impedance offered by a given medium t o a wave of given type is closely related t o the energy flow, but t o bring out its complex nature we may best start with an analogy from a onedimensional transmission line, as docs Schelkunoff. Let z measure length along a n electric transmission line and let V = VoePiwt,I = Ifleczwtbe respectively the voltage across and the current in the line at any point z. The quantities V0, loare functions of z alone. The resistance of the line per unit length is €2 and its inductance per unit length is L. There is a leakage across the line a t each point represented by the conductance G and a shunt capacity C. The series impedance Z and the shunt admittance Y are, thercfore,
SCHELKUNOFF, Bell System Tech. J., 17, 17, January, 1938.
SEC.5.61
283
IMPEDANCE
while t,hc voltage and current are found to satisfy the relations'
These equations are satisfied by two independent solutions which represent waves traveling respectively in the positive and negative directions.
I 1 I, =
(78) where
Alekkziwt
v1 = ZOI', Vf = Z,I2,
>
Aze~kziW1,
(79)
k is the propagation constant and 2 0 the characteristic impedance of the line. Consider now a plane electromagnetic wave propagated in a direction specified by a unit vector n. Distance in this direction will again be measured by the coordinate { arid we shall suppose that the time enters only through the factor ezwt. Whereas voltage and current are scalar quantities, E and H are of course vectors. To establish a fixed convention for determining the algebraic sign, the field equations will be written in such a way as t o connect the vector E with the vector H x n which i s parallel to E and directed in the same sense. (80)
aT
=
a
i w p (x~n),

aT
(H x n)
=
i(we+ w ) ~ .
By analogy with (77)
(81)
z=
i
Y = i(w€
WPC,
+ iu).
The propagation constant is k
(82)
=
i
dY2
= dw%p
+ iwpu
(B), page 273, whilc the intrinsic impedance of the medium f o r plane waves is defined by Schelkunoff as the quantity as in
___
ZO++ (83)
W€
+ xu =
eiy* i+p*
1 See for example, Guillemin, " Communication Networks," Vol. 11, Chap. IT, Wiley, 1935. 2 These results differ in the algebraic sign of the imaginary component from those usually found in the literature on circuit theory due to our choice of erwt rather than e"f. Obviously this choice is wholly nrbitrary. Although it is advantageous t o use erwt in circuit theory, we shall bf concerned in wave theory primarily with the space rather than the time factor. Thr eupansions in curvilinear coordinates to be carried out in the next chaptcr justify the choice of etwt.
284
PLANE W A V E S I N UNBOUNDED, ISOTROPIC M E D I A
[CHAP.V
I n free space this impedance reduces to
(84)
Zo =
t
=
376.6 ohms.
Assuming n to be in the direction of propagation so that the distinction between positive and negative waves is unnecessary, lbe relation between electric vectors becomes (85)
n x E
=
Z,H,
E
=
Z,H x n.
There is an intimate connection bet,wcen the intrinsic impedance and the complex Poynting vector.
and, consequently,
GENERAL SOLUTIONS OF THE ONEDIMENSIONAL WAVE EQUATION
I n the course of this chapter we have investigated certain particular solutions of the field equations which depend on one space variable and the time. Owing to the linear character of the equations these particular solutions may be multiplied by arbitrary constants and summed to form a general solution, about which we now inquire. I n virtue of the infinite set of constants a t our disposal it must be possible to construct solutions that satisfy certain prescribed initial conditions. We shall show, for example, that the distribution of a field vector as a function of l may be specified a t a stated instant t = t o , and that the field is thereby uniquely determined a t all subsequent times; or the variation of a field vector as a function of time may be prescribed over a single plane { = lo and the field thus determined a t all other points of space and time. The means a t our disposal for the investigation of general integrals of a partial differential equation of the second order fall principally into two classes: the methods of Fourier and Cauchy, and those of Riemann and Volterra. The latter have come to constitute in recent years the most fundamental approach to the theory of partial differential equations, and their application to the theory of wave propagation has been developed in a series of brilliant researches by Hadamard.l But although the method of characteristics, first proposed by Riemann, affords a deeper insight into the nature of the problem, it proves less well adapted as yet HADAMARD, “Lepons sur la propagation des ondes,” A. Hermann, Paris, 1903; “Lectures on Cauchy’s Problem,” Yale University Press, 1923
285
ELEMENTS OF FOURIER ANALYSIS
SEC.5.71
to the requirements of a practical solution than the methods of harmonic analysis which will occupy our attention in the present section. 6.7. Elements of Fourier Analysis.For the convenience of the reader, who presumably is to some degree conversant with the subject, we shall set forth here without proof the more elementary facts concerning Fourier series and Fourier integrals. Consider the trigonometric series (1)
2+
(ul cos x
+ bl sin x) + . 
(a,cos nx
+ b, sin nx) + 
..,
with known coefficients a, and b,, and let it be assumed that the series converges uniformly in the region 0 I x I27r. Then (1) converges uniformly for all values of x and represents a periodic function f(x) with period 2 ~ .
+ 27r)
f(x
(2)
= fb).
The coefficients of the series may now be expressed in terms of f(x). In virtue of the assumed uniformity of convergence, (1) may be multiplied by either cos nx or sin nx and integrated term by term. I n the domain 0 < x I 27r, the trigonometric functions are orthogonal; that is to say, cos nx cos m x dx
=
6’“
sin nx sin m x dx
0,
cos nx sin m x dx
=
0,
=
0, (m #
n).
Since cos2nx dx
(4)
it follows that
1 1
1 a, = ; (5)
b,
=
1
=
2r
JzT
sin2 m x dx =
7r,
j(x> cos nx cis, (m= 0, 1 , 2

. ),
2*
; f(x) sin nx dx.
Inversely let us suppose that a function f(x) is given. Then in a purely formal manner one may associate with f(x) a “Fourier series” defined by
+ 2 (a, cos nx + b, sin ax), 00
f(x>
=
n=l
286
P L A N E W A V E S I N U N B O U N D E D , ISOTROPIC M E D I A
[CiwP. V
the coefficients to be determined from ( 5 ) . The righthand side of (6), however, will converge to a representation of f(s) in the domain 0 5 x 5 2~ only when the function f(x) is subjected to certain conditions. The whole question of the convergence of Fourier series is an extremely delicate one, and even the continuity off (2) is not sufficient to ensure it. It turns out, fortunately, that under such circumstances the series can nevertheless be summed to represent accurately the function within the specified interva1.l A statement of the least stringent conditions to be imposed on an otherwise arbitrary function in order that it may be expanded in a convergent or summable trigonometric series would necessitate a lengthy and difficult exposition; since we are concerned solely with functions occurring in physical investigations, we can content ourselves with certain suficient (though not necessary) requirements. We shall ask only that in the interval 0 5 x 5 2r the function and its first derivative shall be piecewise continuous. A function f(s) is said to be piecewise continuous in a given interval if it is continuous throughout that interval except at a finite number of points. If such a point be so,the function approaches the finite valuef(zo 0) as so is approached from the right, and f(s0  0) as it is approached from the left. A t the discontinuity itself the value of the function is taken to f b o 0) f(.o  . 0) be the arithmetic mean ___ 2 The Fourier expansion is particularly well adapted to the represrntation of functions which cannot be expressed in a closed analytic form, but which are constituted of sections or pieces of analytic curves not necessarily joined a t the ends. I n view of the admissible discontinuities in the function or its derivative, the fact that two Fourier series represent the same function throughout a subinterval does not imply at all that they represent one and the same function outside that subinterval. I t is thus essential that one distinguish between the representation of a function and the function itself. The expansion in a trigonometric series indicated on the righthand side of Eq. (6) represents and is equivalent to the piecewise continuous function f(x) within the region 0 5 x I2 r . Outside this domain the values assumed by the series are repeated periodically in x with a period 2r, whereas the functionf(x) may behave in any arbitrary manner. Only when f(s) satisfies the additional relation (2) will it coincide with its Fourier representation over the entire domain  co < x < 00. If within a specified interval the Fourier series of f(s)is known to be uniformly convergent, it may be integrated term by term and the series thus obtained placed equal to the integral of f(x) between the same limits.
+
+ +
WHITTAKER and WATSON,“Modern Analysis,” 4th ed., Chap. IX, Cambridge University Press, London.
SEC. 5.71
28 7
ELEMENTS OF FOURIER ANALYSIS
The uniform convergence of a Fourier series is in itself not sufficient to justify its differentiation term by term and the equating of the derived series to dfldx. If furthermore f(z) is discontinuous a t some point within an interval, its Fourier series certainly is not uniformly convergent everywhere within t h a t interval, and consequently the Fourier expansions of its integral and of its derivative demand special attention.’ I t is usually advantageous to replace (6) by an equivalent complex series of exponential terms of the form m
f(x)
(7)
=
2
n=
c,eim, m
whose coefficients arc determined from 2T
cn
=
_I_
f(a)ei*a da,
2iT
(n= 0,
1, 5 2
. . ).
Since (9)
eanx =
cos nx
+ i sin nx,
it is clear that the complex coefficients cn are related to the real coefficients a, and b, by the equations 2cn
=
a,,  ib,,
2c0 = ao,
2cn = a%
+i L ,
By an appropriate change of variable the Fourier expansion (7) may be modified to represent a function in the region 1 5 x 5 1.
Now this stretching of the basic interval or period from 27r to 21 suggests strongly the possibility of passing to the limit, as the basic interval becomes infinite and thereby obtaining a Fourier representation of a nonperiodic function for all real values of x lying between  w and 03. We shall assume t h a t throughout the entire region  < x < w , f(x) and its first derivative are piecewise continuous, that a t the discontinuities the value of the function is to be determined by the arithmetical
+
0 ~ )
mcan, and further that the integral Jm w or in other words, that the integral 1
J m m
f(x) dx is absolutely convergent, If(x)l dx exists. Let all
= Au.
These questions are thoroughly treated by Carslaw, “Introduction to the Theory
of Fourier’s Series and Integrals,” Chap. VIII, Macmillan, 1921.
288
PLANE WAVES I N U N B O U N D E D , ISOTROPIC M E D I A
[CRAP.~
Then (11) may be written
jmm +(u)d u is defined as the
On the other hand the definite integral limit of a sum as Au + 0:
As 1 approaches infinity, Au approaches zero and we may reasonably expect the limit of (12) to be (14)
f(.)
=
& s_* J:= du
f(a)eiu(za) da.
This is the Fourier integral theorem according to which an arbitrary function, satisfying only conditions of piecewise continuity and the
$p
If(z)l dz, may be expressed as a double integral.’ existence of I t will be noted that the result has been obtained by a purely formal transition to the limit which makes the existence of the representation appear plausible but does not confirm it. A rigorous demonstration is beyond the scope of these introductory remarks. If f(z) is real, the imaginary part of the Fourier integral must vanish and (14) then reduces to
Since (16)
cos u ( x  a ) = cos ux cos ua
+ sin ux sin ua,
it is apparent that the Fourier integral of a real, even function f ( z) $(XI is (17)
L
f(z) =
cos ux
1
f(a) cos ua d a du,
and that of a real, odd functionf(z) (18)
f(x)
=
27r
O0
sin u z
=
=
1
f(z) is
f(a)sin ua d a du.
As a general reference on thc subject the reader may consult “Introduction to the Theory of Fourier Integrals,” by E. C. Titchmarsh, Oxford University Press, 1937.
ELEMENTS OF FOURIER ANALYSIS
SEC. 5.71
289
From these formulas we see that the Fourier integral of a function may be interpreted as a resolution into harmonic components of frequency u/2~ over the continuous spectrum of frequencies lying between zero and infinity, In (18), for example, one may consider the function
;1
as the amplitude or spectral density of f(z) in the frequency interval u to u du. Then
+
_I
g(u) sin usdu.
The relation between f(x) and g ( u ) is reciprocal. g ( u ) is said to be the Fourier transform of f ( x ) , and f(z) is FIG.48.Step function. likewise the transform of g ( u ) . In the more general case of Eq. (14) one may write for the spectral density of f(x) the function
and hence for f(z) the reciprocal relation
An extensive table of Fourier transforms has been published by Campbell and Foster.’ The application of the Fourier integral may be illustrated by several brief examples of practical interest. Consider first the discontinuous step function defined by when 121 < I , Ax) = 1, (23) f(x> = +, when 1x1 = I, when 1x1 > 1. f(x> = 0, The function is real and even, so that we may employ (17). The transform is 2 sin ul g(u) = cos uz d x =
$1’
and the Fourier integral
(25)
f(x)
=
$ km
g ( u ) cos ux d u = 2 r
sorn
sin u l y ux du.
1 CAMPBELL and FOSTER:“Fourier Integrals for Practical Applications,” Bell Telephone System Tech. Pub., Monograph R584, 1931. Appeared in earlier form in Bell System Tech. J., October, 1928, pp. 639707.
290
P L A N E WAVES I N U N B O U N D E D , ISOTROPIC M E D I A [CHAP.V
Let us take next for f(z) the “error function” f(z)
=
c
a2x2 _ 2
.
To find its Fourier transform we must calculale the integral
which is clearly equal to the real part of the complex integral
On completing the square, (28) may be written
The path of integration follows the imaginary axis from iu/a2 to 0 and then runs along the real axis from 0 to co. The imaginary part of the integral arises solely from the interval iu/a2 5 /3 < 0, and since we are concerned only with the real part the lower limit may be taken equal to zero. The result is a definite integral whose value is well known:
a222
so that the transform of eT
is
Then f(x), which is reciprocally the transform of ~ ( u )is,
I n the particular case a
=
1, Eq. (32) becomes a homogeneous integral _ XZ _
equation satisfied by the function e 2 . The pair of functions defined by (31) and (32) have other properties more important for our present needs. Let us modify (31) slightly to form the function
(33)
SEC.5.71
ELEMENTS OF FOURIER ANALYSIS
29 1
The area under this curve is equal to unity, whatever the value of the parameter a ;
J*S(Z, a) ax = 1,
(34)
as follows directly from (30). Now let a become smaller and smaller. The breadth of the peak grows more and more narrow, while a t the same time its height increases in such a manner in the neighborhood of x = 0 as to maintain the area constant, as indicatcd in Fig. 49. I n the limit as a+O, the curve shrinks to the line II: = 0 where it attains infinite amplitude. A singularity has been generated, an impulse function bounding unit area in the immediate neighborhood of x = 0. A unit
X
FIG.49Thc
impulse function S(s, a ) .
impulse function which vanishes everywhere but a t the point x = xo is represented by
(35) Any arbitrary function F ( $)) subject to the usual conditions of continuity, can now be expressed as an infinite integral.
It is apparent from (32) that the transform of the impulse function S o ( x ) is a straight line displaced by an amount 1/45from the horizontal axis. As a final illustration of the Fourier integral theorem consider a harmonic wave train of finite duration. Such a pulse might result from closing and then reopening a switch connecting a circuit to an alternating current generator, or represent the emission of light from an atom in the course of an energy transition. Let
(37)
f(t>= 0,
when
It1
T > 2’
f(t)
when
It1
T < 3.
=
cos mot,
292
P L A N E WAVES I N IJNBOUNDED, ISOTROPIC M E D I A
CHAP.^
To facilitate the integration we shall take f(t) equal to the real part of T eiwotin the domain It/ <  and use Eqs. (21) and (22). 2'
The Fourier integral of f(t) corrcsponds now to the real part of (22), or
(wo 
_
sin
2
(0°
+
wo  w
'
wo+w
w ) T ] cos wt dw.
Equation (39) may be interpreted as a spectral resolution of a function which during a finite interval T is sinusoidal with frequency wo. The amplitude of the disturbance in the neighborhood of any frequency w is determined by the function sin
(40)
A(u)
=
 
7r
(a0

w)T
2 wow
The amplitude vanishes a t the points w = wo 
%n 7
T
(n = *l, +2
*
*
*
),
in the manner indicated by Fig. 50, and has its maximum value a t w = w0. As the duration T of the wave train increases, the envelope of the amplitude function is compressed horizontally until in the limit, as T f W , the entire disturbance is confined to the line w = w,, on the frequency spectrum. The simple harmonic variations that enter into so many of our discussions are mathematical ideals; the oscillations of natural systems, whether mechanical or electrical, are finite in duration, and the associated waves are periodic only in an approximate sense. We shall discover shortly that in the presence of a dispersive medium the entire character of the propagation may be governed by the duration of the wave train. 6.8. General Solution of the Onedimensional Wave Equation in a Nondissipative Medium.To acquire some further facility in the use of the Fourier integral it will repay us to pause for a moment over the elementary problem of finding a general solution to the wave equation in a
SEC.5.81
293
G E N E R A L SOLUTION
nonconducting medium. Let II. represent either the x or ycomponent of any electromagnetic vector. Then satisfies
+
According to Sec. 5.2, a particular solution of (42) is represented by .W
+ = ( A e r + Be*Tz)e&t. 12
(43)
The coefficients A and B are arbitrary and may depend on the frequency w ; that is to say, we associate with each harmonic component an appropriate amplitude which may be indicated by writing A(w) and B(w).
FIG.50.The amplitude function A ( w )
 ~(wo_ _ w~) T/ 2 .
1 sin
= 7r
wow
Now the general solution of (42) is obtained by summing the particular solutions over a range of w . In case # is to be a periodic function, the sum will extend over a discrete set of frequencies. I n general the wave function is aperiodic in both space and time, and the frequency spectrum is consequently continuous. (44)
Let us suppose that over the plane z = 0 the values of the function 4 and of its derivative in the direction of propagation are prescribed functions of time. (45)
294
PLANE WAVES' I N UNBOUNDED, ISOTROPIC MEDIA [CHAP.V
The problem is to find the coefficients A ( w ) and B(w) such that these two conditions are satisfied, and thereby to show that the prescription of the function and its derivative at a specified point in space (or time) is sufficient t o determine $(z, t ) everywhere. If we assume provisionally th at the integral on the right of (44) is uniformly convergent, it may be differentiated under the sign of integration with respect to the parameter z, yielding
On placing z
=
0 in (44)and (46), we obtain
upon comparison wit,h (21) and (22) it is immediately evident th a t the coefficients of the factor ePiwt in the integrands of (47) are Fourier transforms.
A (48)
iw
+B =zt,
(A 21
B) =
Lmm
1 J 2a 
f(t)eiwtdt, F(t)eiwt dt.
m
Solving these two relations simultaneously for A and B and substiiuting' (Y for t as a variable of integration, we obtain
and on substitution into (44)there results:
Now the first of these double integrals is the Fourier expansion of a function which on reference to (14) may be written down a t once. I n 1 The variables (Y and 6 appearing in the next few pages obviously have no connection with the real and imaginary parts of k = CY ip defined on p. 273.
+
295
G E N E R A L SOLUTION
SEC.5.81
the second let us invert the order of integration.
The imaginary part of the last integral vanishes because 1

w
sin
w
2,
z sin w ( a
 t)
integrates to an even function. Therefore,
=
Jn"
[sin w ( a
+
 1)  sin w ( a

g.
 1):
By a slight modification of the conditions which defined the step function (23), i t is easy t o show that each of the last two integrals represents a function with a single discontinuity. In fact a
when
p
0,
when
p = 0,
2
sin px
(53)
dx
=
Hence (52) will vanish whenever the arguments a are of the same sign.
> 0,
1
Changes of sign occur a t a
+ z  t and 
LY
U
= t 
z

U
and LY
 z  i U
=t
+ Z.U
The integral (52) has, therefore, a nonvanishing value only in the interval 2
t 21
ionsarc now to be multiplied by g(P) and integrated with respect to P. With regard to the limits we note the following. Suppose that t is positive. If z  at > 25 or z at < 25, the resultant field +(z, t ) is zero. If at 25 < z < at  25, the partial fields g ( z at) and g(z  at) are zero but the integrals do not vanish, and this no matter
+
+
+
t0.G
z in meters FIG.53n.
how large t may become. A residual field lingers in the region previously traversed by wave pulses. If z at > 25 and z  at < 25, thc limits of integrat,ion may be fixed a t k25, for beyond thcsc values g ( P ) is zero. Such limits are, therefore, appropriate for the entire region between the pulses, at 25 < z < at  25. The results of the integration are plotted in Figs. 53a and 538. The effective field +(z, t ) is indicated by a hcavy black outline; the partial
+
+.
1
10
t in microseconds FIG.53h.
fields expressed by the various terms of (102) are shown dotted. Figure 53a illustrates the distribution along the zaxis a t the instant t = 3.75 microscc. We note the beginning of a deformation of the initially rcctangular pulse, the top sloping forward and the residual tail trailing
308
P L A N E W A V E S I N UNBOUNDED, ISOTROPIC M E D I A
[CHAP,V
behind. As the wave progresses, the sharp front decreases in height, so that tail and pulse eventually merge into a rounded contour. I n Fig. 5% observations are made a t the fixed point z = 125 meters. At times anterior to t = 3 microsec. a field existed here, a field which was wiped out by the passage of the negative pulse. At t = 3 microsec. the positive wave arrives, followed by its tail which persists after the passage of the pulse. The essential difference between the solutions represented in Eqs. (84) and (102) is due to an interchange of the roles of space and time. To illustrate the behavior of the field, let it be assumed that the initial time function f ( t ) in Eq. (84) is a rectangular pulse defined by when Jtl < 1 X sec., when It/ > 1 X sec., for all values of t.
f(t) = 1, f(t) = 0, F(t)
= 0,
_ bZ The functions f t  and f t   are plotted in Figs. 54a ea 2 e 2a and 54b at various values of z and t. In virtue of the definition off@) i t is clear that the third term on the righthand side of (84) contributes b
2
( + :)
( :)
* 8; fcti)
;2
4
f(t+S)
,
I
.=lo meters
I
z=100 meters
t=O 1.0
t=0.5 x 1f6 sec.
t = 2 x 1 C 6 sec.
 1.0
t = 8 x 1 f 6 sec.
t
,300 200
t in microseconds
FIG.54a.
z
nothing if t  a
> 1 or
t
0
100
2
+ az < 1.
100
200
300
in meters FIG.54b.
2
If 1  a
< t < az  1, thc
vanish but the integral does not.
func
In this
interval, which lies between the pulses on the taxis, the limits of integra
309
THEORY OF LAPLACE TRANSFORMATION
SEC.5.121
tion are +1. In Figs. 55a and 55b are shown the deformations of an initially rectangular pulse a t fixed points in time and space due to the tail of the wave. t = 3.75 x 106
Sec.
 0.4  0.2 ,
I
f
,
I
,
,
,
,
,
,
,
FIG.55a.
When studying these figures it must be kept in mind that two preexisting waves coming from sources at infinity have been assumed which superpose to give just lbe correct distribution at t = 0 or z = 0.
,I : ;p,, 11.0
z = 125 meters
, , , , , , , 12
10
8
6
,
4
_
~,
b , Ib,
;2
_
2
t in microseconds FIG.55b.
A single wave whose amplitude is zero for all values of t and z less than zero can be generated only by a source a t the origin constituting a singularity in the fielda problem we have not yet considered. 5.12. Elementary Theory of the Laplace Transformation.Among the oldest and most important of the various methods devised for the solution of linear differential equations is that of the Laplace transformation. If in the equation
the dependent variable w is transformed by the relation (104)
W(Z) =
J u ( t ) o P dt,
it will be found in many cases that u(t) satisfies a differential equation which is simpler than (103), and in fact that u(t) is frequently an elementary function whereas w(z) cannot, in general, be exprcssed directly in terms of elementary transcendentals. 1 When the coefficients p ( z ) and q ( z ) are functions of the independent variable z, it is necessary to 1 INCE,“Ordinary Differential Equations,” Chaps. VIII and XVIII, Longmans, London, 1927.
310
P L A N E WAVES I N UNBOUNDED, ISOTROPIC M E D I A
[CHAP.V
choose the path of integration properly in the complex plane of t in order thsl; (104) shall be a solution of (103);the independent solutions are then distinguishcd by the choice of path. The result is a representation of the particular solutions as contour integrals. If, however, the coefficients p and q are constant, the equation has no other singularities than the essential one at infinit,y and the paths of integration can be chosen t o coincide with the real axis. The solutions are t'hen expressed in terms of infinite irilegrttls intimately related, if not equivalent, to a Fourier integral representation. I n recent years the application of the Laplace transformation to linear equations with constant coefficients has been considered with growing interest, on the part of physicist,s and engineers, and its orderly and rigorous procedure appears to be rapidly replacing the quasiempirical methods of the Heaviside operational calculus. Although the Laplace met'hods lead to no results unobtaiiiable by direct applicat,ion of Fourier integral analysis, they do offcr certain definite rtdvaritagcs from the st'andpoint of convenience. They are particularly well adapted t o the treatment of functions which do not, va'nish a t ~0 a,nd which consequently fail to satisfy the condition of absolute convergence, and they lend in a simple and direct manner t o the soliit>ionof a n equation in terms of its initial conditions. On the other hand, they can be applied only t o problems in which the field may be assumed to be zero for all negative values of the independent variable; in short, the Laplace integral is applicable to problems wherein all the future is of importance, but the past of no consequcnce. We shall approach the theory of the Laplacc transform from the standpoint of the Fourier integral atid suggest rudimentary proofs for the more important theorems. Let us consider a function f(t) which vanishes for all negative values of t . Then apart from a factor 1,'the Fourier transform of f(t) is
+
F(w) = provided only that the integral
f ( t ) e c i W tdt,
6"if(t)l
dt exists.
In case f(t) does not
vanish properly a t infinity thc integral fails to converge, but under certain circurnstanccs absolute conwrgerice can be restored by introThe Fourier transform of e  Y t f ( t ) is then ducing a factor eYt.
If there exists a real number y,such t h a t
Next Page SEC.
5.121
311
T H E O R Y OF LAPLACE T R A N S F O R M A T I O N
then f(t) will be said to be transformable. The lower bound y a of all the 7’s which satisfy (107) is called the abscissa of absolute convergence Suppose, for example, that f(t) = eht when t 2 0 and f(t) = 0 when t < 0. Thenf(t) is transformable in the scrise of (106) and y > b = y a . On the other hand a function f ( t ) = et2 is not transformable, for there exists no real number y leading to the corivergence of (106). The inverse transformation follows from thc reciprocal propertics of the Fourier transforms, Eqs. (21) and (22).
where y > y a . The function defined by thc intcgrd on the right is equal to f(t)eYt only for positive \ d u e s of t , and vanishcs whenever t < 0. To give explicit cxprrssion to this €act wc introduce the unit step function u(t) which is zero when t < 0 and c.qi,:tl to unity when t > 0. The inverse transformation may thus be writ tcn
If y a 5 0, one may pass to the limit as y + 0 aftcr the integration3 indirated in (106) or (109) have bccn effected and thus determine the Fourier transforms of functions not othcrwise integrable. Such a step is unessential, and the Laplace transform contains this convergence factor implicitly. Let us introduce the complcx variable s = y iw. Then the Laplace transform of f ( t ) , henceforth dcsignsted by the operator L[f(t)I, is
+
(110)
~ [ f ( t )= l
f(t)e’i at
=
~(s),
Ws)
> Ya,
where Re(s) is the usual abbreviation for “ t h e real part of s,” and y a is determined in each case by the functional properties of f ( t ) . The inverse transformation is reprcwntcd by a n integral taken along a path in the complex plane of s. (111)
1 L1[F(s)l = 27Fi
1
+zm
z
F(s)ets as = f ( t > . u(t),
where y > ya. The Laplace transformation can be interpreted as a mapping of points lying on the positive real axis of t onto that portion of the complex plane of s which lics to the right of the abscissa y a . The domains of f(t) and its transform F ( s ) are indicated in Fig. 56. From the linearity of the operators it follows that if L[f,(t)]= F l ( s ) , U fs(t)l = F ~ ( s )then , (112)
L[fl(t) t
fi(t)l
= Fl(S)
k Fds).
Previous Page
312
P L A N E W A V E S I N U N B O U N D E D , ISOTROPIC M E D I A
[CHAP.V
If, furthermore, a is any parameter independent of t and s, then L[uf(t)]= uiqs).
(113)
The application of the Laplace transform theory to the solution of differential equations is based on a theorem concerning the transform of d a derivative. Assume that f(t) and its derivative f(t) are transformable dt
in the sense of (107), and that f(l) is continuous at t = 0. Then by partial integration
hence, if L[f(t)]
=
F ( s ) , it follows that
L
K2)I 
=
sF(s)  f(0).
The importance of this thcorem lies in the fact that it introduces the initial valuef(0) of the functionf(t). Integrating partially a second time
t Donlain of f ( t )
FIG.56.A
Laplace transformation effects a mapping of points on the positive real axis of t onto the shaded portion of the splane.
leads to a theorem for the transform of a second derivative which involves not only the initial value f(0) but also the value of the derivative (df / d t ) t=o.
Thus the transform of the dependent variable of a secondorder differential equation is expressed in terms of the initial conditions. These results have no analogue in the theory of the Fourier transform. The “ Faltung ” or “folding” theorem is a particular application of a wellknown property of the Fourier integral.’ Suppose again that 1 See for example Bochner “Vorlesungen uber Fouriersche Integrale,” and Wiener, “The Fourier Integral,” p. 45, Cambridge University Press, 1933.
313
THEORY OF LAPLACE TRANSFORMATION
SEC. 5.121
L[f1(2)] = F l ( s ) , L[fz(t)]= Fz(s), and that we wish to determine the Laplace transform of the product function fl(i!) fz(t). (116) L[fl(l) * fdt)l =
fz(t)e"f dt
=
fl(1)
fl(t) .
[L 2 ~ i
1
y2
1
Fz(a)eutda e*tdt
Jy*'im i m
y2+i
m
da F,(a)
=2?ri LJ
O1
fl(t)e(*u)L dt,
y2j
and, hence, the first theorem: yz+i
(117)
L[fl(t) .f 4 ) l YZ
2 ~J i
= 
> 7aZl
F1(s
 u ) . Fz(a) du,
y?im
Re(s  a)
> Yale
Of more frequent use is a second Faltung theorem on the inverse transformation of a product of transforms. We wish to determine a function whose Laplace transform is F,(s) . Fz(s).
=
!L
27ri
Y+i

Fl(s) [Jmfz(7)esr d ~ esf ] ds
yim
=
d7 fi(T) . =
som
Fl(s)e*(fr)ds
fz(.) . fl(t

T)
. u(t

7)
dr.
Since the unit function vanishes for negative values of the argument, the upper limit of 7 must be t and hence
or
Geometrically, the product F l ( s ) F z ( s ) represents the transform of an area constructed as follows: The function f l ( 7 ) is first folded over onto the negative half plane by replacing T by 7, and then translated to the right by an amount i!. The ordinate fl(t  T ) is next multiplied by fZ(.) dr and the area integrated from 0 to t .
314
PLANE W A V E S IN UNBOUNDED, ISOTROPIC MEDIA
[CHAP.V
A set of operations applied repeatedly in the course of our earlier discussion of the Fourier integral may be referred t o as translations. Let a be a parameter independent of s and t. (121)
L[eatf(t)] =
f(t)e(sa)tdt
=
F ( s  a).
Equation (121) is obviously valid for both positive and negative values of a. Next consider the transform of f ( t  a). (122)
L[f(t  a)]
f ( t  a)est dt =
=
eOS
J:j(7)ecsr d7.
Now i t has been postulated throughout that the function f ( t ) shall vanish for all negative values of the argument arid hence the lower limit a may be replaced b y zero. As a consequence, we obtain
L [ f ( t  a)] = ecaaF(s), Likewise, we may write
a > 0,
f(t>= 0
when t
< 0.
when t
< a;
but one must note with care that this last result applies only to functions
f ( t ) which by definition vanish for t < a. A replacement of t by t  a or t a rcprescnts a translation of f ( t ) a distance a to the right or left
+
respectively. I n addition t o thc foregoing fundamental theorems there exist a number of clernentary but useful relations which may be deduced directly from (110). The following formulas are set down for convenience and their proof is left t o the reader.
f(0)
=
lim f ( t ) t
i
m
lim s F ( s ) ; s
im
=
lim sF(s). s+o
Evaluation of the contour integrals that occur in the inverse transformations (111) is most easily effected by application of the theory of residues. The functions f ( t ) with which one has t o deal are analytic
SEC.5.121
THEORY OF LAPLACE TRANSFORMATION
315
functions of the complex variable t except at a finite number of poles. It will bc recalled that in the neighborhood of such a point, say t = a, the function may be expanded in a series of the form
where @(t)is analytic near and a t a and, therefore, contains only positive powers of t  a. The singularity is said to be a pole of order m if m is a finite integer. If, however, an infinite number of negative powers is necessary for the rcprcsentation of j ( t ) , the singularity is said to be essential. If now the function f ( t ) is integrated about any closed contour C in the tplane which encircles the pole a but no othcr singularity, it can be shown that all the terms in the expansion (131) vanish with the exception of the one whose coefficient is bkl. The result is f(t) dt
(132)
=
27rib1.
The coefficient bkl is called the residue a t the pole. In case the contour encloses a number of poles, thr: value of the integral is equal to 27ri times the sum of the residues. In particular, if f(t) is analytic throughout the region bounded b y the contour C, the integral vanishes. T h w e results are a consequence of a fundamental theorem of function theory. If j ( t ) is analytic on and within the ( 0.
It will be noted in passing that (134) does not fall within the province of functions admitting a Fourier analysis, for the integral is nonconvergent. The Laplace transform of (134) is
J If(t)l dt
(135) We verify this result by applying the inverse transformation (111).
Now F ( s ) has a pole of first order at s = 0. The abscissa of absolute convergence is zero, whence it is necessary that Re(s) > 0. The integration is to be extended along any line parallel to and to the right of the imaginary axis. The integrand of (136) satisfies the conditions of Jordan’s lemma and the path of integration may be closed by an infinite circuit. If t > 0, the path may be deformed to the left so that Re(s) + a. The closed contour y  i.o t o y i m to  00 im to  m  iw to y  im contains the pole at s = 0. The residue is unity, and hence by (132) the righthand side of (136) is also unity for t > 0. Since F ( s ) is only defined in the right half plane, a question may arise as to the justice of extending the integration to the left half plane. One must be careful to distinguish between a function and the representation of that function in a restricted domain. l/s is a representation of F ( s ) when Re(s) > 0, but in view of possible discontinuities in F ( s ) the two functions need not coincide a t all when Re(s) < 0. Now along the line y . i to y i m we are integrating the function l / s ; for purposes of integration we may make use of any of its analytic properties. The analytic continuation into the left half plane is of the function l / s , and not of F ( s ) .
+
+
+
317
THEORY OF LAPLACE TRANSFORMATION
SEC. 5.121
When t < 0, the vanishing of the integrand may be assured by deforming the path to the right. Within the closed contour y  im to y i co to m i to co  ia) to y  i.o there are no singularities and the righthand side of (136) is zero. We have verified that the unit step function may be expressed analytically by
+
+
u(t) =
(137)
 est ds,
~
y
> 0.
It is interesting to apply the translation theorem (123) to this result, putting a = z/v.
[(
31
L u tor
_ 2 = e
V'FF(S)=
1
;s,
e S
(139) From the preceding discussion it is apparent that (139) vanishes when 2
t 
21
< 0, and is equal to unity when t  22, > 0.
I n Figs. 57a and 57b
is represented as a function of t and z respectively. u (t Z/U)
u (t  Z / U )
t'= z/u u(tz/u)=O, t < Z / U , ~
u ( t  z / u ) = l , t>z/u.
FIG.67a.
t
z'= ut
I
u(tZ/U)=O, z> ut, u ( t  z / v ) = l , z< ut. FIG.57b.
As a second example let us imagine that a harmonic wave is switched on suddenly a t t = 0 and continues indefinitely. when t when t
< 0, > 0.
As in the previous case we can verify (141) by applying the inverse transformation w
2 0.
318
P L A N E WAVES I N UNBOUNDED, ISOTROPIC M E D I A
[CHAP.V
If t > 0, the path is again deformed to the left. The enclosed pole is located a t s =  i w ; on applying the Cauchy theorem we see that (142) is indeed equal to rtwt. On the other hand if t < 0, the path is deformed t o the right and the integral vanishes. I n general,
6.13. Application of the Laplace Transform to Maxwell's Equations.Let E, = f(z, t ) and 11, = g(z, t ) be the components of a plane electromagnetic field. The functions f(z, t ) and g(z, t ) are assumed to be transformable in the sense of (107) a d t o vanish for all negative values of t ; moreover, they are related t,o one another by the field equations 3.f dz
(144)
+ p %d t
=
0,
ddz! ' + €  +df +f=O. dt
Treating z as a parameter, the Laplace transform of this system with respect to t proves to be
Upon elimination of G(z, s) an ordinary, inhomogeneous equation for F ( z , s) is obtained in which s enters only as a parameter.
Since the derivative of g(z, t ) with respect to x a t the instant t = 0 can be expressed in terms of f(z, 0) and (dj/dt)t30 through (144), this last result is equivalent to
Byf(z, 0) one means strictly the limit of f(z, t ) as t fl(z)
(148)
=
lim f(z, t), t0
f
0.
The functions
8.f (2, t ) , f 2 ( z ) = lim t+O
dt
represent the initial state of the field and are assumed to be known. writing h2 = pes2 pus, wc obtain finally
+
(149)
d2F  h2F = ~
dz2
_ h2 _ fd.4  pdz(z) s
=
Z(Z, s>.
Upon
SEC.5.131
319
A P P L I C A T I O N OF T H E L A P L A C E TRANSFORM
The general solution of (149) consists of a solution of the homogeneous equation
containing two arbitrary constants, t,o which is addcd a particular solution of (149). One can verify casily enough that (149) is in fact satisfied by (151) F ( z , s) = Aeh8
+ Behz +
in which h is that root of v'hz which is positive when h2 is real and positive. The constants A and B are to be determined such as to satisfy specified boundary conditions at the points z = z1 and x = z2. After A and B have been evaluated in terms of s, a n inverse transformation Scads from (151) back to a solution f(z, t ) of (144) satisfying both initial and boundary condition?. T o illustrate, consider first the elementary case in which the field is zero everywhere a t the initial instant t = 0. Then Z = 0 and we are concerned only with the solution of th e homogeneous cquation. Let us suppose that a t z = 0 there is a source switched on a t the instant t = 0 which sends out a wave in the positive zdirection and whose intensity at the origin is f(0, t ) = f 3 ( t ) . Thus the second boundary condition a t z = z 2 is replaced in th e present example by the stipulation that the field shall be propagated to the right along the zaxis. The transform of the boundary value is represented by (152)
F ( 0 , s)
=FdS) =
L[fa(t)l.
Since the wave is to travel to the right, the constant A must be placed equal to zero. This is clear nhcn u = 0, for then h = s/a, and by the translation theorem of page 314 (1Ti3) ( t 2 0,
(154)
2
> 0).
If the conductivity is not equal to zero, the solution is more difficult In this case we shall write h2 =:
(s
~ +G
$
')u

$ and apply th e
320
PLANE WAVES I N UNBOUNDED, ISOTROPIC M E D I A
N CHAP.^
integral
~ ____
Upon substitution of 2 / p z
+ z2 = at, this reduces to
where u = 1 / 6 , b = a / 2 ~ as , defined on page 297. Finally, by differentiation we obtain
where d(z, t ) is a function defined by =
(159)
0,
when 0
0. DISPERSION
6.14. Dispersion in Dielectrics.A pulse or “signal” of any specified initial form may be constructed by superposition of harmonic wave trains of infinite length and duration. The velocities with which the constantphase surfaces of these component waves are propagated have been shown to depend on the parameters E , p , and CT. In particular, if the medium is nonconducting and the quantities E and p are independent of the frequency of the applied field, the phase velocity proves to be constant and the signal is propagated without distortion. The presencb of a conductivity, on the other hand, leads to a functional relation between the frequency and the phase velocity, as well as to attenuation. Consequently the harmonic components suffer relative displacements in phase in the direction of propagation and the signal arrives at a distant point in a modified and perhaps unrecognizable form. A medium in which the phase velocity is a function of the frequency is said to be dispersive. At sufficiently high frequencies a substance may exhibit dispersive properties even when the conductivity u due to free charges is wholly negligible. In dielectric media the phase velocity is related to the index At frequencies . less than of refraction n by v = c/n, where n = Gm lo8 cycles/sec. the specific inductive capacities of most materials are substantially independent of the frequency, but they manifest a marked dependence on frequency within a range which often begins in the ultrahighfrequency radio region and extends into the infrared and beyond. Thus, while the refractive index of water at frequencies less than los is about 9, it fluctuates at frequencies in the neighborhood of loLo cycles/sec. and eventually drops to 1.32 in the infrared. Apart from solutions or crystals of ferromagnetic salts, the dispersive action of a nonconductor can be attributed wholly to a dependence of K~ on the frequency. All modern theories of dispersion take into account the molecular constitution of matter and treat the molecules as dynamical systems possessing natural free periods which are excited by the incident field. A simple mechanical model which led to a strikingly successful dispersion formula was proposed by Maxwell and independently by Sellmeyer. A further advance was accornplished by Lorentz who extended the theory of the medium as a finegrained assembly of molecular oscillators and who was able to account at least qualitatively for a large number of electrical and optical phenomena. According to Lorentz, however, these
322
P L A N E W A V E S I N U N B O U N D E D , ISOTROPIC M E D I A
N CHAP.^
molecular systems obeyed the laws of classical dynamics; it is now known that they are in fact governed by the more stringent principles of quantum mechanics. Following upon the rapid advances of our knowledge of molecular and atomic structure revisions were made in the dispersion theory which a t present may be considered to be in a very satisfactory state. Both the classical and the quantum theories of dispersion undertake to calculate the displacement of charge from the center of gravity of an atomic system as a function of the frequency and intensity of the disturbing field. After a process of averaging over the atoms contained within an appropriately chosen volume element, one obtains an expression for the polarization of the medium; that is to say, the dipole moment per unit volume. The classical result corresponds closely in form to the quantum mechanical formula and leads in most cases to an adequate representation of the index of refraction as a function of frequency. We shall confine the discussion, therefore, to the case in which the electric polarization in the neighborhood of a resonance frequency can be expressed approximately by the real part of1 a2
P= 
w ;  a2 
iwg
eoE.
By the electric field intensity, we shall now understand the real part of the complex vector (2)
E
=
Eo eiut.
The constant a2 is directly proportional to the number of oscillators per unit volume whose resonant frequency is WO. The constant 20 is related to this resonant frequency by (3)
2;
= W;  +a2,
such that WO + wo a t sufficiently small densities of matter. The constant g takes account of dissipative, quasifrictional forces introduced by collisions of the molecules. The constants Wo and g which characterize the molecules of a specific medium must be determined from experimental data. At sufficiently low incident frequencies W, the polarization P according to (1) approaches a constant value (4) 1 For the derivation of this result, see Lorentz, “Theory of Electrons,” or any standard text on physical optics such as for example Born, “Optik,” Springer, 1933, or Forsterling. “Lehrbuch der Optik,” Hirzel, 1928.
SEC.5.141
DISPERSlOfi I N DIELECTRICS
3 2.1
and, since the specific inductive capacity is related to the polarization by
P = ( K  l)~oE, (5) one may express K in terms of the molecular constants.
When, however, the incident frequency is increased, further neglcct of tlie two remaining terms in the denominator becomes inadmissible. In that case we shall define by analogy a complex inductive capacity K' through either of the equation:.;
(7)
P
= (K'
 l)eoE,
D
=
K'QE,
whence from (1) we obtain
In terms of this complex parameter the Maxwell equations in a medium whose magnetic permeability i;i po are (9)
as a consequence of which the rectangular components of the field vectors satisfy the wave equation
A plane wave solution of (10) is represented by
9
(11) where (12)
k
=
=
+oeik"iwt
,
*+ w c
=a
iB,
so that (13)
The wave is propagated with :a velocity v = w / a = c/n, but a and the refractive index are now explicit functions of the frequency obtained by introducing (8) into (13). I n gases and vapors the density of polarized molecules is so low that K' differs by a very small amount from unity. The constant a2 is thercfore small, so that 60 differs by a negligible amount from the natural
324
P L A N E W A V E S I N UNBOUNDED, ISOTROPIC M E D I A
[CHAP.
V
frequency wo and the root of K' can be obtained by retaining the first two terms of a binomial expansion. Thus
When the impressed frequency w is sufficiently low, the last two terms of the denominator may be neglected so that
The index of refraction and consequently the phase velocity in this case are independent of the frequency; there is no dispersion. If the impressed frequency w is appreciable with respect to the resonant frequency wo but does not approach it too closely, the damping term may still be neglected. \WE  w21 >> wg,
The attenuation factor is zero and the medium is transparent, but the refractive index and the phase velocity are functions of the frequency. If w < wo, n will be greater than unity and an increase in w leads to a n increase in n and a decrease in v. If w > wo, the refractive index is less than unity but an increase in w still rcsult,s in an increase in the numerical value of n. The dispersion in this casc is said t o be normal. Finally let w approach the resonance frequency w 0 . Upon resolving (14) into its real and imaginary parts one obtains
InqFig. 58 are plotted the index of refraction and the absorption coefficient /?c/o of a gas as functions of frequency. The absorption coefficient exhibits a rather sharp maximum a t wo, so that in this region the medium is opaque to the wave. As w rises from values lying below wOl the index n reaches a peak at w1 and then falls off rapidly to a value less than unity at 02, whence it again increases with increasing w ) eventually approaching unity. Whenas in the region wlw2an increase of frequency leads to a decrease in refractive index and an increase of phase velocity, the dispersion i s said conventionally to be anomalous. According to this definition dispersion introduced by the conductivity of the medium and discussed in the earlier sections of this chapter is anomalous. In fact it might
SEC.5.151
DISPERSION I N METALS
325
almost be said that the “anomalous” behavior is more common than the ancc. We have seen that the Fouricr integral of a nontjt:rminating wave train does not converge, but this difficulty may be circumveiitcd by deforming thc path of integration into the complex domain of the variable w. On the other hand, the theory of the IJaplace t'ransform was introduced for the very purpose of treating such functions; and indeed, i t may be shown by a simple change of variable t'hat, t,ho complex Fourier int>egralemployed by Sommerfeld is exactly the transform defined in Eq. (143) of Xec. 5.12. It will facilitate the prescnt discussion t o modify the original analysis accordingly. At x = 0 the signal is defined by the function
The representation of this field a t any point z wit,hin the medium is obtained by extending (20) to a solution of the wave equation, which by (lo), page 323, is
1 SOMMERFELD, Ann. Physik, 44, 177202, 1914. BRILLOUIN, ibid., 203240. A complete and more recent account of this work was published by Brillouin in the reports of the Congrks International d'EZectricitd, Vol. 11, 1'" Sec., Paris, 1932.
S E ~5.181 .
335
W A V E FRONT A N D SIGNAL VELOCITIES
Equation (21) is satisfied by cilementary functions of the form
from which we construct our signal. y
> 0.
I n a nondispersive medium, K' is constant and (22) is ident,ical with (143), page 318. If, howevcr, the medium is dispersive, K' is a function of s. We shall suppose that, the dispersion can be represented by a formula such as (8), page 325, and upon replacing  i w by s we obtain
To include metallic conductors we need only put G i = 0. The wave function which for t > 0 reduces to a harmonic oscillation a t x = 0 is, therefore, y
(24)
As s + l i m , p ( s ) + 1. If now
7
=
2
t c
< 0,
> 0.
the contour may
be closed, according to Jordan's lemma, page 315, by a semicircle to thc right of infinite radius. This path excludes all singularities of the integrand and consequentlyfl:~,t ) = 0. Thus we have proved that at a point z within the medium the field is zero as long as t < z / c , and hence that the velocity of the wave front cannot exceed the constant c.
If
7 =
t 
5 > 0, the path may be closed only t o the left. The C
singularities thus encircled occur a t the pole s =  i w and a t the branch points of p. These last are located a t the points where /3 = 0 and p = a . If P ( s ) is written in the form
wc see that
1
P=
00,
P
0,
(26) =
i
when s =  g l  d 4 G , 2  g z , 2 2 1 i when s = g t 4 4 ( 4 a2)  g2. 2 2 1
+
The disposition of these singularities in the complex splane is indicated
336
P L A N E W A V E S I N U N B O U N D E D , ISOTROPIC M E D I A [CHAP.V
in Fig. 60a, where 1 i (27) u* =  3 g L 3d,
b*
1 i & 22 / 4 ( 4 + a2)  g. 2
= g
Now if one encircles a branch point in the plane of s, one returns to the initial value of 0 but with the opposite sign. This difficulty is obviated by introducing a ( ( c u t ” or barrier along some line connecting a+ and b+, and another between a and b. Over this rrc u tl’planewhich in the functiontheoretical sense represents one sheet of the Riemann surfacc of b(s)the function P(s) is singlevalued; for, since it is forbidden to traverse a barrier, any closed contour must encircle a n even number of branch points. The path of integration in (24), which follows the line drawn from y  i m to y i 00 and is then closed b y an infinite semi
+
+
_______ I I
b
of integration in the splane.
circle to the left, may now be deformed in any manner on the “ c u t” plane without altering the value of the integral, provided only that in the process of deformation the contour does not sweep across the pole a t s = iw or either of the two barriers. In particular the path may be shrunk to the form indicated in Fig. 60b. The contributions arising from a passage back and forth along the straight lines connecting Co and C1 and Co and Cz cancel one another and (24) reduces to three integrals about the closed contours Co, C1, and Cz. The first of these three can bc evaluated a t once.
whence by Cauchy’s theorem
SEC.5.181
W A V E FRONT AND SIGNAL VELOCITIES
33 7
BY (25)
wh _
fo(z, 1)
=
e
+
e
Thc remaining two integrals which are looped about the barriers cannot be evaluated in any such simple fashion and they shall be designated by
Thus the resultant wave motion a t any point within the medium can be represented by the sum of i.wo terms,
Physically these two components may be interpreted as forced and free vibrations of the charges that constitute the medium. The forced vibrations, defined by fo(z, t ) , arc undamped in t’ime and h a w thc same frcqucncy as the impinging wave train. Thc free vibrations flz(z, t ) arc damped in time as a result of the damping forces acting on the oscillating ions and their frequency is determined by the elastic binding forces. The course of the propagation into the medium can be traced as follows: Up t o the instant t = z / c , all is quiet. Even when the phasc velocity u is greater than c , no wave reaches z earlier than t = z/c. At t = z / c the integral fI2(z, t ) first exhibits a value other than zero, indicating that t,he ions have been set into oscillation. If by the term “wave front” we understand the very first arrival of the disturbance, then the wave jront velocity i s always equal to c, no matter what the medium. It may be shown, however, that at t,his first instant t = z / c the forced or steadystate term fo(z, t ) just cancels the free OK’transient term fI2(z, t ) , so that the process st,arts always from zero amplitude. The steady st,ate is then gradually built up as the transient dies out, quite in the same way that the sudden application of an alternating o.m.f. to an electrical network results in a transient surge which is eventually replaced by a harmonic oscillation. The arrival of the wave front and the role of the velocity v in adjusting the phasc are illust>ratedin Fig. 61, which however shows only the steadystat,e term. The axis x / c is drawn normal to the axis of t. The line at 45 deg. then determines the wavefront velocity, for i t passes through a point z a t the instant z / c . The line whose slope is t a n 0 = c/v determines the time t = z/v of arrival a t z of a wave whose velocity is v. Actually the phase velocity has nothing to do with the propagation; i t gives only t’he arrangement of phases and, strictly speaking, this only in the case
338
P L A N E W A V E S I N UNBOUNDED, ISOTROPIC ME’DIA [CHAP.V
of infinite wave trains. The phase of the forced oscillation is measured from the intersection with the dotted line at t = x/v, and the phase at the wave front t = 2/c is adjusted to fit. If v > c, 0 < 45 deg. The phase of the steady state is again determined by the intersection with the line z / c a t t = z/v, but the wave front itself arrives later. The transition from vanishingly small amplitudes a t the wave front to the relatively large values of the signal have been carefully examined
FIG 61.Determination
of the phase in the steady state.
by Brillouin. This investigation is of a much more delicate nature and we must content oiirsclves here with a statement of conclusions which may be drawn after the evaluation of ( 3 2 ) . According to Brillouin a signal is a train of oscillations starting a t a certain instant. In the course of its journey the signal is deformed. The main body of the signal is
FIG. 62a.Illustrating
the variation in amplitude of the first precursor wave.
FIG. 62b.Illustrating the arrival of the first and second precursors and the signal.
preceded by a first forerunner, or precursor, which in all media travels with the velocity c. This first precursor arrives with zero amplitude, and then grows slowly both in period and in amplitude, as indicated in Fig. 62a; the amplitude subsequently decreases while the period approaches the natural period of the electrons. There appears now a new phase of the disturbance which may be called the second precursor, GO traveling with the velocity ____ c. This velocity is obtained by
4 
assuming in the propagation factor P (  i w ) of Eqs. (30) and (32) that the impressed frequency w is small compared to the atomic resouance frequency Go. The period of the second precursor, at, first very large,
SEC.5.181
W A V E FRONT A N D SIGNAL VELOCITIES
339
decreases while the amplitude rises and then falls more or less in the manner of the first precursor. With a sudden rise of amplitude the main body, or principal part, of the disturbance arrivcs, traveling with a velocity w which Brillouin defines as the signal velocity. An explicit and simple expression for w cannot be given and its definition is associated somewhat arbitrarily with the method employed Lo evaluate the integral (32). Physically its meaning is quite clear. In Fig. 62b are shown the two precursors and the final sudden rise to Lhe steady state. When restricted to this third and last phase of the disturbance the term “signal” represents that portion of the wave which actuates a measuring device. Under the assumed conditions a measurement should, in fact, indicate a velocity
.h. 1 42
x ....
\ \ /
.
s 20
w
ido
C / U , u = group velocity
 C/U, FIQ. 63.Behavior
u = phase velocity
of the group, phase and signal velocities in thc neighborhood of a resonance frequency.
of propagation approximately equal to w. It will bc noted, however, that as the sensitivity of the detector is increased, the measured velocity also increases, until in the limit of infinite sensitivity we should record the arrival of the front of tho first precursor which travels always with the velocity c. Qualitatively a t least we may imagine the medium as a region of free space densely infested with electrons. An infinitesimal amount of energy penetrat,cs the empty spaces, as through a sieve, traveling of course with the velocity c. Each successive layer of charges is excited into oscillation by the primary wave and reradiates energy both forward and backward. By reason of the inertia of the charges these secondary oscillations lag behind the primary wave in phase, and this constant retardation as the process progresses through successive layers results in a reduced velocity of the main body of the disturbance. The wavefront velocity defined here is thus always equal to the constant c. The phase velocity v is associated only with steady states and may be either greater or less than c. The group velocity u differs from the phase velocity only in dispersive media. If the dispersion is normal,
340
P L A N E W A V E S I N UNBOUNDED, ISOTROPIC M E D I A
[CHAP.V
the group velocity is less than v; it is greater than the phase velocity when the dispersion is anomalous. I n the neighborhood of an absorption band i t may become infinite or even negative. The signal velocity w coincides with the group velocity in the region of normal dispersion, but deviates markedly from i t whenever u behaves anomalously. The signal velocity is always less than c, but becomes somewhat difficult and arbitrary to define in thc neighborhood of an absorption band. In Fig. 63 the velocities which characterize the propagation of a wave train in the neighborhood of a n absorption band are plotted as ratios of c. Problems 1. Let E a he the amplitudt: of the electric vcctor in volts per nieter and 8 the mean energy flow in watts per square nieter of a plane wave propagated in free space. Show that =
1.327 X
EO = 27.45 4
E; wntt/nietcr2,
3 volts/meter.
2. Let E Ohe the amplitude in volts per meter of the electric vector of a plane wave propagated in free space. Show that the amplitudes of the magnetic vectors H n arid B , are related numerically to E , by
+
E a ampereturn/meter = X lo* En oersted, H , = 2.654 X EOweber/meterz = X lop4E , gauss, Ro = f X
+
where the oersted is the iinr:tt,iori:rlized c.g.s. electromagnetic unit. in statvolts per centimeter, sliow~that
Ho
=
E , oersted,
B,
=
If E , is expressed
Eo gauss.
If a particle of charge q moves with a velocity v in the field of a planc electromagnetic wave, show that the ratio of the forccs exerted by the magnetic and electric components is of the order v / c . 3. The theory of homogeneous, plane waves developed in Chap. V can be extended to inhomogeneous waves. If II. is any rectangular component o f an electromagnetic vector, we may take as a general definition of a plane wave the expression $ = $, ei+,
4 = wt
+ k . r,
in which $z0 is a complex amplit.ude : ~ n d4 the complex phase. k is now a complex vector which wc shall writc as k
=
kl
The propagation factor
+ ZIZ + ipnz, = 0ln1
where n I and nz are unit vectors. Show that 01 is the real phase constant; that the surfaces of constant, real phase are planes normal to the axis nl, and the surfaces of constant amplitude are planes normal to n z ; and that B is the factor measuring attenuation in the direction of most rapid change of amplitude. The true phase velocity v is in the direction nl. Show that the operator V can be replaced by zk and the field equations are
k X E   w B =O, k X H i.wD =  iJ,
k * B = 0, k D = 0.

341
PROBLEMS Show also that, if the medium is isotropic,

Let kl kz = klkz cos 0 and find expressions for kl and kz in terms of p , E, U, and the angle 0. 4. Continuing Problem 3, assume that the electric vector is linearly polarized. Show that E is perpendicular to both kl and kz and that H lies in the plane of these two vectors. What is the locus of the magnetic vector H? Inversely, show that if H is linearly polarized i t is perpendicular to both kl and kz, while E lies in the kl, kzplane. These inhomogeneous plane waves are intimately related to the socalled “ H waves” and “I3 waves,” Sec. 9.18. 6. The theory of plane waves can be extended to homogeneous, anisotropic dielectrics. Let u = 0, p = pol and the properties of the dielectric be specified by a tensor whose components are t j k . Choosing the principal axes as coordinate axes, we have
Dj
=
0’
cOK~E~,
=
1, 2, 3).
The field equations relating E and D are then
k X (k X E)
+ ~*poD
=
0,
k * D = 0.
Let k = kn, where n is a unit real vector whose components along the principal axes are nl, n2, n 3 . Let u = w / k and v j = c/&. The v j arc the principal Velocities. Show that the components of E satisfy the homogeneous system
and write down the equation which determines the phase velocity in any direction fixed by the vector n. Show that this equation has three real roots, of which one is infinite and must be discarded. There are, thcrefore, two distinct modes of propagation whose phase velocities are u‘ and u“ in the direction n. Correspondingly there are two types of linear oscillation in distinct directions, characterized by the vectors E’, D’ and E”, D”. Show that thme are related so that
.
= D’!. E’ = D’ . D” = 0, E’ . E” # 0, E * n # 0.
D’ E”
6. Show that in a homogmeou:, anisotropic dielectric the phase velocity satisfies the Fresnel relation
$y&
50,
j=1
where the n1 are the direction cosines of the wave normal n with respect to the principal axes, and the vl are the phase velocities in t b directions of the principal axes. (See Problem 5 . ) Show also that
342
P L A N E WAVES I N UNBOTINDED, ISOTROPIC M E D I A [CHAP.V
where the yi are the direction cosines of the vector D with respect to the principal axes. 7. If in Problem 6 V I > 02 > 213, show that there are two directions in which v has only a single value. Find these directions in terms of ul, v2, va, and show that in each case the velocity is v 2 . The directions defined thus are the optic axes of the medium which in this case is said to be biazial. What are the necessary conditions in order that there be only one optic axis? 8. Show that for a plane wave in an anisotropic dielectric
where v is the phaso velocity defined in Problem 5. A vectorial velocity of energy propagation u in the direction of S is defined by the equation S
=
hu.
The velocity u is related to the phase velocity v by
Show t h a t
and, hence, that the energy flow is equal t o the total energy density times the velocity
u, with II=
E2n  (n E)E E 2  (n E)2 v,
or in magnitude
9. From the results of Problem 8 show that the magnitude of the velocity u of energy propagation in a plane wave can be found in terms of the constants U I , u t , v3 from the equation
in which the lj are direction cosines of the vector u. Show also that there are, in general, two finite values of u corresponding to each direction. I n what directions are these two roots identical? What is the relation of these preferred axes of u to the optic axes? 10. A nonconducting medium of infinite extent is isotropic but its specific inductive capacity K = t/en is a function of position. Show first t h a t the electric field
343
PROBLEMS vector satisfies the equation
where k 2 = w2ep and the time enters only through the factor exp( id). Assume now that the spatial change of c per wave length is small, so that IVK~ h 1. Show t h a t the righthand term can be neglected and t h a t the propagation is determined approximately by
0.
eBz,
Show that
16. Obtain a Fourier integral representation of the function f ( t ) defined by f ( t ) = 0, f ( t ) = ebl cos w t ,
t
t
0.
Plot the amplitudes of the harmonic components as a function of frequency (spectral density) and discuss the relation between the breadth of the peak and the logarithmic decrement. 17. The surges on transmission lines produced by lightning discharges are often simulated in electrical engineering practice by impulse generators developing a unidirectional impulse voltage of the form
where 01 and p are constants determined by the circuit parameters. This function represents a pulse or surge which rises sharply to a crest arid then drops off more slowly in a long tail. The wave form can be characterized by two constants t i arid L 2 , where t l is the time required to reach the crest, and 1 2 is the interval from zero voltage to that point on the tail a t which the voltage is equal to half the crest voltage. A standard set of surges is represented by the constants (a) t i ( b ) tl (c) t i
= =
0.5, 1.0,
t Z
=
1.5,
t z = 40,
tz = =
5, microseconds.
10,
From these values one may determine the constants
a
m(1 0
346
PL,ANE ILJAVES I N UNBOUNDED, ISOTILOPIC M E D I A
[CHAP.V
Make a spectral analysis of the wave form and determine the frequencies a t which thc spectral density is a maximum for thc three standard cases. 18. The “telegraphic equation”
has the initial conditions i)U

u = f(z),
at
=
gb),
when t
=
0.
Show t h a t u =
arrcz
+ at) + f ( x
where
This solution was obtained independcntly by Hraviside and by Poincari.. 19. An electromagnetic pulse is propagated in a homogmeous, isotropic medium whose constants are c, p, u in the direction of the positive zaxis. At the instant t = 0 the form of the pulse is givrn by
where 6 is a parameter. Find a n expression for t h r pulse f(z, 1) a t any subsequent instant. 20. Wherever tho disp!:mcment current is negligible with respect to the conduc1), the field satisfirs approximately the equations tion current (u/ew
>>
an
VXEI/.lO, at Show that t o the same approxiriiation
nA E=. J
H
at
aA V’A  / . l ~ = 0, at
and that the current density J satisfies
VzJ  pu
(3J 
il1
=
0
’
347
PROBLEMS
What boundary conditions apply to the current density vector? These equations govern the distribution of current in metallic conductors a t all frequencirs in the radio spectrum or below 21. The equation obtained in Problcm 20 for the current distribution in a metallic conductor is identical with that governing thc diffusion of heat. Consider the onedimensional case of a rectangular component J propagated along the zaxis
Show that
is a solution such that at t = 0, J .= f(z), and which is continuous with continuous derivatives for d l values of z when t > 0. 22. In Problem 21 it was found that the equation v
av a22
a$
=
at
is satisfied by ( a  z ) 2
f(a)e
4ut
dor.
From the theory of Fourier transforms show t h a t this can be written in the form
Obviously the equation is also satisfied by
Compare these solutions and from thc uniquencss theorem demonstrate the identity
This relation has been applied by Zwald to the thcory of wave propagation in crystals. 23. Show that if radio waves are propagatcd in an electron atmosphere in the earth's magnetic field, a resonxncr phcnonlenon may be expected near a wave length of about 212 meters. Marked selrriive ahsorption is actually ohserved in this rrgion. Assume e = 1.60 x 1019 coulomb/(Llcctron, m = 9 X 1031 kg /electron, and for the density of the magnetic field of the cmth B , = 0.5 X lop4weber/mcter2. 24. A radio wave is propagated in an ionized atmosphere. Calculate and discuss the group velocity as a function of frequency, assuming for the propagation constant
348
P L A N E W A V E S I N UNBOUNDED, ISOTROPIC M E D I A
CHAP.^
the expression given in Eq. (36), page 329. The concentration of electrons in certain regions of the ionosphere is presumed to be of the order of 1OI2 electrons/meter3. 25. A plane, linearly polarized wave enters an electron atmosphere whose density is 1012 electrons/meter3. A static magnetic field of intensity B O = 0.5 X lo' weber/meter2 is applied in the direction of propagation. Obtain a n expression representing the change in the state of polarization per wave length in the direction of propagation. 26. A radio wave is propagated in a n ionized atmosphere in the presence of a fixed magnetic field. Find the propagation factor for the case in which the static field is perpendicular t o the direction of propagation. 27. As a simple model of an atom one may assume a fixed, positive charge to which there is bound by a quasielastic force a negative charge e of mass m. Neglecting frictional forces, the equation of motion of such a system is m'i + f r = 0,
where r is the position vector of e and f is the binding constant. A static magnetic field is now applied. Show that there are two distinct frequencies of oscillation and t h a t the corresponding motions represent circular vibrations in opposite directions in a plane perpendicular to the axis of t h e applied field. What is the frequency of rotation? This is the elementary theory of the Zeeman effect. 28. T h e force per unit charge exerted on a particle moving with a velocity v in a magnetic field B is E' = v X B. The motional electromotive force about a closed circuit is
V'=$E'.as
=J[vX(vXB)].ndu.
At any fixed point in space the rate of change of B with the time is aB/at. If by E we now understand the total force per unit charge in a moving body, then
aB
V X E =  + V X
at
(V
XB).
Show that the righthand side is thc negative total derivative
so that the Faraday law for a moving medium is
The displacement velocity is assumed to be small relative t o the velocity of light,
CHAPTER VI
CYLINDRICAL WAVES An electromagnetic field cannot, in general, be derived from a purely scalar function of space and time; as a consequence, the analysis of electromagnetic fields is inherently more difficult than the study of heat flow or the transmission of acoustic vibrations. The threedimensional scalar wave equation is separable in 11 distinct coordinate systems,' but complete solutions of the vector wave equation in a form directly applicable t o the solution of boundaryvalue problems are a t present known only for certain separable systems of cylindrical coordinates and for spherical coordinates. It will be shown that in such systems an electromagnetic field can be resolved into two partial fields, each derivable from a purely scalar function satisfying the wave equation. EQUATIONS OF A CYLINDRICAL FIELD
6.1. Representation by Hertz Vectors.We shall suppose that one set of coordinate surfaces is formed by a family of cylinders whose elements are all parallel to the zaxis. Unless specifically stated, these cylindrical surfaces are not necessarily circular, or even closed. With respect to any surface of the family the unit vectors il, iz, i3 are ordered as shown in Fig. 64. il is, therefore, normal to the cylinder, i3 tangent t o it and directed along its elements, and i, is tangent to the surface arid perpendicular t o il and is. Position with respect to axes determined by the three unit vectors is measured by the coordinates ul, u2, x, and the infinitesimal line element is (1)
ds
=
ilhl dul
+ i,h, du2 + i3dz.
Now let us calculate the components of the electromagnetic field associated with a Hertz vector n directed along the zaxis, so that II1 = 112 = 0, IIz # 0. We shall assume in the present chapter that the medium is not only isotropic and homogeneous but also unbounded in extent. Then by (63) and (64), page 32, the electric and magnetic vectors of the field are given by
E(')= v x v x n,
H(l) =
EISENHART, Annals of Math., 36, 284, 1934. 349
(c$
+ .) v x n;
350
[CHAP.VI
CYI,INDRICAI; TTAVES
since both 111and 112arc‘ zero, it is a simplc matter to calciilate from (81) and (85), page 49, tlic componcnts of E(’) and H(I).
Thus we have derivcd from a scalar function IL = $, an electromagnetic field characterized by the absencc of an axial or longitudinal component
X
V
lo FIG.64.Relation
Y
of unit vectors to a cylindriral suiface. arc parallel t o the zaxis.
The generators of the surface
of the magnetic vector. Since n is the elect>ricpolarization potential, this can bc called a field of electric type (pagc 30), but in the present instance the term transverse magnetic field proposed recently by Schelkunoff seems more apt. Since nZis a rectangular component, it must satisfy the scalar wave equation, (5)
V’* 
21c.
a$
p € 7 pu  =
dt
at
0,
1 SCHELKUNOFF, Transmission Tlicory of Plane Electromagnetic Waves, Proc. Inst. Radio Enyrs., 25, 14571492, Novcrnbcr, 1037.
SEC.6.21
SCALAR AND VECTOR POTENTIALS
351
or by (82), page 49,
The elementary harmonic solutions of this equation are of the form
(7)
*
= j(u1, u2),kihziot,
whcre j ( u l , u Z )is a solution of
I n quite the same way a partial field can be derived from a second Hertz vector II* by the operations
if
n*is directed along the xaxis, the components of these vectors are
The scalar function n,“is a solution of ( 5 ) , and the field of magnetic type, or transverse electric, derived from it, is characterized by the absence of a longitudinal component of E. The electromagnetic field obtained by superposing the partial fields derived from II, and 11; is of such generality that one can satisfy a prescribed set of boundary conditions on any cylindrical surface whose gcnerating elemcnts are parallel to the zaxis; i.e., on any Coordinate surface defined by u1 = constantor prescribed conditions on cither a surface of the family u2 = constant or a plane z constant. The choice of these orthogonal families, however, is limited in practice to coordinate systems in which Eq. (8) is separable. 6.2. Scalar and Vector Potentials.The transverse electric and transverse magnetic ficlds defined in the preceding paragraph have interesting propertjies which can be made clear by a study of the scalar and vector potentials. Consider first the transverse magnetic field in which
[CHAP.V I
In the present instance
+ = 11, and, hence,
The components of E(’) are then
for the components of BC1), we obtain
in which A without a subscript has been written for A,. Now it will be noted that in the plane z = constant the vector E(’) is irrotational and therefore in this transverse plane, the line integral of E(‘)between any two points a and b is independent of the path joining them; for an element of length in the plane z = constant is expressed by ds = ilhl dul izhzdu2 and, consequently,
+
The difference of potential, or voltage, between a n y two points in a transverse plane has a dejinite value at each instant, whatever the frequency and whatever the nature of the cylindrical coordinates. Next, one will observc that the scalar function A plays the role of a stream or flow function for thc vector B(l). Let the curve joining the points a and b in Fig. 65 represent the trace of a cylindrical surface intersecting the plane z = constant. We shall calculate the flux of the vector B(‘) through the ribbonlike surface element bounded by the curve ab whose width in the zdirection is unity. If n is the unit normal to this surface and i3 a unit vector directed along the zaxis, we have (18)

B(’) n d a
=
B(’)

(i3
x ds)
=

i3 (ds x B(l)),
SEC.6.21
353
SCALAR AND VECTOR POTENTIALS
where d s is again an element of length along the curve. Upon expanding (18) we obtain (19)

B(1) n d a
hence,
=
hlBhl) d u l  h2Bi1)d u 2 =  d A ;
lb B(’J  n da
(20)
=
A(a)  A@).
T h e magnetic $ux through a n y unit strip of a cylindrical surface passing through two points in a plane z = constant i s independent o j the contour of the strip. If the components of E(’) and B(I) are expressed in terms of the scalar function $, i t is easy t o show that &l)B\l) + E‘1)B‘” = 0 (21) 2 2 7 and consequently the projection of the vector E(l) o n the plane z = constant i s everywhere Y FIG. 65Thc rurvc ah renormal to B(IJ. Moreover, in the transpresents thc intcrscction of a cyverse plane the families of curves, = con lindrical surface with the zyplane stant, A = c o n s t a n t , c o n s t i t u t e a n which contains also the vectors n and B1. orthogonal set of equipotentials and streamlines. When the time variation is harmonic, it is clear from (14) and (7) that the equipotentials are the lines
+
f(ul, u2) = constant,
(22)
in whichf(ul, u2)satisfies Eq. (8). The transverse electric field has identical properties but with the roles of electric and magnetic vectors reversed. According t o ( 3 5 ) , page 27,
(23)
D(2J
=
If now we let $
v x
=
I:,
II:
H(2) =
A*,
=
IIg
=
v+*  dA* ~
0, then
(25) The components of the field vectors are, therefore,
dt
 u A * , 6
354
CYLINDRICAL WAVES
[CHAP.V I
(27)
a 23’ H12’ =
a21C,  pu.atC.
dz2  ptat2
at
The projection of H(2)on the transverse plane is irrotational and, consequently, the line integral representing the magnetomotive force between two points in this plane is independent of the path of integration.
Likewise the flux of the vector D(Qcrossing a strip of unit width as shown in Fig. 65 depends only on the tcrminal point,s. (29)
lb
.n da = L b d A * = A * @ )  A*(a).
D(2)
The projection of H(*)on the plane x = constant is everywhere normal to D(2);hence, the families +* = constant,, A * = constant are ort>hogonal. The field D(2),H(2)is in 1his sense conjugate to the field E(I),B(l). 6.3. Impedances of Harmonic Cylindrical Fields.We shall suppose that the time variable enters only in the harmonic factor ecWt. Then the potentials and field components of a transverse magnetic field are
+
in which k 2 = pew2 ipuw. The upper sign applies to waves traveling in the positive direction along the xaxis, the lower sign to a negative direction of propagation. A set of impedances relating thc field vectors can now be defined on the basis of Sec. 5.6. I t is apparcnt Ilmt the value of t,he impedance depends upon the direction in which it is measured. By definition, (33)
The intrinsic impedance of a homogeneous, isotropic medium for plane waves is by (83), page 283,
SEC.6.41
ELEMENTARY WAVES
355
so that (33) can be expressed as
zp
(35)
=
h t2,. k
Likewise impedances in the dirrctions of the transverse axes can be defincd by the rclat,ioris
(36)
&1’
= Z(1)[j(1) 1
2
1
EL”
=
Z(l)H(l). 2 1
The correlation of the components of electric and magnetic vectors and the algebraic sign is obvioiisly determined by the positive direction of the Poynting vector. Sincc Hi1)is zero, there is no component of flow represented by the terms EL1)HI1), E$l)HL1), and the associated impedances conscquently are infinite. Upon introducing the appropriate expressions for the field components into (%), we obtain
The determination of corresponding impedances for the harmonic components of transverse clectric fields nceds no further explanation. The potentials and field vectors are in this case given by
From these relat,ions are calcu1:rtcd the various impedances.
There emerges from these results the rather curious set of relations: (44)
zpzi2’ = z&”z‘,2)= Z(l)Z(Z) == 2 ‘ 2 ’ z z 0 . WAVE FUNCTIONS OF THE CIRCULAR CYLINDER
6.4. Elementary Waves.By far the simplest of the separable cases is that in which the family ,ul= conslant is represented by a set of
356
CYLINDRICAL WAVES
coaxial circular cylinders. (1)
Then by 1, page 51,
u2 = 8,
u1 = r,
[CHAP.VI
hl = 1,
hz = T ,
and Eq. (8) of Sec. 6.1 reduces to
which is readily separated by writing f ( r , 8) as a product
f
(3)
= fl(T)fZ(@,
wherein fl(r) and f 2 ( 8 ) are arbitrary solutions of the ordinary equations
(4) (5) The parameter p is, like h, a separation constant; its choice is governed by the physical requirement that a t a fixed point in space the field must be singlevalued. If, as in the present chapter, inhomogeneities and discontinuities of the medium arc excluded, the field is necessarily periodic with respect to 8 and the value of p is limited to the integers n = 0, *I, &Z, . . . . When, on the other hand, a field is reprcscnted by particular solutions of (4) and (5) in a sector of space bounded by the planes 8 = 81, 8 = 82, it is clear that nonintegral values must in general be assigned to p . Equation (4) sahfied by the radial function f l ( r ) will be recognized as Bessel's equation. Its solutions can properly be called Bessel functions but, since this term is usually reserved for that particular solution J p ( d mr ) which is finite on the axis r = 0, we shall adopt the name circular cylinder function to denote any particular solution of (4) and shall designate it by the letters f l = Z , ( v " ~r). The order of the function whose argument is d k 2  h2r is p . Particular solutions of the wave equation ( 5 ) , page 350, which are periodic in both t and 8, can therefore be constructed from elementary waves of the form (6)
=
ein6
z,(42
r ) ,+ihziot.
The propagation constant h is, in general, complex; consequently the field is not necessarily periodic along the zaxis. An explicit expression for h in terms of the frequency w and the constants of the medium can be obtained only after the behavior of $ over a cylinder r = constant or on a plane z = constant has been prescribed.
SEC.6.51
357
PROPERTIES OF THE FUNCTIONS Z&)
6.6. Properties of the Functions Z,(p).Although it must be assumed that the reader is familiar with a considerable part of the theory of Bessel’s equation, a review of the essential properties of its solutions will prove useful for reference. If in Eq. (4) the independent variable be changed to p = d2 T, me find that Z p ( p ) satisfies (7) an equation characterized by a regular singularity at p = 0 and a n essential singularity a t p = m. The Bessel function J , ( p ) , or cylinder function of the first kind, is a particular solution of (7) which is finite a t p = 0. The Bessel function can, therefore, be expanded in a series of ascending powers of p and, since there are no singularities in the complex plane of p other than the points p = 0 and p = m , it is clear that this series must converge for all finite values of the argument. For any value of p , real or complex, and for both real and complex values of the argument p we have
When p in (7) is replaced by  p , the equation is unaltered; consequently, when p is not an integer, a second fundamental solution can he If, however, p = n is a n obtained from (8) by replacing p by  p . integer, J,(p) becomes a singlevalued function of position. The gamma function r(n m 1) is repheed by the factorial (n m ) ! ,so that
+ +
+
(9)
(n = 0, 1,2,
The function it by
J,(p)
*
*
*
).
is now no longer independent of (9) but is related to
(10)
Jn(Pj
= (l)nJn(P),
so that one must resort to some other method in order to find a second solution. The Bessel function of the second kind is defined by the relation (11)
NAP)
1
=
[J,(P)
cos p n  J*(P)l.
This solution of (7) is independent of J,(p) for all values of p , but the righthand side assumes the indeterminate form zero over zero when p
358
[CHAP.
CYLINDRICAL WAVES
VI
is a n integer. It can be evaluated, however, in the usual way by differentiating numerator and denominator with respect to p and then passing to the limit as p + n. The resultant expansion is a complicated affair1 of which we shall give only the first term valid in the neighborhood of the origin.
(n = 1, 2,
*

. ),
where y = 1.78107 and [pi
(34)
When init,ial conditions arfl prescribed over given plane or cylindrical surfaces, a solution is constructed by snpcq~ositionof clementary wave functions. For fixed values of w (or k ) and h, one obtains for the resultant field in cylindrical coordinates the equations m
n=
m
m
n=
m
where a, and bn are coefficicnfs to be determined from initial conditions. The direction of propagation is positive or negative according t o the sign of h. INTEGRAL REPRESENTATIONS OF WAVE FUNCTIONS
6.7. Construction from Plane Wave Solutions.If { measiires distance along any axis whose direction with respect to a fixed reference system ( 2 , y, z ) is determined by a unit vector n, then the most elementary type of plane wave can be represented according to Secs. 5.1 to 5.6 by
362
CYLINDRICAL W AVES
[CHAP.V I
the function
9 = eikfiut
(1)
?
in which the constants k and w are eithcr real or complex. Let R be the radius vector drawn from thc origin to a point of observation whose rectangular coordinates are 5 , y, z. The phase of the wave function a t a given instant is then measured by
5
(2)
=

n R
=
n,z
+ n,y + n,z.
The direction cosines n,, nu,n, of the vector n are best expressed in terms of the polar angles CY and p shown in Fig. 66.
(31
n, = sin a cos
p,
n,
=
sin a sin p,
n,
=
cos a ;
hence,
(4)
$=
As t,he paramekrs
CY
e i k ( z sin
a
cos j3+y sin a sin B f z 00s a) i d
are varied, the axis of propagation can be oriented a t will. With each direction of propagation one associates an amplitudc g(a, p) depending only on the angles a and p ; since tho field equations are in the prescrit case linear, a solution can be constructed b y mpcrposing plane waves, all of the same frequency but traveling in various dircct,ions, each with its appropriat,e amplitude.
and
Y
(5) +(x, 2 / ,
2,
t ) = e'"'JdaJdPg(a,
&(z
sin
(I
coa B+y sin
( I
P)
sin B f z cos (I).
FIG.tXThc phase of an elementary plane wave is measured along tlie ritzis whose direction is detcrmined by the unit vector n. A fixed point of observation is located by the vcctor R.
If the angles are real, the limits of integration for a are obviously 0 and r ;p goes from 0 t,o 2n. But such a solution is mathernatically by no means the most general, for (5) satisfies the wave equation for complex as well as real values of the parameters a and /3 and we shall discover shortly that complex angles must in fact be included if we are t o represent arbitrary fields by such an integral. When o is real, the wave funct'ion defined by (5) is harmonic in time. To represent fields whose time variation over a specified coordinate 1 The general theory of such solutions of the wave equation has been discussed by Whittaker. See WHITTAKERand WATSON,"Modern Analysis," 4th ed., Chap. XVIII, Cambridge University Prcss, 1927.
PEC.6.71
CONSTRUCTION PROM PLANE W A V E SOLUTIONS
3 63
surface is more complex i t will be necessary t o sum or integrate (5) with respect to the parameter w. Wc shall define a vector propagation constant k = k n whose rectangular components are
(6)
kl
=
k sin a cos p,
kz
=
k sin a sin 0,
k 3 = k cos a,
so that the elementary plane wave function can be written
+ = ezk.Rzwt (7) It follows upon introduction of (7) into the wave equation dz+
(8)
ax2
+ az+ + dz2 a2+
@
!Je
az+
a+
z 2  PJ d t = 0,
that the components of k must satisfy the single relation
(9)
k?
+ ki + k2, = pew2 + ipuw = k 2
and are otherwise completely arbitrary. Thus of the parameters kl, k 2 , k3,W , any three may be chosen arbitrarily whereupon thc fourth is fixed b y (9). Let 11s suppose then that ovcr the plane x = 0 thc function is prescribed. We shall say that +(z, y, 0, t ) = f(z,y, t ) . Thc desired solution is
+
in which kl, k2 and w are real variables and mined by
(11)
k;
= w2p€
k 3
is a romplex quantit,y detcr
+ ipuw  k::  k,2.
The amplitude function is t,o he such that
If f(z, y, t ) and its first derivatives arc pirccwise continuous and alxolutely integrable, then g(kl, k,, w) is its Fourior transform and is given by
When (r = 0, each harmonic component is propagatrd along the xaxis ._ with a velocity v = w / k 3 , but since k a = d i & L  k2,  k ; is not a linear combination of w, kl, and k,, it is apparent that the initial disturbance ~
~
364
[CHAP.V I
CYLINDRICAL WAVES
f(z, y, t ) does not propagate itself without change of form even in the absence of a dissipative term.
More precisely, there exists no general
 t>. ( Equally one might prescribe the function
solution of (8) of the typef
2, y,
1

#(s, y, z, t ) throughout all space a t an initial instant t = 0. Let us suppose, for example, that #(z, y, z, 0) = f(z,y, z). The field is to be represented by the multiple integral
in which ?GI, kp, JC3 arc real variables and o is a complex quantity determined from (9).
+ Li + k:)  b2, 
(15)
io = b
f i \/a2(&
where a = l/dG, b = a/2e as defined on page 297. weighting function g(k1, k z , JCS) is to be such that
The amplitude or
If f(z, y, z) has the necessary analytic properties, the Fourier transform exists and is given by
Only positive or outward waves have been considered in the foregoing and initial conditions have been imposed only upon the function # itself. If both # and a derivative with respect to one of the four variables are to satisfy specified conditions, it becomes necessary to include negative as well as positive waves according to the methods described for the onedimensional case in Secs. 5.8, 5.9, and 5.10. 6.8. Integral Representations of the Functions Z,(p).In any system of cylindrical coordinates u1, u2, x , the wave equation (6), page 351, is satisfied by (18)
#
= f(u1,
u2) f +  ~ t ,
where h and o are real or complex constants. Following the notation of ~ _ _ the preceding section h = ks = k cos and, since L = d p c w 2 i p u w , it (Y
+
SEC.6.81
365
INTEGRAL REPRESENTATIONS
follows that the angle a made by the directions of the component plane waves in (5) with the zaxis is also constant. I n other words, the elementary cylindrical wave function (18) can be rcsolved into homogeneous plane waves whose directions form a circular cone ahout the zaxis, but the aperture of the cone is in the general case measured by a complex angle. (19)
in which x and y are to be expressed in tcrms of the cylindrical coordinates
u',u2. I n the coordinat,es of the circular cylinder we have x
=
r cos 0,
y = r sin 8, and, hence,
PO)
x cos p
+ y sin p = r cos ( 6  p).
I n the notation of the preceding paragraphs we have also
(21)
kr sin a
=
r 4 F h Z = p,
so tha t
We now change the variable of integration in (22) from p to + =  8, and note that, since the equation for f(r, 0) is separable, it must he possible to express g(+ 0) as a product of two functions of one variable each.
+
(23)
g(P) = g(+
+ 0)
= g1(+>g2(0);
hence,
The angle function gZ(0) must obviously be some linear combination of the exponentials e i p O and eciPo, and the amplitude or weighting fact,or gl(+) must be chosen so that the radial function fl(r) satisfies (4), page 356, which in terms of p is written
Let us substitute into (25) the integral
366
CYLINDRICAL WAVES
[CHAP.
VI
Differentiating under the sign of integration, we have (27)
hence from (25) (28)
(p2
sin2 4
+
ip
cos 4
 p2)gl(+)e i p C o s
0
d+
=
0.
This equation is next transformed by an integration by parts. (28) is evidently equivalent to
Equation
If P and Q are two functions of +, then
this applied t o (29) gives
The first of these two ntcgrals can be made to vanish if the contour of integration C is so chosen t h a t the differential has the same value a t both thc initial and terminal points; the sccond is zero if the intcgrand vanishes. Therefore (26) is a solution of the Bessel cquation if gl(+) satisfies
and if the path C of intcgrat,ion is such that
Equation (32) is evidently satisficd by However, we shall find that, if fl(p) is to be identical with the cylinder functions Z,(p) defined 1 ip; in Sec. 6.5, a constant factor  e must be added. Then by choosing el*+.
lr
SEC.6.81
367
I N T EGR A L REPRES E N T ATIO NS
we obtain Sommcrfeld's integral rr:prcsentat.ion of the cylinder functions,' . a
P (P ) =
(35)
e
apz
.I
e i b cos ++P+)
P
the contour C to be such that
&,
I
( p sin 4 + p ) e i ( p coa ++p+) z 0. (36) C The distinction between the various particular solutions J , ( p ) , N,(p), H,(p) is now reduced to a distinction in thc cont,ours of integration i n the complex plane of 4. Consider first the most elementary case in which p = n, an integer. Then clearly, if the path of integration is extended along the real axis of 4 from  P t'o P, or any other segment, of length 2 ~ the , condition (36) is fulfilled and the definite integral
in
(37)
JJP)
=
7&
eip
cos ++in+
d4
is a solution of the bessel equation. That the function defined in (37) is actually identical with the particular solution (9), page 357, can be verificd by expanding e i p C o s 9 in a series about the point p = 0 and integrating term by t'erm. I n the general case when p is not, an integer, or t o obtain the independent solution, one must choose complex values of 4 in order that (36) shall vanish a t the terminal points. Let 4 = y in. Then
+
+
+
+
i p 4 = p sin y sinh r]  pr] i p cos y cosh q i p y . (38) ip cos 4 If p is complex we shall assiime that p = a ib, where a is a n essentially positive quantity. Then by an a'ppropriate choice of y and r] the real part ip4) 4 0 . of (38) can be made infinitely ncgative, while exp(ip cos 4 This condition will be satisfied, for example, if we let 7 + 00 and choose for y either  ~ / 2or + 3 ~ / 2 ;but the exponential also vanishes when 7 +  provided y = + ~ / 2 . I n order that (35) shall represent a solution of the Bessel equation, it is only necessary that the contour C connect any two of these points. Sommerfeld has chosen as :L pair of fundamental solutions the two int'egrals
+
+
SOMMERFELD, Math. Ann., 47, 335, 1896.
+
3 68
CYLINDRICAL WAVES
[CHAP.VI
The contour C1 followed by the integral (39) starts a t q = 00, y =  ~ / 2 , crosses both real and imaginary axes a t C#I = 0, and terminates eventually a t r ] =   m , y = a/2. The contour C z followed by (40) starts a t the terminal point of C1, crosses the real axis a t y = T , and comes to an end at q = a, y = 3n/2. Both contours are illustrated in Fig. 67. The point at which the crossing of the real axis occurs is, of course, not essential to the definition. The contours may be deformed a t will provided only they begin a t an infinitely remote central point of one shaded area and terminate a t a corresponding point in a second shaded area.
FIG.67.Contours
of integration for the cylinder functions.
The advantages of carrying C1 and Cz across the real axis a t thc particular points y = 0 and y = P become apparent when the integrals (39) and (40) are evaluated for very large values of p. For if the real part of p is very large, the factor cxp(ip cos 4 ) becomes vanishingly small a; all points of the shaded domains in Fig. 67 with the exreption of the immediate neighborhood of the points q = 0, y = 0, * T , * 2 ~ , . . . . At these points the real part of i p 30s 4, Eq. (38), is zero however large p ; consequently, if C1 and Cz are drawn as in Fig. 67, the sole contribution to the contour integrals will be experienced in the neighborhood of the origin and the point q = 0, y = a. Out of a shaded region in which the values of the integrand are vanishingly small, the contour CI leads over a steep “pass” or “saddle point” of high values a t the origin, and then abruptly downwards into another shaded plane where the contributions to the integral are again of negligible amount. The contour 6 2 encounters a similar saddle point a t q = 0, y = P. To confine the integration to as short a segment of the contour as possible, one must approach the pass by the line of steepest ascent, descending quickly from
SEC.6.91
FOURIERBESSEL INTEGRALS
369
the top into the valley beyond by the line of steepest descent. This means in the present case that the contours C1, Cz must cross the axis at a n angle of 45 deg. The behavior of the integrals (39) and (40) in the neighborhood of a saddle point has been uscd by Debye’ to calculate the asymptotic expansions of the functions H Y ) ( p ) and H g ) ( p ) . When p is very large, both with rcsprct to unity and to the order p , one obtains the relations (22) and (23) of page 359, thereby identifying the integrals (39) and (40) with the Hankel functions defined in See. 6.5. Debye considered also the case in which p was larger than the argument p, but his results have been improved and extended by more recent investigations. A contour integral represcntation of J p ( p ) , when p is not an integer, follows directly from the relation
The contour C S shown in Fig. 67 represents a permissible deformation of c1 6.9. FourierBessel Integrals.It has been shown how in cylindrical coordinates the two fundamental types of electromagnetic field can be derived from a scalar function y5. I n case the cylindrical coordinates arc circular, y5 is in general a function obtained by superposition of elementary waves such as the particular solutions (28) and (29) of page 360. Our problem is now the following: a t the instant t = 0 the value of +(r, 0, x, t ) is prescribcd over the plane x = 0; to determine J, for all other values of z and t. Let us suppose then that when t = x = 0, $ = f ( r , 0). We shall assume that f ( r , 0) is a bounded, singlrvalued function of the variables and is, together with its first derivatives, piecewisc continuous. Then f ( ~0), must be periodic in 0 aiid can be expanded in a Fourier series whose coefficionts are functions of r alone.
+ cz.
If now f ( r , 0) vanishes as r vergence of the integrals
so“
+ m
in such a way as to ensure the con
IfvL(r)/ & dr, then each coefficientf,(r) can
be represented by a modified Fourier integral.2 DEBYE,Math. Ann. 67, 535, 1909. See also WATSON,Zoc. cit., pp. 235$. Tho FourierBessel integral is established by more rigorous methods in Chap. XIV of Watson’s “Bessrl Functions.”
3 70
C YLZNDRZCAL II'A VES
[CHAP.VI
Consider a function f(x, y) of two variables admitting the Fourier integral representation
A transformation to polar coordinates is now made both in coordinate space and in k space.
x
(45)
kl
=
y = r sin 8, kz = X sin p,
r cos 8, cos p,
= X
so that in the notation of earlier paragraphs X = k sin OL = 42. Thcn klx k2y = Xr cos ( p  8) and (44), as a volume integral in k space, transforms to
+
(46)
f ( r , 8)
=
dh
27r
Lzr
do g(X, p)
eiiy
co3 ( 6  0 ) .
The physical significance of this representation is worth noting. The function exp[iXr cos (0  e)  id] represents a plane wave whose propagation constant is X, traveling in a direction which is normal to the zaxis ?nd which makes an angle p with the xaxis. Each plane wave is multiplied by an amplitude factor g(X, p) and then summed first with respect to p from 0 to 2a and then with respect to the propagation constant, or space frequency A. The transform of f(x, y) is
which when transformed t o the polar coordinates f ieads to
(48)
g(x,
p)
p
=
Suppose finally that f(r,
e)
dp
so2'
d,u j ( p , p)
= fn(r) eins.
Then
= p cos p, 7 = p
e  i x p cos
(0p).
sin p,
SEC.6.101
REPRESENTATION OF A PLANE W A V E
371
Likewise (46) becomes now
or upon placing 4 (52)
=
fi
jn(.)

0,
einQ =
einQ
Thus we obtain the pair of FourierBesscl transforms (53)
(54) The function
is a solution of the wave equation in circular cylindrical coordinates, which a t t = 0 reduces tof(r, 0) on the plane x = 0. But this obviously is not the most general solution satisfying these conditions, for we have still at our disposal the two parameters w and k which are subject only to the relation k 2 = pew2 i u p w . If w is real, harmonic wave functions such as (55) may be supcrposcd to represent an arbitrary time variation of over the plane z = 0. If both positive and ncgative waves are considered, i t is permissible also to assign values to both $ and d+/dz at z = 0. The treatment of such a problem has been adequately illustrated in Chap. V. 6.10. Representation of a Plane Wave.The FourierBessel theorem offers a very simple means of representing an elementary plane wave in terms of cylindrical wave functions. Let the wavc whose propagation constant is k travel in a dircction defined by the unit vector n p whose spherical polar angles with respect to a fixed reference system are a and p as in Fig. 66, page 362. Thcn
+
+
(56)
+
= ezk
sin a ( z cos @ + y sin @)
. ezka cos
and our problem is t o represcnt the function f(z, y) = eik sin u ( z cos @+y 8in 8) = (57)
ezkr sin a cos ( 8  0 )
as a series of the form m
(58)
j ( r , 0) =
C *=
jn(r)cine. 00
crwt
372
CYLINDRICAL W A V E S
By (43)we have (59)
l=
f&> = 1 2n
eikr sin a cos
[CHAP.VI
( p  8 )  in8
Upon replacing fl  8 by 9, this becomes equal to
and we obtain thereby thc useful expansion (61)
sin a cos ( p  8 ) =
2
n=
inJn(kr sin a ) ein(B0). cc
Several other wellknown series expansions are a direct conscqiicnce of this result. Thus, if we put
p =
kr sin a, 0  p
=
9
7r )
2
(61)
becomes m
which upon separation into real and imaginary parts lcads t o m
THE ADDITION THEOREM FOR CIRCULARLY CYLINDRICAL WAVES
bol,\
6.11. Several important relations pertaining to a translation of the axis of propagation parallel to itself can be derived from the formulas of the foregoing section. I n Fig. 68 0 and 01 denote the origins of two rectangular r e f e r e n c e s y s t e m s . The sheet of the figure coincides 1_ _ _ _ _ _ XI with the zyplane of both systems 01 and the axes 21,y ~ z1 , through O1 are r0 e s: respectively parallel to 2,y, 2. The 0 X function Jn(Xr,) einel, when multiFIG. tXTranslation of the referonce plied by exp( +ihzl  iwt), represystem. sents a n elementary cylindrical wave referred to the zlaxis. We wish to exprcss t,his cylindrical wave in terms of a sum of cylindrical wave functions referred to the parallel zaxis through 0.
373
A D D I T I O N THEOREM
SEC. 6.111
We write first
From the figure it is clear that 81 (2)
Tl COS
rl sin
= 0
+ II. and that
II. = rl cos (el  e) = T  yo cos (e  e,), II. = r1 sin (el  0) = ro sin (e  e,).
Moreover, in virtue of the periodic properties of the integrand, (1) is equivalent to
It follows from (2) that
(4)
r1 cos (+ 
4) =
r cos
4  r0 cos
(+
+ 0  eo);
hence,
By (61), page 372, we have
and in virtue of the uniformit,y of convergence one may interchange the order of summation and integration. m
m=
Upon replacing (8)
m
el by e + II. this result becomes
J,(XrI) ein* =
5
m=
J,(xro)J,+,(xr)
eim(000).
m
An analogous expansion for the wave function Hil)(Xrl)einsl can be obtained by writing (9)
3 74
CYLINDRICAL WAVES
[CHAP.VI
where C1 is the contour described in Fig. 67, page 368, with a shift of amount $ along the real axis to ensure the vanishing of the exponential a t the terminals. Upon expanding the exponent by (2), we find
If now Irl > / r o c,os (0 is led to the expansion
&)I,
one can proceed exactly as above and
m
m= 
m
If ro = 0, the two centers coincide; J,(O) = 0 for all values of m except zero, and J o ( 0 ) = 1. The angle II, is zero and the right and left sides of '(11) are clearly identical. The expansion can be verified also in the case when r and r1 are very large, for then the Hankel functions can be replaced by their asymptotic representation, (22), page 359. The angle is approximately zero and rl N r  ro cos (6  0,). The amplitude factor 1/2/arl can be replaced by m r without appreciable error, but the term ro cos (6  0,) must be retained in the phase. Then (11) reduces to
+
and the righthand side of (12) in turn is by (61), page 372, the correct expansion of the plane wave on the left. Indeed as r1 becomes infinite, the expanding cylindrical wave function designated by (11) must become asymptotic to a plane wave. When Irl < Ir0 cos (0  @,)I the expansion (11) fails to converge and is replaced by
(13)
H c ) ( l r l )ein*
=
3 m=
H2)(Aro)Jn+m(Ar) eim(eeo),
m
which is finite a t r = 0. At this point Jn+,(Xr) is zero unless m = n. Moreover N ?r eo  e and, since IIYA(p) = enTiHk1)(p),the right and left sides of (13) are clearly identical. I n the other limiting case of r0 very large, we write rl = ro  r eos (6  0,) and find by means of the asymptotic representations of the Hankel functions that (13) approaches
+
+
a planewave function propagated from 0 1 towards 0 along the line joining these two centers.
SEC.
375
ELEMENTARY WAVES
6.121
WAVE FUNCTIONS O F THE ELLIPTIC CYLINDER
6.12. Elementary Waves.Cirrular wave functions are in a sense a degenerate form of elliptic wave functions obtained by placing the eccentricity of the cylindcrs equal to zero. The analysis of the ficld and the properties of the functions must inevitably prove more complex in elliptic than in circular coordinates, but the results are also fundamentally of greater interest. From 3, page 52, we take
f(E,
upon introducing these into (8), page 351, we find that satisfy the cquation
7) must
+ Co”(k2 h2)(f2  q2)f This in turn is readily separated by writing f
d2fi (t2 1) 7 + E (if1dE f [c;(k2  h2)E2 b]f l dE
(5)
(1  Vz)
d*fz
&T  7
!fi + [b  C i ( k 2  h2)72] (ill
f2
0
and leads to
= fl(()fz(q)
(4)
=
=
0,
=
0,
where b is an arbit>rary constant of separation. Thus fI(E) and f Z ( 7 ) satisfy the same dzflerential equation. Equations (4) and ( 5 ) are in fact special cases derived from the associated Mathieu equation1
(6)
( 1  9)w”  2 ( a
+ 1)zw’ + ( b  czz2)w= 0
by putting the parameter a = 4. These equations are characterized by an irregular singularity at infinity and regular singularities a t x = k 1. When (>> 1, (4) goes over to a Bessel equation. A certain simplification of (4) and (5) can be achieved by transformation of the independcnt variables. Let
E
(7)
=
cosh
U,
7 =
cos
V,
so that the transformation to rectangular coordinates is expressed by
(8)
x
=
cg cosh u cos v,
y
= co sinh
u sin v.
1 WHITTAKER and WATSON, Zoc. cit., Chaps. X and XIX; INCE, “Ordinary Differential Equations,” Chap. XX, Longmans, 1927.
3 76
[CHAP.VI
CYLINDRICAL WAVES
Then in place of (4) and (5), we obtain
d”fz + (b  c;x2 dv2
COY2 V ) f i =
0,
where as in the preceding sections X = d k 2  h2. The distance between the focal points on the xaxis, Fig. 9, is 2c0 and the eccentricity of the confocal cllipses is e = l/cosh u. The transition to the circular case follows as the limit when co 4 0 and u 3 m . Then co cosh u + co sinh u +r , and v is clearly the angle made by the radius r with the xaxis. For this reason we shall refer to fl(u) as a radial function and f2(v) as a n angular function. When u = 0, the eccentricity is one and the ellipse reduces to a line of length 2c joining the foci on the xaxis. The Mathieu equations (9) and (10) have been studied by many writers. We shall consider first the angular functions f2(v). There are, of course, solutions of (10) whatever the value of the separation constant b. But the electromagnetic ficld is a singlevalued function of position and hence, if the properties of the medium are homogeneous with respcct to the variable v, i t is necessary that fi(v) be a periodic function of the angle v. Now Eq. (10) admits periodic solutions only for certain characteristic values of the parameter b. These characteristic values form a denumerable set bl, bz, . . . , b,, . . . . Their determination is a problem of some length, so we shall content ourselves here with a rcferencc to tabulated results1 and proceed directly to the definition of the functions satisfying (9) and (10) when b coincides with a characteristic value b,. For the proper values of b, Eq. (10) admits both even and odd periodic solutions. The denumerable sct of characteristic values leading to even solutions may be designated as bg) and the associated characteristic 1 A very readable account of the theory of the Mathieu functions has been given by Whittaker and Watson, Zoc. cit., Chap. XIX; further details with extensive references to the literature have been published by Strutt in a monograph entitled “Lam& sche Mathieusche und verwandtr. Funktioncn in Physik und Technik,” in the collection ‘i Ergebnisse der Mathematik,” Springer, 1932. ‘rables of Mathieu functions and characteristic values have been published by Goldstein, Trans. Cambridge Phil. SOC., 23, 303336, 1927. The functions Se, and So, defined in the text differ from the CF, and se, of these authors only by a proportionality factor. Whereas Goldstein chooses his coefficients such that the normalization fartor is we have found i t advantageous to normalize in such a way t h a t the even function and the derivative of the odd function have unit value a t thc pole v = 0. See Stratton, Proc. Nut. Acad. Sci. U . S., 21, 5156, 316321, 1935; and MORSE,ibid., pp. 5662. Extensive tables of the expansion coefficients I):, the chararteristic values b , and the normalization factors have been computed by P. M. Morse and will appear shortly.
4,
SEC.6.121
377
ELEMENTARY W A V E S
functions can then be represented by the cosine series (11)
Se,(coX,
cos v)
=
c' D:: cos nu,
(m = 1, 2, 3,

*
*
),
n
where the primed summation is to be extended over even values of n if m is even and over odd values of n if m is odd. The recursion formula connecting the coefficients Q(c0X) can be found by introducing (11) into (10). All coefficients of the series are thus referred to an initial one which is arbitrary. It is advantageous to choose this initial coefficient in such a way that the function itself has unit value when v = 0, corresponding to 9 = cos v = 1. To this end we impose upon the D; the condition n
The odd periodic solutions of (10) are associated with a second set of characteristic values to be designated by b:). These functions can be represented by a sine series So,(coX,
cos v)
=
2' F;
sin nu
n
upon whose coefficients F;(coX) we impose the condition
c' nF;
=
1.
n
Conscquently the derivativc of So,(coX, = 0.
cos v) will have unit value a t
2,
;[
So,(coX,
cos v)
= 1.
The characteristic functions Se, and So, constitute a complete orthogonal set,. Let by) and by) be two characteristic values and Se,, Sei the associated functions. They satisfy the equations (15)
d2Sej dv2
F _
+ (6je)  c;h2 cos2 v)Sei = 0.
Multiply (15) by Sei, (16) by Sei, and subtract one from the other.
Upon integrating (17) from 0 to 27r and taking account of the periodicity of the functions, we obtain Sei(coX, cos v) Sei(coX, cos v) dv = 0,
Np,
i. # j ., z = 3.
3 78
CYLINDRICAL WAVES
[(:HAP.
v1
The normalization factors N g ) can be computed from the series expansions of the functions. In identical fashion one deduces that
su”” SOi(COX, cos v) S O i ( C O X , cos v) dv and (20)
I”’
Se,(coX, cos v) X o , ( c o X , cos
=
0,
Ay’,
i # j, i = 3.
v) dv = 0,
the last for i = j as well as i # j . With each characteristic value bg) there is associated one and only one periodic solution Se,. For this same number bg) there must exist, however, an independent solution of (10). Since the second solution is nonperiodic it is unesscntial in physical problems so long as the medium is homogeneous with respect to the angle v. When, on t8heother hand, there are discontinuities in the properties of the medium across surfaces v = constant, the boundary conditions may require use of functions of the second kind. We turn our attention now to the radial functions. It can be shown without great difficulty that Eq. (9) is satisfied by an expansion in Bessel functions whose coefficients differ only by a factor from those of (11) and (13). Thus, when the parameter b assumes one of the characteristic values bg), an associated radial function is
where [ = cosh u, imn=
. Since all solutions of
Bessel’s equation satisfy the recurrence relations (24) and (25), page 359, the even radial functions of the second kind are defined in terms of the Bessel functions of the second kind by
The convergence of such series of functions is not easy to demonstrate but appears to be satisfactory in the present instance. The great advantage of these representations is that they lead us at once to asymptotic expressions for very large values of coXE. By (18) and (19), page 359, we have
SEC.6.121
379
ELEMENTARY WAVES
By analogy with the Hankel functions, we are led to form the linear combinations
The function Re; satisfies the same equation as the angle function Se,; neither has singularities other than that at infinity. The two must, therefore, be proportional to one another. (29)
Se,(coX, 5)
=
6l g ) R e h ( c o X , E ) ,
2'
1 1:) =
inm
__ D::
D$
ninm
(30)
2 D"
( m odd).
Df
COAT
On the left, t replaces cos o.
(m even),
.Lz'
lr
Consequently when u = 0, 5
=
1, we have
Ret,(coX, 1) =
A corresponding set of relations can be obtained for the odd radial functions. We define:
+
Rok(coX,[) = ROT, iRo2,. (33) Rof(coX, E) = Rot, iRoi, The asymptotic expansions of the odd functions are identical with those of the corresponding even functions. Finally, (34) i Xo,,(coX,
.F)
=
6zg)Rot,(coX, t ) , 4
__
12) =
7rCiX2
~
2
*COX
3'FFznE n
7'
in,
inm
(meven) , (modd).
380
[CHAP.VI
CYLINDRICAL WAVES
At u = 0,
= 1, we have
With these functions a t our disposal we are now in a position to write down elementary wave functions for the elliptic cylinder. The even and odd wave functions, which are finite everywhere and in particular along the axis joining the foci, are constructed from radial functions of the first kind. (36)
$t = Sem(coX,cos v) Rek(coX, E ) e+idkZzXi._iwt,
(37)
$& =
Xo,(cok, cos
v) Rot,(coX, l) e f i d k.~ 2   X Zczot.’
When it is known that at great distances from the axis of the cylinders the field is traveling radially outward, the elementary wave functions will be constructed from radial functions of the third kind. (38)
J.,,, = Xe,(coX, cos v) Ref(coX, [) e*idEzLzht,
(39)
+m :
= SO,(C~X, cos
t)
v)
~ o k ( c ~ ~e *, i
d m z  i w t .
The components of the electric and magnetic vectors can now be found by the rules set down in See. 6.3. 6.13. Integral Representations.According to (19), page 365, we have in any system of cylindrical coordinates
f(u1,
(40)
~
g(p)
= )
2
eik sin 4 z cos 8+2/ sin 8)
dp.
If now the rectangular coordinates are replaced by elliptic coordinates through the transformation x = co cosh u cos v, y = cu sinh u sin v, we can write (40) in the form f(u,v>
(41)
=
J g(P>
eihp
dp,
___
where as in the past X = d k 2  h2 = k sin a, and where
+ y sin
co(cosh u cos v cos /3 sinh u sin v sin @.. This function j(u,v) is to satisfy the equation
(42)
p(u,v, 0)
(43) but sincef (44)
=z
cos /3
3 + i3U2
3+
= ji(u)ji(v), it is
=
ciX2(cosh2u  cos2 v)f
=
0,
clear t,hat it must also satisfy
azf + (ciX2 cosh2 u  b)f aU2
+
= 0,
SEC.6.131
INTEGRAL REPRESENTATIONS
obtained by multiplying (9), page 376, by fz(v). (41) with respect to u,we have
38 1
Upon differentiating
(45) if, therefore, (41) is to satisfy (44), it is necessary that
[ 2  ?$) + 2
(46)
g(p) i x
A2
cgX2 cosh2 u  b
Now p is a function of 0 as well as of u and it can be easily verified that (47)
Consequently (46) is equivalent to (49)
Finally by (30), page 366,
hcnce, (41) is a solution of (44) provided g(p) satisfies
3 + (b do2
c;x2
cos2 P)g
=
0,
and the path C of integration is so chosen that
It will be noted that the equation of the transform g(p) is identical with that of the angular function j2(v),and in fact differs unessentially from that for fl(u)with which we set out. This property is common to the transforms of all solutions of equations belonging to the group defined by (6), page 375. The results just derived are valid whatever the value of the separation If, however, we confine ourselves to the even and odd sets constant b,. of characteristic values bg) and b z ) , it will be advantageous to choose for g ( p ) the periodic solutions of (51) which have been designated by Se,(coX, cos p), Xo,(cGX, cos 0). Since p(u, v, p) is also periodic in p with period 27r,it is apparent that (52) will be fulfilled when the path of integration
382
CYLINDRICAL WAVES
[CHAP. VI
is any section of the real axis of length 2a. An integral representation of f(u, v) which holds when b belongs to the set b z ) is then
(53) From this an integral representing j l ( u ) can be found immediately by placing v = 0.
jE(u) =
(54)
L2r
cash
Sc,(coX, cos p)
co8
B dj3.
There remains the task of identifying the particular solution defined by (54). Upon introducing the cosine expansion of Se, and abbreviating C O X cosh u = p, we obtain (55) f E ( u ) =
2'D; Szr
eip
cos B
0
n
cos np dp
=
27r
2'inD:Jn(p). n
Next we note that in = (l)"i", and remember that the summation extends over even values of n if m is even; over odd values only if m is odd. Therefore (l)n = (  l ) m ,and in virtue of the definition (21)) page 378, it follows that
where = cosh u. Moreover, Sen,(coX, cos p) is proportional to ReA(coX, cos a), in which [ has been replaced by cos 0,so that by (29)) page 379, the function Rei(coX, t ) satisfies an integral equation,
Still another representation of RE: can be found by expanding the integrand of (57). Thus (58) Re:(coX, t ) =
4
Zz)
i"D;
lzr
J,(coX cos p) eicaX[
B
dp.
n
But (59) whence,
la'
(60)
J,(coX cos p)
eicoXE cos
@
dp
=
i"
cos n+ Jo[coX(E
+ cos +)I d4.
SEC.6.131
383
INTEGRAL REPRESENTATIONS
Remembering once more that (  1)” = ( l)m, we obtain (61) Rei(coX, E )
=
& lzw+
JO[COX(E cos +)] Sem(coX,cos 4) d+.
If’(  l)m
Integral representations of the radial functions of the third and fourth kinds can be obtained by a variation in thci choice of contour. To simplify matters a little, let us take eosh ZL = .$,v = 0, cos /? = t . Then the several radial functions are represented by integrals of the form
$$(E>
(62)
=
L Sem (coX , t ) ezcoAft(l t2)f
dt,
where the contour C is such as to ensure the vanishing of the socalled “bilinear concomitant” (52), which now assumes the form (63)
d dt
(1  t’)i[ icoX&!5‘e,(coX, t )   Sem(coX,t )
When t becomes very large, the asymptotic expansion of Xe, is
consequently, the vanishing of (63) at infinity is governed by a factor exp icoX([  1)t. If, therefore, the real part of c(,X(E  1) is greater than zero, the contour can begin or end at t = i m , while for the other limit one may choose either f l or 1. As a result we have
provided the real part of [cox((  l ) ] 0plane, we find (65) equivalent to
(67)
Ref(coX, t ) = im
> 0.
Reverting to the complex
* Se,(coX, cos
a)
eicohf ens
do,
with a corresponding integral for Rei(coX, [). With the help of (67) one may deduce further integrals of the type (61). Thus, in place of (58) we write
(68) Ref(coX,
6) = Z$
62’ n
fi
J,(coX cos
0 ) eicohE
0 dp
384
[CHAP.VI
CYLINDRICAL WAVES
Now by (39), page 367, when p
provided the real part of
p
=
0, we have
> 0; consequently,
provided the real part of [coX(E  l)] (70) leads to
> 0. A
combination of (61) and
A clue to the formulation of integral representations of the odd functions is given by eiCoAi
CO8
B
cos np dp,
the result of a n integration by parts. The proof of the relations
is left to the reader.
6.14. Expansion of Plane and Circular Waves.The periodic functions Xe,(coX, cos v) and SO,(COX, cos v) constitute a complete, orthogonal set. Consequently, if a function f(u, v) is periodic in v with period 2~ and together with the first derivative af/av is piecewise continuous with respect to v, it can be expanded in a series of the form m
(75) j ( u , v)
=
C jg)(u)Se,(:oX,
m=?
cos v)
+
oc
jz)(u) SO,(C~X, cos 01, m=O
SEC.6.141
385
E X P A N S I O N OF P L A N E A N D CIRCULAR W A V E S
whose coefficients are determined from
This expansion may be applied to the representation of a plane wave whose propagation constant is k , traveling in a direction defined by the unit vector n whose spherical angles with respect to a fixed reference system are CY. and /3 as in Fig. 66, page 362. $, =
(77)
ezk sin a(x
30s
0+y sin 0). ezke
cos oriot
Upon expressing x and y in terms of u and v and putting X have
(78) f(u,v, B)
=
ezhp,
p
=
co(rosh u cos v cos /3
=
k sin a , we
+ sinh u sin v sin p).
Since there is complcte symmetry in (78) between v and p, the expansion (75) can be written
(79)
echp
=
2
a,(u> Se,,(ccX, cos
~ e , ( c o ~cos , v)
m=O m
I
2 b,(u) Xo,(coX,
cos p) Som(coX, cos u )
m=O
in which the coefficients a,(u) and b,(u) depend on u alone. of (76), wc have a,(u) NE)Se,(coX, cos p) =
(80)
I n virtue
Xe,(coX, cos v) e z h p dv.
is unaffected by the value of p, we may put P 1) = 1, p = co cosh u cos v. Then by (56),
since a,(u) Se,(coX,
=
0,
Likewise (82)
b,(u) N E ) S ~ , ( ~cos ~ Xp), =
~ o , , ( c ~cos ~ ,v)
eiAp
dv.
We now differentiate this last relation with respect to p and then put 6 = 0. I n virtue of (14) the coefficient b,,(u) proves to be
386
[CHAP.VI
CYLINDRICAL W A V E S
The complete expansion for a plane wave of arbitrary direction is, therefore, m
Sem(coX, cos
P)
1 + NF Ro;(coX,
I
cash U ) SO,(C~X,cos V ) XO,(coX, cos B) .
This last result enables us to write down a t once the integral representations of those elementary elliptic wave functions that are finite on the axis. We need only multiply (84) b y Se,(cd, cos P) and integrate over a period. From the orthogonal properties of the functions it follows that
(85) Rei(cd, similarly, for the odd functions,
With the help of (67) it can be shown also that the elementary functions of elliptic waves traveling outward from the axis are represented by the contour integral
The upper sign in the exponential is to be chosen when  ~ / 2 < v < ~ / 2 , the negative sign when r / 2 < v < 3 ~ 1 2 . A similar relation can be deduced for the odd functions. From the analysis of Sec. 6.8 it is easy to aee th a t the even circular wave function cos n0 Jn(Xr),which remains finite on the axis, can be represented by the integral (88)
2 d cos n0 J,(Xr) =
where p = z cos j3
sozu
cos nB eiX* dp,
+ y sin /3 = r cos (j3  0).
D; and summation over even or odd values of (89)
Rei(coX, cosh
U)
Se,(coX,
cos v )
=
Upon multiplication by one obtains
n,
,&7'
inmDzcos n0 J,(Xr),
which expresses the even elliptic wave function as a n expansion in circular wave functions. A similar expression can be found for the odd functions.
387
PROBLEMS
An addition theorem relating circular and elliptic waves referred to two parallel axes has been derived by Morse.’ Problems
1. Show t h a t the equation
is satisfied by wave functions of the type t,b = Ce’(61+6P)cos (hlx  61) cos (hzy h: hi h: = k2.
+ +
 6z)eihsziwf,
Let il. = II,, II, = II, = 0, be the components of a Hertz vector oriented in the direction of propagation. Show that this leads t o a transverse magnetic field whoso electric components are
E , = ihlh3Cei(61+62)sin (hlx  61) cos (hzy  8z)eih3Zimt, E I  ihzhsCei(61+*2)cos (hlx  61) sin (hzy  62)eih3Liwf, E , = ( k Z  h:)Ceifb1+6z)cos (hlz  61) cos (hzy  6 z ) e i h r z  k f , arid whose impedances are
E.
z= = H,
=
k e  hi u p cotan (hlz ik’hi
&I),
Find the components of a corresponding transverse electric field by choosing n:, n: = 11* = 0. The factor eir61+62) is introduced to include traveling as well as standing waves. Thus, if one chooses 6L = im , the field behaves as eihLz. 2. The equation of a twodimensional wave motion in a nondissipative medium is $b =
Show that
where
x‘  x = r cos 8,
y’
y
=T
sin 8,
and
+
=
dx, Y),
!? at = G(x, y),
when t = 0.
+
This is the PoissonParseval formula. Note that a t time t the wave function is determined by the initial values g and G a t all points on the circumference and the interior of a circle of radius at. Thus even in nondissipative media the disturbance 1
MORSE,loc. cit.
388
CY L IN D R IC A L W A V E S
persists after the passage of the wave front. wave motion in evendimensional spaces. 3. Show that the wave equation
[CHAP.
VI
This behavior is characteristic of
r dr
has a particular solution of the form
iI.
=
1L 2 ” Y
+ ir cos a, t
(2
27r
:’)

sin a
dor,
which reduces to tl. = F ( z , t ) on the axis r = 0, where F is an arbitrary function. Show that another solution of thc same equation is
+=
Jmm
F (z

)
1 
r sinh a,
cosh a
da.
(Bateman)
4. A special case of the solution obtained in Problem 3 is the symmetric, twodimensional wave function. $ =
F ( 1  cosh
2x
a)
da,
which represents a wave expanding about a uniform line source of strength F ( t ) along the zaxis. Show that if F ( t ) = 0 when t < 0, then
which is zero everywhere as long a s t < r / c . If the source acts only for a finite time T , so that F ( t )
0
=
0 except in the interval
< t < T, show that the wave leaves a residue or “tail” whose form, when t
>>
 I: C
is determined by * =
(Lamb, “Hydrodynamics,” 5th cd., p. 279, 1924. 6. Show t h a t the equation
is satisfied by
Cambridge University Press.)
+
sin k ( ~ a ) d [ r+ro d E 2
where r2
6. Show that the equation
= x2

+ y2,
(5
2
ri
2
 (Y
= 2:
 Yo12
+ yi.
(Bateman)
T,
389
PROBLEMS is satisfied by
+
where r2 = 2 2 y2 and f and F are arbitrary functions, T o what limitations is this solution subjected? Obtain in this way the particular solution (Bateman) 7. The equation
is rcduccd by the substitution J. = e*tu(z, y, t )
to
The initial conditions are
1,et r he the radius of a sphere drawn about thc point of observation and a, p , y the direction cosines of r with respect to the x, y, z axes. Show that a solution is represented by u ( z , y, 2) =

t g(z
+ ~ t ay, t uta) cos (ibtr) d Q +2,
SS
1 G(x
+ ata, y + ato) cos (ibty) df2, (Roussincsq)
where d Q is thc solid angle subtended by an clcment of surface on the sphcrc. 8. Show that
is an integral representation of the Bessel function of the first kind valid for all value: of the order p provided E e ( p ) >  i. 9. Derive the integral representations
Imz
> 0.
for the Besscl function of the second kind and zero order, and
Imz 0 and
It is assumed t h a t Re
/arg ZI
T
< . 2
12. Show that for very large values of the argument the asymptotic behavior of the modified Bessel functions is given by m
n
provided 2
3 < arg z < r, 2
larg zI m=O
where
(P’m,
r(p + m + +L m + 3)
= m ! r(p
(4p2  1)(4p2  32)
..
. [4p2  (2m  1)2]
22mm!
3
< 2 r ,
391
PROBLEMS Retaining only the first term,
.
13. Prove that if R
= d r 2
+ z2 is the distance between two points, the inverse
distance is represented by
R
=
2 x
cos ,8z Ko(gr)d p ,
where Ko(pr) is the modified Bessel function defined in Problem 10. 14. As a special case of Problem 4,show that the equation
is satisfied by where k = w / c and K Ois the modified Bessel function defined in Problem 10. This solution plays the same role in twodimensional wave propagation as does the function eikH/R in three dimensions. 16. Show that the equation
is satisfied by
n
n
m=l
which has a period of 2nrr in 0 and hence is multivalued about the axis. this solution can be represented by the integral
Show t h a t
iu
following a properly chosen contour C in the complex orplane. This solution is everywhere finite and continuous. $ behaves like a plane wave a t infinity provided I 6 I < P , but vanishes (is regular) a t infinity if 0 lies between any pair of the limits T
< B < 3rr,
3u
< e < h, . .
I
& < e
< ",
which define the 2nd, 3rd, . . . nth sheets of a Riemann surface. Such multiform wave functions have been applied by Sommerfeld to diffraction problems. 16. Derive explicit expressions for the radial and tangential impedances of elliptic cylindrical electromagnetic waves. 17. Show that the integral
is a cylindrical wave function which is equal to e i k x / R , R2
=
r2
+ z2, when f(h)  1.
CHAPTER VII
SPHERICAL WAVES I n the preceding chapter it was shown how the analytic difficulties involved in the treatment of vector differential equations in curvilinear coordinates might be overcome in cylindrical systems by a resolution of the field into two partial fields, each derivable from a purely scalar function satisfying the wave equation. Fortunately this method is applicable also to spherical coordinates to which we now give our attention. The peculiar advantages of cylindriral and spherical systems are a consequence of the very simple character of their geometrical properties. A deeper insight into the nature of the problem and of the difficulties offered by curvilinear coordinates in general will be gained from a brief preliminary study of the vector wave equation. THE VECTOR WAVE EQUATION
7.1. A Fundamental Set of Solutions.Within any closed domain of a homogeneous, isotropic medium from which sources have been excluded, all vectors characterizing the electromagnetic fieldthe field vectors E, B, D, and H, the vector potential and the Hertzian vectorssatisfy one and the same differential equation. If C denotes any such vector, then dC = 0. 5  pu v2c  p€ d2C dt Because of the linearity of this wave equation, fields of arbitrary time variation can be constructed from harmonic solutions and there is no loss of generality in the assumption that the vector C contains the time only as a factor ctwt. By the operator v2acting on a vector one must understand V2C = V v C  V x v x C ; therefore, in place of (1) we shall write

(2)
v v  c v x v x
+
C+k?C
=
0,
where k 2 = epu2 iupw as usual. Now the vector equation ( 2 ) can always be rcplaced by a simultaneous system of three scalar equations, but thc solution of this system for any component of C is in most cases impractical. [Cj. (85), page 50.1 It is only when C is resolved into its rectangular components that three 392
SEC.7.11
A FUNDAMENTAL SET OF SOLUTIONS
393
independent, equations are obtained and in this case V2Cj
(3)
+ kZCi = 0,
( j = r , Y,z).
The operator V2 can then be expressed in curvilinear as well as rectangular coordinat'es. Very little attention has been paid to the determination of independent vector solutions of ( Z ) , but the problem has been attacked recently by Hansen' in a series of interesting papers dealing with the radiation from antennas. Let the scalar function $ bct a solution of the equation
v:'++ k2$
(4)
=
0,
and let a be any constant vectJor of unit lengt,h. We now construct three independent vector solut'ions of (2) as follows:
(5)
L = V+,
M
=:
V x a+,
N
=
1  V x M. k
If C is placcd equal to L, M, or N, one will vcrify that ( 2 ) is indeed satisfied identically by (5) subject to ( 4 ) . Since a is a constant vector, it is clear t>hatM can be written also as M
(6)
=
1; x a
=
1
 V x N. k
For one and the same generating function $ thc vector M is perpendicular t3 the vector L, or
(7)
LM=0.
Thc vector fiinct,ions L, M, and N have ccrtain ridable properties that follow directly from thcir definitions. Thus
vxL
(8)
=
0,
vL
=
V2$
= k2+,
whereas M and N are solenoidal.
(9)
V  M = 0,
V * N = 0.
The particular solutions of (4) which are finite, cont'inuous, and singlevalued in a given domain form a discrete set. For the moment wc shall ¬e any one of these solutions by Associated with each charactmistic function are three vec:t)or solutions L,, M,, N, of (2), no two of which are colinear. Prcsum:tbly any arbitrary wavc funct,ion can bc rcpresented as a linear combinat,ion of the characteristic vect,or funct'ions; since the L,, M,, N, possess ccrtain orthogonal properties which me shall demonstrate in due course, t'he coefficients of the expansion can be deter
+,.
%$1 HANSEN, Phys. Rev., 47, 139143, January, 1935; Physics, 7,460465. December, 1936; J . A p p l i e d Phys., 8, 284286, April, 1937.
394
[CHAP.V I I
SPHERICAL WAVES
mined. When the given function is purely solenoidal, the expansion is made in terms of M, and N, alone. If, however, the divergence of the function does not vanish, terms in L, must be included. The vectors M and N are obviously appropriate for the represent>ation of the fields E and H, for each is proportional to the curl of the other. Thus, if the time enters only as a factor ei”t and if the freecharge density is everywhere zero in a homogeneous, isotropic medium of conductivity u jwe have 1 H=VXE. WJ Suppose then that the vector potential can be represented by an expansion in characteristic vector functions of the form
the coefficients a,, b,, c,, to be determined from the current distribution. By ( 8 ) , (lo), and the relation pH = V X A, we find for the fields (12)
E
=
3
(a,M,
+ b,N,),
H=
(a,N,
+ b,M,).
n
The scalar potential
+ plays no part in the calculation, but can, of course,
be determined directly from (11). For by (27), page 27,
v A
i
=  k2+; w
hence, by (8) n
Then V+ =
2 cnLn and clearly the relation E = V+  aA  leads dt n
again t o (12). Finally, if we recall that under the specified conditions an electromagnetic field can be represented in terms of two Hertz vectors by the equations (14)
E
V X V X
+iwpV X
n*,
H
=
k2
V X rI + V x V x II*. ZWP
it is immediately apparent that
where a is a constant vector. Before applying these results t o cylindrical and spherical systems, let us consider the elementary example of waves in rectangular coordinates.
APPLICATION TO CYLINDRICAL COORDINATES
SEC.7.21
395
A plane wave whose propagation vector is k = kn is represented by $ = eik * R  i d
(16)
J
where R is the radius vector drawn from a fixed origin. Then since k R = k,x k,y k,z, it is easy to see that (17)
+
+
.
L
=
i$k,
M
= i$k
x a,
.
N
=
1
$(k k
x a) x k.

I n this particular case L M := M N = N L = 0; all three vectors are mutually perpendicular and L i s a purely longitudinal wave. Now the polar angles dctermining the direction of k are a! and p, as defined in Fig. 66, page 362, and general solutions of (2) can be constructed by integrating these planewave functions over all possible directions. If g(a, p) is a scalar amplitude or weighting factor, one may write for L in any coordinate system, subject to convergence requirements,
L
= ieiwt
da!
d p g ( a , @)k(a!, 8) eik.R;
da
d/3 g ( a , 8)k
likewise for M and N
M = ieiwt
x a eikeR,
7.2. Application to Cylindrical Coordinates.'The scalar eharacteristic functions of the wave equation in cylindrical coordinates were set down in Eq. (6), page 356. Therc are certain disadvantages in the use of complex angle functions exp(in0) and it will prove simpler to deal with the two real functions cos no and sin no which may be denoted simply as even and odd. Cylindrical wave functions, constructed with Bessel functions of the first kind and hence finitc on the axis, will be denoted by $i'), while the wave functions formed with N,(Xr) or HL')(hr) will be called $i2) and $3 respectively. Thus
where X = d k 2 1
 h2 = k sin a! as usual.
Functions of the first kind are
Compare the results of this section with Eqs. (36) and (37), p. 361.
396
SPHERICAL WAVES
[CHAP.VII
represented by the definite integrals
and representations of other kinds differ from these only in the choice of the contour of integration. From (23) we now construct integral representations of the vector wave functions. Thus (24) Upon differentiating (23) with respect to r , 0, and z and noting that k sin a cos (P  e), k sin a sin (P  O), and k cos CY are the components of k respectively along the axes defined by the unit vectors ill i,, i, so that (25) k = ilk sin
CY
cos (0  0)
+ izk sin a sin (p  6) + iSk cos a,
we find for the even function
which is of the form (18). The corresponding odd function is obtained by replacing cos np by sin np under t,he integral. When applying these characteristic functions to the expansion of an arbitrary vector wave, it proves convenient to split off the factor exp(ihz  i w t ) , and we shall define the vector counterparts of th e scalar functionf(r, 0) of Chap. VI by (27) L, = lneihziiot M, = mneihaiiwt N I t  nneihziiwt. From (26) we have
I n like fashion one finds integral representations of the independent solutions. For the constant vector a we choose the u.nit vector is directed along the xaxis. Then
whence it is easily deduced that
SEC.7.21
since
397
APPLICATION TO C Y L I N D R I C A L COORDINATES
v x M,
=
kN,, we havc
From these integrals one can easily calculate the rectangular components of the wave functions. The vector functions of angle in the integrands are resolved into rectangular components which can then be combined with cos no or sin no. Thc resulting scalar integrals for the components arc evaluatcd by comparison with (23). To expand a n arbitrary vector function of r and 8 in terms of the l:,LA,mznx,nEnh, one must show that these functions are orthogonal. Let
il, i2,is be unit base vectors of a circularly cylindrical coordinate system, Fig. 7, page 51, and Z,(Xr) a cylindrical Bessel function of any kind. Then by (5) we obtain
a
z,(Xr) (32) le,,A= dr 0
=
(33)
me,x
'Ps n8 il
SlIl
7; n Z,(Xr)
f
zn(Xr)sin n8 i, 4 ih z,(Xr) cos no is, cos sin n8 il  d z,(Xr) '0' n8 i,, dr sin
g3
'?'
sin
n8 is.
Now it is immediately evident that the scalar product of any two functions integrated over 8 from 0 to 2n must vanish if the two differ in the index n. (35)
if n # n'.
Let us understand by i the function 1 with the sign of ih reversed.' the normalizing factor is to br found from the integral
Then
where 6 = 0 unless n = 0, in which case 6 = 1. The first two terms on the right can be combined with the aid of the recurrence relations (24) and (25), page 359, giving 1
This is not necessarily the conjugate, since X and h may be complex quantities.
(42)
J z W0
.I 2*
.
l 0e n i m;n,,
=
mE"h
";.A.
de
= J2* lenA 0 *
fiZnxt
dB
=
0,
and by use of the recurrence relations once more 2a

do = m:nA no, de = 0. mp' e" There remains only one other combination, and this, unfortunately, leads to a difficulty. (43 )
(44)
':"A
JZr lenA *
0
0
*
tienx, dB = i(1 0
+ 6) ={  hX'* Z,(Xr) k
XX'h
 2[Z,,(Xr) Zl(Xfr) a quantity not identically zero.
Z,(X'r)
+ Zn+l(Xr) Zn+1(X'r)1
The set of vector functions lenh,menh, 0
0
nenh,therefore, just fail to be completely orthogonal with rcspect to e 0
because of (44). I n many cases this is of no importance, for if a vector function is divergencelessas are t,he electromagnetic field vectors in the absence of free chargeit can be expanded in terms of menh,ne*x 0
0
alone. A complete expansion of the vector potential, on the other hand, requires inclusion of the lenhel 0
The completeness of the set of vector functions has not been demonstrated but presumably follows from the completeness of the scalar set
I&:.,.
SEC.7.31
E L E M E N T A R Y SPHERICAL W A V E S
399
I n the case of functions of the first kind the FourierBessel theorem has been applied by IIansen to complete the orthogonalit,y and to simplify the normalization factors. From ( 5 3 ) and (54), page 371, after an obvious interchange of variable, we have
(45)
f(h) =
J “ r dr
A’ dX’f(X’)J,(h’r)J,(Xr).
This is applied to (37), (39), (41), and (44) when Z,(Xr) is replaced by ,Jn(X~). For f(X’) we shall h:ive A’, h‘ = d m ,h’X’ and the like. Now one must note that the validity of the FourierBessel integral has
J
been demonstrated only when
If(X)\fi
dX exists, and that this con
dition i s not fuljilled in the present instance. The artifices employed to ensure convergence involve mathematical difficulties beyond the scope of the prcsent work. With certain reservations one may introduce an exponential convergence factor exactly as in the theory of the Laplace transform on page 310. By 1;: let us undcrstand henceforth the limit

0
approached by l$~:e’slAas s 0
0.
Then
Consequently the normalizing factors reduce to
(49) while for the troublesome (44) we find’
THE SCALAR WAVE EQUATION IN SPHERICAL COORDINATES
7.3. Elementary Spherical Waves.Since an arbitrary time variation of the field can be represented by Fourier analysis in terms of harmonic components, no essential loss of generality will be incurred hereafter by the assumption that (1)
+ = f ( R , 0, 4)eht.
1 The reader must note that the limit approached by (46) as s + 0 is not necessarily equal t o the value of the integral at s = 0, for this would imply continuity of the function defined by the integral in the neighborhood of s = 0. This point underlies the whole theory of the Laplace transform. Cf. Carslaw, “Introduction to the Theory of Fourier’s Series and Integrals,” 2d ed., p. 293, Macmillan, 1921.
400
SPiiERICAL W A V E S
[CHAP.V I I
I n a homogeneous, isotropic medium the function f ( R , 0, +) must satisfy
(2)
v2f+ky=0,
which in spherical coordinatcs is expanded by (95), pagc 52, to
The equation is separable, so that upon placing f finds
= fl(r)fL(0)fS(+)
one
The parameters p and q arc separation constants whose choice is governed by the physical requirement that a t any fixed point in space the field must be singlevalued. If the properties of the medium are independent of equatorial angle +, it is necessary that f3(+) be a periodic function with period 2n and (I is, thcrcfore, restricted to the integers m = 0, i1, & 2, . . . . To determine p we first identify the solution f 2 as an associated Legendre function. Upon substitution of 7 = cos 8, Eq. ( 5 ) transforms to
(7) a n equation characterized by rcgular singularities at the points 7 =  1, 9 = + I , 7 = to, and no others. I t s solutions arc, therefore, hypergeometric functions. Now when m = 0, ( 7 ) reduces to the Legendrc equation. There are, of course, two independent solutions of thc Legendre equation which may be expanded in ascending power series about the origin 7 = 0. These series do not, in gcncral, converge for r] = _+1. If, however, we choose p 2 = n(n l), where n = 0, 1, 2, . . . , then one of the scries breaks off after a finite number of terms and has a finite valuc a t the poles. Thcse polynomial solutions satisfying the equation d 2v  2 7 dU (1  72) n(n l)v = 0 d 7. d7
+
+
+
are known as Legendrc polynomials and are designated by P,(q). now we differentiate (8) m times with respect to 7, we obtain
If
SEC.7.31
where w
ELEMENTARY SPHERICAL WAVES = dmu/dqm,and
401
upon making a last substitution of the dependent m
variable, w
=
(1  qz)5zlfp(v), we obtain (7) in the form
The solutions of (10) which arc’ finite a t the poles 7 = rt 1 and which are, consequently, periodic in 0 arc’ the associated Legendre polynomials fi(9) =
P;(,)
=
(1 
”dP ( ) Ad 9
92)Z
For each pair of integcrs there exists a n independent solution &;(q) which becomes infinite a t 7 = I l , and which consequently does not apply to physical fields in a complete spherical domain. The definition of the associated Legcndre polynomial as stated in (11) holds only when n and m are positive integers. The functions of negative index are related in a rather simple fashion to thosc of positive index, but we shall have no need of them here. To obviate any confusion on this matter we choose for particular solutions of (6) thz real functions cos m+, sin md, and restrict ?n and n to the positive integers and zero. It is clear from (11) that P;(v) vanishes when m > n, for Pn(q)is a polynomial of nth degree. We have in fact m
The properties of the hypergeometric functions, of which the Legendre functions are the best known c.xample, have been explored in great detail and i t would be scarcely possible to give an adequate account of the various series and integral representations within the space a t our disposal. We shall set down only the indispensable recurrence relations,
402
[CHAP.VII
SPHERICAL WAVES
T o these may be added the differential relations
1 2
=  [ ( n m
+ l ) ( n+ m)P:’
 Pm+l n I.
I t may be remarked in passing that these relations are satisfied also by functions of the second kind Qz. The functions cos rn4 P;(cos 6 ) and sin m4 P;(cos 6) are periodic on the surface of a unit sphere and the indices m and n determine the number of nodal lines. Thus when m = 0, the field is independent of the equatorial angle 4. If n is also zero, the value of the function is everywhere constant on the surface of the sphere. When n = 1, there is a single nodal line at the equator 6 = ~ / along 2 which the function is zero. When
1
FIG.6g.Nodes
of the function sin 3rp Pi(cos 0) on the developed surface of a sphere. The function has negative values over the shaded areas.
n = 2, there are two nodal lines following the parallels of latitude at approximately 6 = 55 deg. and 0 = 125 deg., so that the sphere is divided into three zones; the function is positive in the polar zones and negative over the equatorial zone. As we continue in this way, it is apparent t ha t there are n nodal lines and n 1 zones within which the function is alternately positive and negative. For this reason the P,(cos 0) are often called zonal harmonics. If now m has a value other than zero, it will be observed upon examination of Appendix IV that
+
m.
the function is zero at the poles due to th e factor (1  q2)H, and that the number of nodal lines parallel to the equator is n  m. Moreover, the function vanishes along lines of longitude determined b y the roots of ccs rnb and sin md. There are obviously m longitudinal nodes that intersect the nodal parallels of latitude orthogonally, thus dividing the
403
ELEMENTARY SPHERICAL WAVES
SEC.7.31
surface of the sphere into rectangular domains, or tesserae, within which the function is alternately positive and negative. For this reason the functions cos m4Pz(cos 6) arid sin m+P:(cos 6) are sometimes called tesseral harmonics of nth degree and mth order. There are obviously 2n 1 tesseral harmonics of nth degree. This division into positive and negative domains is illustrated graphically in Fig. 69 for the function sin 34 P,3(cos 6). If the tesseral harmonics are multiplied by a set of arbitrary constants and summed, one obtains the spherical surface harmonics of degree n with which we have already had something to do in Chap. 111.
+
Yn(6,4)
(15)
(anr,, cos
=
+ b,,
m4
m=O
sin rnd)P;(cos 6).
The tesseral harmonics form a complete system of orthogonal functions on the surface of a sphere.l It is in fact easy to show through an integration of (12) by parts that
From these relations follows the fundamental theorem on the expansion of an arbitrary function in spherical surface harmonics: Let g ( 6 , 4) be a n arbitrary function o n the surface of a sphere which together with all its first and second derivatives i s continuous. Then g ( 6 , 4) oan be represented by a n absolutely convergent series of surface harmonics,
(18)
g(6,
4)
=
m
n
n=O
m=l
2 [anoPn(cOs6) + 2 (anrncos m4 + b,,
PF(cos 611,
whose coeficients are determind by ano = 2n __ + Qr
(19) an, b,,
+
izr
l g ( 6 , 4) P,(cos 19) sin 6 d6 d4,
2n 1 (n  m ) ! ___2r (n m ) !J zoU
+ 2n + 1 (n  m ) ! 2 r (n + m ) !
= 
= __
sin m$)
g (6, 4) P;(cos 0) cos rn4 sin 0 do d4,
Jg(o,4) P;(cos
6)
sin m+ sin e de ci4.
1 The proof of this statement and of the expansion theorem which follows will be found in CourantHilbert, “ Methoden der mathematischen Physik,” 2d ed., Chap.
404
SPHERICAL JBAVES
[CHAP.
lrII
In the case of a function th at depends on 0 alone, the conditions determining the convergence of a n expansion in Legendre polynomials are identical with those governing the convergence of a Fourier series. Under such circumstances it is sufficient that g(0) and its first derivative be piecewise continuous in the interval 0 5 0 I 27r. The convergence theory of the expamion (18) in surface harmonics, on the other hand, presents definitely greater difficulties and the extension of the theorem to discontinuous functions involves considerations beyond the scope of the present outline. Tliere remains the identification of the radial function f l ( R ) satisfying (4). If for f1 we write f l = (kR)+v(R),it is readily shown that v(R) satisfies (20)
R2d2v dR2
+ R dR dv f [k2R2 ( n
+3”]
v
= 0,
and hence, by Sec. 6.5, is a cylinder function of half order.
The characteristic, or elementary, wave functions which a t all points on the surface of a sphere are finite and singlevalued are, therefore, (22)
f e Om“
=
1
dlcR
cos
Z,+,(kR) Pg(c0s 0) sin m4.
As in the cylindrical case, we choose for Z,++(kR) a Bessel function of the first kind within domains which include the origin, a function of the third kind wherever the field is to be represented as a traveling wave. 7.4. Properties of the Radial Functions.Various notations have been employed a t one time or another t o designate the radial functions (kR)*Z,++(kR)but none appears to have gained general acceptance.’ What seems t o be a logical proposal has been made recently by Morse2 and will be adopted here. Accordingly we define the spherical Bessel functions by ,
.
,
.
VII, $5, 1931. For greater detail see Hobson, “ T h e Theory of Spherical and Ellipsoidal Harmonics,” Cambridge University Press, 1931. 1 See WATSON, “Bessel Functions,” p. 55, Cambridge University Press, 1922. MORSE:“Vibration and Sound,” p. 246, McGrawHill, 1936.
SEC.
7.43
PROPERTIES OF THE RADIAL FUNCTIONS
405
Series expansions of these functions about the point p = 0 can be obtained directly from (8:) and (11), page 357. If we recall that I'(x 1) = xr(z), I?(+) = arid make use of the dupZicaEion formula
+
6,
whrnce it is apparent that j n ( p ) is an integral function. Likewise from the rcllation N,+g(p) = (l)nlJ.n;(p), we obtain for the function of the second kind
We turn next t o representations when p is very large, and discover a t once a notable property of the Bessel functions whose order is half an odd integer. On page 358 expansions in descending powers of p were given for the functions of arbitrary order p . Such series satisfy the Bessel equation formally, but are only semiconvergent. We obscrve now, however, that when p == n 1/2, the series P,+i(p) and &,+;(p) break off, so that there is no longer question of convergence. Consequently (13) and (14) of page 358 are analytic representations of J,+;(p) and Nn++(p); furthermore, i t is apparent that these functions of hulf order can be expressed in finite terms. From (23) above and Eqs. (13) to (17) of page 358, we obtain
+
where (29) Pn++(p)
=
+
n(n2  l)(n 2) 1 _ _  ~ 22. 2! p2 n(n2  l)(n2  4)(n2  9)(n 24.4! ,,4
+
+ 4)  . . .
For the functions of the third and the fourth kind this leads to (;)n+l h'l'(P) =  e"[Pn++(P) iQn+:(~)li P (31) ;n+ 1 h;''(p) = __ e  i ~ [ P n + t ( ~ ) i&n+t(~)I.
+
P
1
406
SPHERICAL WAVES
[CHAP.V I I
These series converge very rapidly so that for large values of p the first term alone gives the approximate value of the function. Thus, when
The recurrence relations satisfied by the spherical Bessel function follow directly from Eqs. (24) to (27) of page 359.
Having defined the radial functions, we can a t last write down the general solutions of (3) as sums of clementary spherical wave functions. I n case f(R, 0, 4) is to be finite a t th e origin, we have
(36) f"'(Rj
$1
4) =
2 j n ( k R ) [asd'n(cos n=O
0)
+2
cos m4
( ~ n m
m=l
+ b,,
sin m4)P;(cos O)],
while a field whose surfaces of constant phase travel outward is represented by f("(R, 0, $), obtained by replacing j,(kR) by hhl)(kR)in (36). A spherically symmetric solution results when all coefficients except a00 are zero. Then apart from an arbitrary factor, we have
7.6. Addition Theorem for the Legendre Polynomials.If g(0, 4) is any function satisfying the conditions of the expansion theorem (18), its value a t the pole, 0 = 0, must be
sin 0 d0 &, since P,(l) = 1, P,m(l) = 0. In particular let us take for g(0, 4) any surface harmonic Yn(6, +) of degree n expressed by (15). Then in
407
ADDITION THEOREM
SEC. 7.51
virtue of the orthogonality relations, the sum in (38) reduces to a single term and we obtain the formula
We shall apply this result to the problem of expressing a zonal harmonic in terms of a new axis of reference. Let P in Fig. 70 be a point on a sphere whose coordinates with respect to a fixed rectangular reference system are 0 and 4. A second point Q has the coordinates a and 8. The angle made by the axis OP with the axis OQ is y. The zonal Y harmonics a t P referred to the new polar axis OQ are of the form P, (cos y ) and our problem, therefore, is to expand P,(cos y ) in terms of the coordinates 0, 4, and a,8. It is apparent that on a unit FIG. 70.Rotation of the axis of reference from OP to OQ for a system of sphere cos y is the projection of the harmonica. line OP on the axis OQ. If 2, y, are the coordinates of P, x’, y’, z‘ those of Q, then
+ +
a) + cos 0 cos a.
(40) cos y = xx’ yy’ zz’ = sin 0 sin a cos (4 We now assume for P,(cos y) an expansion of the form n
multiply both sides by P::(cos 0 ) cos m+, and integrate over the unit sphere. With the help once more of the orthogonality relations, we obtain (42)
12= 0
J’P,,(eos r)~
( C O S0)
0
cos m+ sin e de d+ (n + m)! + I (n  m)!
 27r 2n
‘me
But by (39) (43)
s,”
P,(cos y)~;(cos e> cos rn4 sin e do d 4

47r
2n
+ 1 [P;(cos
0) cos m d ~ ] , ,=~
4* ~ ( c o as) cos mp, ml
488
SPHERICAL W A V E S
since when
y =
0, 8 = a , 4 = p.
2
Cm =
(44)
[CHAP.VII
Consequently,
(n  m ) ! P:(cos a ) cos m/3. (n m)!
Likewise,
+
(n  m ) ! PF(cos a ) sin mp. ( n m)! The desired expansion, or addition formula, is therefore d,
(46) P,(COS
y) =
=
2
+
P,(COS a ) P,(COS
e)
+ 2 2 ~( n+~mP) !~ ( c o as ) P:(COS
8) cos m(4  p).
m=l
This result leads to an alternative formulation of the expansion theorem stated on page 403. If g ( B , 4) and its derivatives possess the necessary continuity on the surface of a sphere, its Laplace series (18) is m
7.6. Expansion of Plane Waves.It is now a relatively easy matter to find a representation of a plane wave propagated in a n arbitrary
direction in terms of elementary spherical waves about a fixed center. The direction and character of the plane wave will be determined by its propagation vector k whose rectangular components are (49) kl
=
k sin a cos p,
kz
=
k sin a sin 0,
k3
=
k cos a,
while the coordinates of any arbitrary point of observation are
(50) x = R sin 0 cos 4, Then the phase is given by (51)

k R
=
y
=
R sin 8 sin 4,
kR[sin a sin 0 cos (4  p)
x = R cos 0.
+ cos a cos 61 = k R cos y
The function f ( l ) ( R ,8, 4) = exp(ikR cos y) is continuous and has continuous derivatives everywhere including the origin R = 0. It can, thcrefore, be expanded according to (36), page 406. Consider first an axis that coincides with the direction of the wave. Since the wave is symmetrical about this axis, we write m
SEC. '7.71
409
INTEGRAL REPRESENTATIONS
The coefficients are determined in the usual manner by multiplying both sides by P,(cos y) sin y and integrating with respect to y from 0 to T. Then by (17)
To rid this relation of its dependence on R, differentiate both sides with respect to p = kR and then place p = 0. From (25) it follows that
(54) hence, (55)
2n(n!)2 a, (2.n l)!
+
=
2n
+
in
2
1%
cosn y ~',(cos y> sin y dy.
Thc integral on the right of (55) is readily evaluated and we obtain a, = (2% l)i", or
+
(56)
+
?:"(2n l)j,(kR) P,(cos 7).
eikR cos y = n=O
I n case the zaxis of the coordinate system fails to coincide with the direction of the planc wave, one may use the addition theorem (46) to express (56) in terms of an arbitrary axis of reference. Then
+22
I
(n  m)! ~ ~ ( c a> o ~s ~ ( c 0)o cos s m(4  p> .
m=l
7.7. Integral Representations.We have found it convenient on various occasions to represent a wave function as a sum of planc waves with appropriate weighting or amplitude factors. The direction of each component wave is determined by the angles a and p. Integration is extrndpd over all real directions in space, and may in certain cases include imaginary values of a and /3 as well. As in Xec. 6.7, we wish to find representations of the form
(58)
f(z, y, z) =
or, when x (59)
=
dp g(&, p)
R sin 0 cos 4, y f ( ~0,, 4)
=
=
&k(Z sin a cos
R sin 0 sin 4, z
B f u sin a slnB+z cos a)
=
J' dor 1cip g(a, a) eikn
R cos 0, y,
where a and /3 are the angles shown in both Figs. 66 and 70. I n the case of cylindrical waves a fixed frequency and a fixed wave length along
410
SPHERICAL WAVES
[CHAP.V I I
the zaxis lead to a constant value of a , so that the directions of the plane waves representing a cylindrical function f(ul, u2) constitute a circular cone. Since for spherical wnvcs there exists no such preferred axis, the integration must be extcndcd t o both a and p . An integral representation of tJlemcntary spherical wave functions can be obtained directly from the expansion (57). If both sides of that equation are niultiplicd by P:(cos a ) cos mp or by ~ ( C Oa)Ssin mp, one obtains, thanks to the orthogonality relations (19), the result
sin a d a dp. Upon multiplying both sides by the arbitrary constants anm,b,, and summing over m, this can be written also as
(61)
j,(kZ2)%,(8, 4)
=in
47r
Lz"l
eikn cos ?Yn(a,p) sin a d a do.
From these general formulas may be derived a number of useful special cases. Thns by choosing m = 0, 0 = 0, we obtain a representation of the spherical Bcssel function j,(kR).
j,(kR)
7.To'
=
eikR
a
Pn(c0S a) sin a da,
It is in fact easy t o show that Eq. (4), page 400, is satisfied by the integral
provided the contour C is such that the bilinear concomitant
vanishes at, the limits. In place of r) = i 1, we may choose a value that will cause the expoiiential t o vanish. Thus if lc is real, we take for the upper limit r) = iw and obtain the rcprcsentations
SEC.7.81
41 1
A FOURIERBESSEL INTEGRAT,
If on the other hand k is complex due to a conduct,ivity of the medium, one may choose t h a t root of k 2 whose imaginary part is positive and replace the limit 17 = i cc) by 17 = a . Again from (60) and (54), placing 4 and R equal to zero, one obtains a representation of the function Pz(cos 0) :
The integration with respect t o a cannot be carried out easily with only the help of formulas from the present section at our disposal, but by olher means i t may be shown that (67) reduces t o J 2 r (cos 0 (68) P?(cos 0) = (n + nL)!iPm 2mz !
+ i sin 0 cos p)" cos mp do.
Finally, one will note that by making usc of the integral representation (37), page 367, for a Bessrl function, the righthand side of (60) can be reduced t o a single integral When 4 = 0,
(69) j,(kR) P,"(cos 0)
= 
l
eikR co8 a cos eJ,(kR sin 0 sin a )
P;(cos a ) sin a d a .
7.8. A FourierBessel Integral.It will be assumed that the arbitrary function f(x, y, z) and its first dcrivativm are pitcewist. continuous and that the integral of the absolute value of the function extended over all space exists. Subject to these conditions the Fourier integral of f(x, y, z ) is
ilk1 d k , dks,
and its transform is
(71)
g ( h , k2, k3) =
(&)
Jm :
y,
, .(m f:J
m : J
~ ) e  ~ ( ~ ~ ~ + ~ ~ y f ~ ~ ~ )
dx dy dz. We now transform these integrals to spherical coordinates under the tacit assumption that a triple integral extended over an infinite cube can bc replaced by an integral over a sphere of infinite radius. Proceeding as in Sec. 6.9 and noting that klx k2y kzz = kR cos y , we obtain
+
(72) f ( R , 0, 4) =
(ky
(73) g ( a , p, k )
(&>'
=
0
+
f r J z r g ( a , p, k ) eikRcos y k 2 sin a dk d a d p ,
. 0
0
lml[f(R,
0,
4)e  i k R c o s y R2sin 0 d R d 0 d 4 .
412
SPHERICAL WAVES
[CHAP.V I I
Next we shall assume that f(R, 8, 4) = fn(R)Y,(6, 4 ) and impose upon the otherwise arbitrary function f n ( R )the condition that it shall be piecewise continuous and have R piecewise continuous first derivative, and that the integral
LmIfn(R)[dR shall exist'.
Then
sin 6 d6 d4, which in virtue of (61) and the fact th at j n (  k B ) over into
=
(l)"j,(kR) goes
or g(a, P, k ) = (1),Yn(a, 0) g , ( k ) . Introducing this evaluation of the transform back into (72) and again interchanging the order of integration, we are led to
and hence to the symmetrical pair of transforms
(77) (78) from which the subscript n has been dropped as having no longer any particular significance. I n the special case n = 0, Eqs. (77) and (78) reduce to the ordinary Fourier integral (18), page 288. I n case the chosen value of R coincides with a point of discontinuity in f ( R ) ,we write (79)
:J
0
k 2 dk
1
 J ( P ) j , ( k p ) j,(kR)
1
p2 dp =
2 [f(R + 0) + f ( R  011.
7.9. Expansion of a Cylindrical Wave Function.When calculating the radiation from oscillating current distributions it is sometimes necessary to express a cylindrical wave function in spherical coordinates. The integral representation of wave functions that are finite on the axis is
Now (81)
k R
= hr
cos (P  4 )
+ ihz = kR[sin a sin e cos ( 4  p) + cos a cos el,
SEC.7.101
413
A D D I T I O N THEOREM FOR z o ( k R )
and in ( 5 T ) , page 409, we have a n expansion in the appropriate spherical wave functions of exp(ik. R). Upon integration with respect to p, we obtain
e) j , ( k ~ ) .
P;(COS a ) P:(COS
Since P; vanishes if m > n the first m terms of (82) are zero and the expansion can be writtcn
2 m
(83)
J,(XT) eihz =
42m
! + 1) (n n 2mp P:+:,,(cos
a)
+
7k=O
P;+,(COS
When a = ~ / 2 h,
=
0) j , , + m ( ~ ~ ) .
0, X = k , and (n
+ 2m  l)!
if n is odl .*
.
The result is an expansion of a cylindrical Bessel function in a series of spherical Bessel functions. (41
(1)l
z=o
+ 2m + 1) (21)! + 2m 1 ! ( + ~ m  I)!
21
2Zz+m1
P ~ + , ( c o s6 ) j Z ~ + & W .
7.10. Addition Theorem for zo(kR).Let P(R0, 19, $) be a point of observation and &(Ill, el, $1) the source point of a spherical wave. The coordinates Bo, 6, # and R1,61, (blare referred to a fixed coordinate system whose origin is a t 0. The distance from & to P is, therefore, R
=
dz++
122,  2RoR1 cos y
where cos y = cos 6 cos 61 sin 6 sin 61 cos (4  41). Our problem is to express a wave function referred to Q as a sum of spherical waves whose source is located at 0. General treatment of this problem demands a lengthy calculation and its practical importance is insufficient to warrant discussion here. However, in the theory of radiation we shall have occasion to apply the special case in which the wave is spherically symmetric about the source &. One finds then without great difficulty the expansions :
414
[CHAP.V I I
SPHERICAL W A V E S
(87)
The proof is left to the reader. THE VECTOR WAVE EQUATION IN SPHERICAL COORDINATES
7.11. Spherical Vector Wave Functions.As in the case of cylindrical coordinates, one can deduce solut,ions of the vector wave equation in spherical coordinates directly from the characteristic functions of the corresponding scalar equation. Following the notation of the preceding section we shall put 4, = f, ewt, where f, is the characteristic Om=
Om*
solution (1)
fp
=
cos
sin m+P:(cos 0) x,(kZZ).
omn
According to See. 7.1, oiie solution of the vector wave equation
v v * c v x v x
(2)
C+kT
=
0
can be found simply by taking the grsdiont of (1). We define L = v+, and split off the time factor by writing L = k w tThen . by (95), page 52, (3)
1,
a
= z,(kR)
aR
cos
P:(cos 0) sill m+ il
cos + IZ1 z,(IcR) dae P,”(cos e) sin rn4 i, 
m
zn(kld)P;(cos Tm
e) sin cos m4 i3,
where il, iz, is are the unit vectors defined in Fig. 8, page 52, for a spherical coordinate system. To obtain the indeperident solutions M and N as described in Sec. 7.1, one should introduce a jixcd vector a. Such a procedure in the present instance is of course permissible, but then M and N will be neither normal nor purely tangential over the entire surface of th e sphere. If in place of a we usc a radial vector il, a vcctor function L X i, is obtained which is tangent to the sphere; but il is not a constant vector and consequently i t by no means follows B priori that it can be employed to generate a n independent solution. However, we shall discover t h a t a tangential solution M can in fact be constructed from a radial vector. Unfortunately the procedure fails in more general coordinate systems.
SPHERICAL VECTOR WAVE FTJNCTIONS
SEC.7.111
415
Let us try to find a solution of (2) in the form of a vector
V x (ilu(R)+) = L x ilu(R), where u(R) is a n unknown scalar function of R. Then if hl,, M 2 , M S are the R, 6, and cp components of M,we have M
(4)
MI
=
=
0,
The divergence of M is wro and we now expand the equation V x V x M  k2M = 0 into R, 8, and C#I components by (85), page 50. The Rcomponent is identically zero, whatever u(l2). The conditions on the 0 and 4 components are satisfied provided u(R)is such that
(The peculiar advantagc of spherical coordinates over more general systems is that the conditions on both tangential components are identical.) If, therefore, we choose u(R)= R, Eq. (5) reduces to the required relation V2$ k2+ = 0. (6)
+
Hence in spherical coordinates, (2) is satisfied by M = V X R# = L
(7)
xR
=
1
V x N, k
whose components are
(8) From the relation kN be easily found.
=
V x M the components of a third solution can
+
Since must satisfy (4), page 400, the radial component reduces t o the simpler form'
1 The first thorough investigations of the clcctroniagnetic pro1)lcin of the sphere were made by Mie, Ann. Physik, 26, 377, 1908 and Dcxhycs, AWZ.I'hyszk, 30, 57, 1909. Both writers made use of a pair of potential functions lcading directly to our vectors M and N. The connection between these solutions and a radial Hertzian vector has been pointed out by Sommerfeld in RicmannWebcr, " Diffcrentialgleichungen der Physik," p. 497, 1927.
416
SPHERICAL WAVES
[CHAP.V I I
To obtain explicit expressions for the vector wave functions M and N, we need only carry out the differentiation of (1) as required by (8) and (9). The time factor is split off by writing M = me&', N = n c i W t , and we find
l a a +kR dR [Rzn(kR)]ae P,m (cos 8) sin rn4 i, 'OS
7.12. Integral Representations.The wave functions 1, m, and n can be represented as integrals of vector plane waves such as (18) t o (20), page 395. For the functions of the first kind, which are finite at the origin, we have by (60), page 410,
First, we calculate the components of the gradient; namely, af/aR,
'a
~ aa' e ~R' sin e a+'
differentiating under the sign of integration. Next we
observe from an inspection of Fig. 70, page 407, th a t if the vector k(a, p) is directed along the line OQ and R(0, 4) along the line OP, then
+
= k cos y = k[sin a sin 0 cos (4  p) cos a cos 01, k iz = k[sin a cos 0 cos (4  p)  cos a sin 01, k i3 = k sin a sin (4  p),
k. il (14)

the unit spherical vectors i,, i,, il referring, of course, to the point of observation at P. Then it follows without further calculation that
To find the corresponding representations of m(') and n(l) we note that R (16)
xk
=
?&{sin a sin (4  B)i,
+ [sin a
COR
0 cos (4 
 cos
(k x R) x k
=
LY
sin
a)
eli,),
k2R(sin2y il  cos y[sin N cos e cos (4  p)  cos a sin O]iz cos y sin a sin (4  /3)i3],
+
and, upon differentiating according to (8) and (9), obtain
QEC.
417
ORTHOGONALITY
713]
e i k R cos
Y
~ ( C Oa)S‘OS m@sin a d a dfi. sin
To obtain the corresponding integral representations of functions of the third and fourth kinds the integration over a from 0 to T must be replaced by a contour in the complex domain as described on page 410. Rectangular components of the vectors 1, m, and n can be calculated without great difficulty with the help of these integral representations, although the resultant expressions are somewhat long and cumbersome. The vector integrands are rcsolved into their rectangular components and become explicit expressions of a, p, 0, 4. Thus, for example, k , = k sin a cos p. The recurrence rtlations for the spherical harmonics are then applied to eliminate these factors and to reduce the integrals to a form that can be evaluated by (13). 7.13. 0rthogonality.The scalar product of any even vector with any odd vector, or with any vector differing in index m, is obviously orthogonal and we need consider only such products as do not vanish when integrated over 4 from 0 to 27r. Thus (19)
s”” 0
1,omn ’ 1,om* , d#$ = (1
a
+ 6)T
a
zn(kR) aiz z , ( k R ) z,(kR) z,(kR)
where 6 formula
=
0 if m
> 0,
1 i
6 = 1 if m = 0. To reduce (19) we apply the
 01
+
2 (n m)! n(n 2n+l(nm)!
when n # n’,
+ l),when n
=
n’.
Integration of (19) over 0 gives, therefore, zero when n # n’, or, when = n’,
n
A final reduction follows from the recurrence relations (33) and (34), page 406, and we obtain the normalizing factor
418
[CHAP.VII
SPHERICAL WAVES
The same formulas lead directly to the integrals
while all products differing in the index n are zero. Upon examining the cross products, we find first (25)
J2m
1,
0'""
mom,, e &J
=
5 (1 3. 6)
?r
m d
 xn Zn'  [P,"P z ] ;
sin
e ae
hence.
for arbitrary values of n and n'. Likewise
The complete orthogonality of the system is spoiled, as in the cylindrica,l n, , whose integral over the sphere does not case, by t,he product 1,
.
Omn
vanish if n
= n'.
Omn
I n that case we find
To complete the orthogonalization one might treat k' as a variahle parameter and integrate over k' and R as in Sec. 7 . 2 ; such a procedure usually proves unnecessary. 7.14. Expansion of a Vector Plane Wave.To study the diffraction of a plane wave of specified polarization by a spherical object one must
E X P A N S I O N OF A VECTOR PLANE N A V E
SEC.7.14j
419
first find a n expansion of the incident vector wave in terms of the spherical wave functions 1, , me , n, omn
onn
0""
.
Consider the vector function f(z) = a e%kz = a &kR (29)
cos 0 7
where a is a n amplitude vector oriented arbitrarily with respect to a rectangular reference system. Let a be resolved into three unit vectors directed along the z, y, zaxis respectively. Then
+ +
a, = sin 0 cos C#J il cos 0 cos cp i,  sin cp il, a, = sin 0 sin cp il cos 0 sin 4 i, cos cp i,, a2 = cos 0 il  sin 0 i,, where il, i2, is are again the unit vectors defined in Fig. (8), page 52, for a spherical coordinate system. Now the divergence of the vector functions a, exp(ikz) and a, exp(ikx) is zero and consequently they may be cxpanded in terms of the ch:tracteristic functions m and n alone. At R = 0 the field is finite and we shall, tLerefore, require functions of the first kind. It is apparent, moreover, that the dependence of (30) on cp limits us to m = 1; whence upon consideration of the odd arid even properties of (11) and (12), we set up the expansion (30)
(31)
a,etkR
cos 0 =
2
+
(unrn::L
+ bnn::b).
n=O
To determine the coefficicntu preceding scction. (32)
s' J'*
a, m:;;
0
eskR
cofi
0
whence by (23), (33)
which in virtue of (24) gives (35) hence,
W I apply ~
the orthogonality relations of the
sin 0 d0 d+ = 2rinn(n
+ 1)[j,(kR)]2,
420
[CHAP.V I I
SPHERICAL WAVES
The same process gives for the wave polarized in the ydirection the expansion
(37) Since the longitudinal wave function a, exp(ikz) has a nonvanishing divergence, its expansion must involve thc 1 functions. It turns out, in fact, that only this set is required and one finds without difficulty that
Problems 1. Show t h a t the spherical Bessel functions satisfy
0, when m # n,
r m
provided n and m are integers satisfying the conditions n 2 0, m > 0. For the application of such integrals to thc expansion of functions in Bessel series see Watson, "Bessel Functions," page 533. 2. Show t h a t the cylindrical Besscl functions satisfy
and with the help of these formulas show that the spherical Bessel functions satisfy j,(z) = (  1 ) " ( 2 2 ) " __
nn(x)
=
dn cos z (1)"'(22)" d(z2)" ~ 
3. Show that if R 2 = R:
(;&)n &). cos x
= Z"
+ R:  2 R o R 1cos 7, sin kH
sin kRo sin kR1
d(c0s y ) =  9
kRo
sin cos _ kR1 _ kRo __ d(c0s y )
=
kRo kRi cos kRo sin kR1 ___ ___ kRo kRi
7
kRi Ro 5 Ri,
Ro 2 RI.
4. Show that in the prolate spheroidal coordinates defined on page 56 the equation
V2$
+ k2$
=
0
421
PROBLEMS assumes the form
Let @
=
$,(.$)$2(v)+3(4), and show that the above separates into
where m and C arc separation constants. wave functions can be constructed from
Next show that a set of prolate spheroidal
m m fimn= ( E *  1)' (I . +)'ScLn(c, v)Ee,,(c,
cos sin
5 ) . m4
in which both the "angular functions" S c and the "radial functions" Re satisfy the equation (1  2')"'  2(m 1 ) ~ y ' (b  c222)y = 0,
+
+
with m an integer or zero, b = C  m, and c = kd = 2ad/h. 6. When the wave functions of Problcin 4 apply to a complete sphcroid one must choose m = 0, 1, 2, . . . , and find the separation constant b such that the field is finite at the poles TJ = 1. A discrete set of values can be found in the form
b, = n(n
+ 1) +fn(c),
n=0,1,2,

'
..
Show that the functions S e and Re can be represented by the expansions m __
Sc:n(c,
2) =
T,"(z)
Z'd,T,"(z),
=
(1 
22)
">&(z)
s
where P:,,, is a n associated Legendrc: function and the priine over the summation sign indicates that the sum is to be extended over even values of s if n is even and over odd values if n is odd. Re:,(c,
z)
' en ~ (am + s)!
=
2vYL !(C.)"
S!
d s j s + , (CZ)
where j s + m ( c ~is) a spherical Bessel function. The independent solution is obtained Show that the coefficients satisfy the recursion hy replacing j8+m(cz) by n,+,(cz). formula (s
(2s
+ 2m + 2)(s + 2m + 1) + 2m + 5)(2s + 2m + 3)
s(s  1)
+ 2m  1)(2s + 2m  3) C2d,z 2s2 + 2s(2m + 1) + 2m  1 c2 + s(s + 2m + 1)  b (2s + 2m  I ) ( % + 2m + 3) C2ds+2
+
(2s
1
d, = 0
and from this relation give a method of determining the characteristic values b,.
422
SPHERICAL W A V E S
[CHAP.
VII
6. Prove t h a t the spheroidal functions defined in Problem 5 satisfy the relations
Se;, = i"2"+lrn! XnRe:, etczt(1
Se:,(c, z ) = A,, S ' l
 tz)%5'e~,(c,
1 ) dt
where X, is one of a discrete set of characteristic values. 7. Write the equation V2$ k2$ in oblate spheroidal coordinates and express the solutions in terms of the functions Se and Re defined in Problem 5. 8. A system of coordinates E, 7, 9 with rotational symmetry is defined by
+
ds2 = hl d f 2
+ hz dq2 + r2 d @
where r is the perpendicular distance from the axis of rotation. If the field has the same symmetry as the coordinate system, its components are independent of 9. Show that the equations
V
)(
E  iwpH = 0,
V X
H
+
( b e
 u)E = 0
break up into two independent groups:
i _a
rhz a?
i _ a _ rhl aE
(rE+) i w p H ~= 0, (rEg)
+ iwgH,,
=
0,
Show that (11) is satisfied by a potcntial Q such that
(I) can be integrated in the same manner. (Abraham) 9. Apply the mcthod of Problem 8 to find a rotationally symmetric electromagnetic field in prolate spheroidal coordinates. I n the case of electric oscillations directed along meridian lines show that Q satisfies
423
PROBLEMS where Q
=
Q~(E)&~(V).
Note that both of these are a special case of the equation
(1  9 ) y ”  2 ( a
when a =  1 .
+ 1)zy’ + ( b 
c222)y =
0
Show that the field is given by EE = imp d2k2
dQz 
&I 
d(p
1)(E2 
?
72)
dq
10. Two types of spherical electromagnetic waves have been discussed i n this chapter: the transverse electric for which E E = 0, and the transverse magnetic for which H R = 0. Obtain expressions for the radial wave impedances as was done previously for cylindrical waves. 11. Prove the expansions (86) and (87) of Sec. 7.10 and show t h a t when el = T , R 1+ m , Eq. (87) goes over asymptotically into the expansion of a plane wave. 12. Obtain expressions for the rectangular components of the vector spherical wave functions 1, m, and n by the method suggested a t the end of Sec. 7.12.
CHAPTER VIII
RADIATION I n the course of the past three chapters we have studied the propagation of electromagnetic fields with no concern for the manner in which they are established. We consider now the sources, and the fundamental problem of determining the intensity and structure of a field generated by a given distribution of charge and current. THE INHOMOGENEOUS SCALAR WAVE EQUATION
the problem 8.1. Kirchhoff Method of Integration.Mathematically of relating a field to its source is t h a t of integrating an inhomogeneous differential equation. Let J/ represent a scalar potential or any rectangular component of a field vector, and let g(z, y, x , t ) be the density of the source function. We shall assume that throughout a domain T’ the medium containing the source is homogeneous and isotropic, and that its conductivity i s zero. The presence of conductivity introduces serious analytical difficulties which can be avoided in most practical applications of the theory. The effect of conducting bodies in the neighborhood of oscillating sources will be treated as a boundaryvalue problem in Chap. IX. Subject to these restrictions, the scalar function $ satisfies the equation
where v = ( ~ p )  iis the phase velocity. The Kirchhoff theory of integration is an extension to the wave equation of a method applied in See. 3.3 t o Poisson’s equation. Let V be a closed domain bounded by a regular surface S and let 4 and be any two scalar functions which with their first and second derivatives are continuous throughout V and on S. Then by (7), page 165,
+
where d / d n denotes differentiation in the direction of the positive, or outward, normal. Let ( d ,y’, z’) be a fixed point of observation within B and
+
+
r = d ( z ’  x)’ (2’  z ) ~ (y’  y)2 (3) the distance from a variable point (2,y, z ) within V or on X t o the fixed 424
SEC.8.11
KIRCHHOFF M E T H O D OF INTEGRATION
425
point. For 6 we shall choose the spherically symmetric solution 4
(4)
=
;r
+ ;)
(t
of the homogeneous equation
(5) where f ( t ment t
+ i)is a completely arbitrary analytic function of the argu
+
As in the analogous problem of the stationary field, 4 has a singularity at r = 0 ; this point must, therefore, be cxcludcd from the domain V by a small sphere S1 of radius drawn about (x’,y’, 2 ’ ) as a center. The volume V is now bounded externally by X and internally by X1. Furthermore, if we denote the left sidv of (2) by I and make use of (1) and ( 5 ) , we obtain
Or, since $a2+  J / a2Q
at2
=
at2
a

at
at
(4 ;;

+a+) , dt
(7)
at
Upon integrating over the ent,irc duration of time this givcs
+ ( To this end let t‘
The next task is to choose the function f t the last term of (8) vanishes. f ( t t’) the impulse function
+
(9)
f(t
+ t’)
1
= 
e
 (t +t‘)2 ____ 2P
~
6
.L/%
=
S,(t
in such a way that =
r/v and take for
+ t’),
defined by (35), page 291, with the property (10)
J*f ( t + t’) d ( t + t’)
=
1.
+
To avoid any question as to the continuit,y of and its derivatives, wc shall imagine 6 t o be exceedingly small but shall not pass a t once to thc limit 6 = 0. Thus defined, S o ( t t’) vanishes everywhere outside an infinitesimal interval in the neighborhood of t =  t’. With this in mind
+
426
RADIATION
[CHAP.V I I I
it is apparent that if F ( t ) is an arbitrary function of time, then
F(t’)
(11)
=
J


S,(t
+ t’)F(t) dt,
and that a f l a t approaches zero with vanishing 6 for all values of t. Since now 4 and a+/at vanish a t the limits t =  m and t = ~0 for all finite values of T , it follows that the last term of (8) must likewise vanish and, consequently, that
= 
sv
g(z, y, z, t’)
dv.
When equated to the righthand side of (2), this lcads to
Over t,he surface SI we know that d / a n = a/ar, so that
Passing to the limit as r l
3 0,
this reduces to
and hence by (11), (16)
Jwm
dt Jsl (4
$  J. %) an da
= 4?nc/(z’, y’,
z’, 0).
Obviously these.limits may now be drawn in t o enclose a n infinitesimal interval containing the instant t =  r / v .
SEC. 8.11
KIRCHHOFF M E T H O D OF I N T E G R A T I O N
427
Substituting this value into (13) gives
To reduce these integrals to a simpler form we notc that
and that
An integration by parts gives
and application of (11) leads to (21)
I)(%’, Y’,z’, 0)
=
g(z,
Y, Z,
t’)
dv
+
According to (21) the valuc of at the point d ,y‘, z’ and the instant t = 0 is obtained by summing the contributions of elcments whose phases have bcen retardcd by a n amount t’ = r/u. But the location of the reference point on the time axis is purcly arbitrary; consequently, the Iiirchhoff formula becomes



in which the symbol
will denote a function with retarded phase. The volume integral in (22) is a particular solution of the inhomogeneous wave equation representing physically the contribution to $(XI, y’, z’, t ) of all sources containcd within V . To this part icu1:tr solution is added a general solution of the homogeneous equation ( 5 ) cxprcssed
428
[CHAP.V I I I
RADIATION
as a surface integral extended over S and accounting for all sources located outside S. In case the values of II, and its derivatives are known over S, the field is completely determined at all interior points, b u t it is no more permissible to assign these values arbitrarily than in the static case. We shall have occasion below to discuss this point further in connection with the Huygens principle. The behavior of a t infinity is no longer obvious since r enters
+
explicitly into [g] through the retarded time variable t 
rU
The field,
however, is propagated with a finite velocity; consequently, if all sources are located within a finite distance of some fixed point of reference and if they have been established within some finite period in the past, one may allow the surface S to recede beyond the first wave front. It lies then entirely within a region unreached by the disturbance at time t and within which II, and its derivatives are zero. I n t,his case
where V now represents the entire volume occupied by sources. The reader has doubtless noted that the choice of the function
f (t
+ :) is highly arbitrary, since the homogeneous equation (5) admits
:>
also a solution  f t   . This function leads obviously to an advanced rl ( time, implying that the quantity II, can be observed before it has been generated by the source. The familiar chain of cause and effect is thus reversed and this alternative solution might be discarded as logically inconceivable. However the application of “logical ” causality principles offers very insecure footing in matters such as these and we shall do better to restrict the theory to retarded action solely on the grounds that this solution alone conforms to the present physical data. 8.2. Retarded Potentials.When (24) is applied to the scalar potential and to the rectangular components of the vector potential we obtain the formulas I#+?,
t)
A(d, t)
=
: :
4tie Jv  p ( z , t*)
4”, Jv
= 
dv,
 J(z,t*) d ~ ,
which enable one to calculate the field from a given distribution of charge and current. Here x’ represents the coordinate triplet x‘, y’, z’, and x r the triplet x, y, x, while the retarded time is expressed by t* = t  . 2,
SEC.8.21
RETARDED POTENTIALS
429
This statement, however, is subject to one condition. We have in fact only shown that (25) and (26) are solutions of the inhomogeneous wave equation. I n order that they shall also represent potent,ials of an electromagnetic field i t is neccwary that thcy satisfy the relation
The prime attached to the operator v denotcs differentiation with PaSpc’ct to x’, y‘, z’ a t the point of observation. We shall show that this condition is fulfilled by (25) and (26) provided the densities of charge and current satisfy an equation of continuity. Since a / a t = a/at* a t a fixed point in space we have
Now when the operator V r n = V’rn.
v
is applied to any power of r , onc may w r i k r The variables x’ occur in J(x, t * ) only through t* = t  . V’
By the divergence theorem (34) since the closed surface S hounding V can be taken so large that J is zero everywhere over S. T h m on combining (28) and ( 3 3 ) , we find
430
[CHAP.V I I I
RADTATION
and this must vanish since thc conservation of charge requires that
(36)
(v . J)t*=ronstant t *ata P
=
0
at any point whose local time i s t*. 8.3. Retarded Hertz Vector.By virtue of (36) it is sufficient to prtscribe the current distribution in space and time. The charge density can then be calculated and the potmtials det crinined by evaluating thc integrals ( 2 5 ) and (26). However, in most cases it is advantageous to derivc: the field from a single source function by means of a Hertz vector. Let us express the current and charge densities in terms of a single vector P defined by ap J = p = V.P. (37) at '
The equation of continuity is then satisfied identically and the field equations are dH (111) V . H = 0, (I) v x E I p Tf = 0,
(11) V x H  t aE

at'
at
v .E
(IV)
=
1 v.P.
As shown in Sec. 1.11, thcse cqiiations zre satisfied b y
where
(39)
n is any solution
of
vxvx
rI 
The rc,ctangular componcni 5 of wave cquation
(40)
v211,
vv en
+
am
I*€
at2
=
1 P.
n, thcrc~fow,satisfy the inhomogencous
a2n,
1
 I*€ 2    P I ,
at
( j = 1, 2, 3).
It need scarcely bc said that the vector P defined by (37) is not identical with the diclectric polarization dcfined first in See. 1.6. The (x,,x,,x,) vector P of the present section measures thc polarization, or moment per unit vol( 1 1 , (2, (3) ume, of the freecharge distribution and R1 consequently is not equal to D  toE. T o facilitatc the calculations that follow a small change in notation will be 0 made. The point of observation in Fig. FIG. 71.h current element is located at tl, t 2 , t 3 ,while xi, x 2 , x 3 71 whose radial distance from a fixed orilocate a fixed Pint of observation. gin 0 is R will now be located by the rectangular coordinates ( 5 1 , z2, Q). The current element located a distance
y
SEC.8.41
43 1
D E F I N I T I O N OF T H E M O M E N T S
R 1 from 0 has the coordinates of observation is r, so that
((1,
El,
[3)
and its distance from the point
We need consider only the harmonic components of the variation, from which general transient or steadystate solutions may be constructed. Let 1 J = Jo(E> cht, P = P O ( $ ) c  ' ~ ~ , PO = .T V JdE), (42) aw
(43)
P
=
PO(l) c  ' ~ ' ,
i
P" = w Jo.
Applying (24) to (40) and noting that in the present case k we obtain the fundamental formula
=w
&= w/v,
(44) A MULTIPOLE EXPANSION
8.4. Definition of the Moments.To evaluate the retarded Hertz integral one must frequently resort t o some forin of series expansion. The nature of this expansion will be governed by the frequency and the geometry of the current distribution. In the present section we shall consider an extension to variable fields of the theory o€ multipoles set forth in Secs. 3.8 to 3.12 and Secs. 4.4 to 4.7 for clectrostatic and magnetostatic fields. According to (87), page 414, when R > El, the integrarid of (44) above can be expanded in the series
eik. = ik 2 (Zn + 1)P,(cos y) jn(lcRl)hL1)(kR) ; r m
ri = 0
consequently, if R is greater than the radius of a sphere containing the entire source distribution. (an n=O
+ l)hk1'(kR)
sv
Po(( ) j,(lcR1) P,(cos y) dv.
Suppose now that P,,([) is expressed as a function of 121, y, 4 , where 4 is the equatorial angle stbout an axis drawn from 0 through the point of observation. Then t,he vector Pc(R1, y, 4) can be resolved into scalar components and each component expanded in a series of sphtric:tl
432
[CHAP.V I I I
RADIATION
harmonics as in (36), page 406. I n virtue of the ortnogonality of the functions P,(cos y ) , the integral on the right reduces essentially to an expansion coefficient times [jn(kRl)l2.The success of this attack will be determined by the nature of the current distribution and the difficulties encountered in computing the expansion coefficients. If the wave length of the oscillating field is many times greater than the largest dimension of the region occupied by current, the series (2) can be interpreted in terms of electric and magnctic multipoles located at the origin. We shall assume that for all values of RI within V
where A is the wave length in the medium defined by e and p. Consequently the function j,(kRl) can be replaced by the first term of its series expansion. By (25), page 405,
(4)
j,(kRl)
Znn! = ____ (kR1)". (2n + l ) !
The total field will be represcnted as thc sun1 of partial fields by m
(5)
ar(Z,
2) =
2
rI(*)(X, t ) ,
n=O
and, hence,
the first term
(7) Let II. bc any scalar function of position.
By the divergence theorem
(8) and, hence,
(9)

If X encloses thc entire source distribution, Po n will be zero over S and the surface integral will vanish. Choose now for # a rectangular component of the radius vector R1. Then sVPoidv= 
S ijv.
Podv;
433
D E F I N I T I O N OF T H E MOII.IENP’S
SEC.8.41 conseauen tlv
The partial Jield n(O)i s generated by a n oscillating electric dipole whose moment pC1j i s defined exactly as in the electrostatic case, Eq. (47), page 179. To obtain the second term of (6) we note that Pl(cos y) = cos y ,
To interpret the integral we first expand the integrand
(13) PoRl cos y
=

P (Ri R) 2
R
_  1 [(RI X PO)X R 2Et
+ P O @ R)~ + R1(R. Po)]. *
The magnetic dipole moment was defined in (28), page 235, as m(1) = :
(14)
a

R1 X
JO
dv
Since R is independent of the variables 4 of integration, one has, therefore,
The electric quadrupole moments of a charge distribution were defined in (48), page 179, as the components of a tensor,
Upon putting #
=
(17) Furthermore,
tiffin
(9) this gives
pij
J (tipoi +
= 3
(18)
R1.R
=
R
x j=1
Consequently,
aR .$I%:
4201)
dv =
P0.R = R
pii.
s
dR Poi.
j=1
ax1
434
R A D I A TION
[CHAP.V I I I
The quantities p:z) arc the componc’nis of a w c t o r (20) p(2’ = 2 p . vn, where 2p is the tensor whosc component., are pt3. [Cf. Eqs. (23), (24), pagc 100.1 The partial ficlcl II(l) rcprrscnts the contrzbutions of a magnrtic dipole and an electric puadrupolc.
Comparison of (21) with (7) shows that the two fields differ in the first place by a factor 6 , or approximately 3 X lQs, and consequently the moments m(l) and pCz) must be very nzucli larger than p(I) if nc1)is to be of appreciable magnitude. On the other hand, it must be remembered that the electric moments of a current distribution arc functions of the frequency a h wc,ll as of the gcometry [Eq. (43), page 4311. If p(?) is expressed in terms of current rather llian charge, wc scc that the magnitude of r P is indcpcndcnt of frequency, whcrcas the magnitude of diminishes as 1/w with incrcasing frequency. The itmaining tmms of the serics in ncn)rrprescnt the contributions of electric and magnetic multipolcs of higher ordcr. These must be cxFIG.73.Coniponerit,s of field about an I)r(>ihedin terms of tensors and the electiic dipole. labor of computation incrcasm rapidly. The magnitudcs of the siiccessivc tcrms progrew csscritially as ( o / c ) ~ . In c a s ~the rate of convcrgcncc is so slow that higher order terms must hc taken into account, one is usually forced to adopt some other method of calculating the ratliat ion. 8.5. Electric Dipole.Let us suppcasc that, the charge distribution is such that only the electric dipole momcnt is of importance. This is the case, for example, of a current oscillating in a straight scction of wire whose length is very small compared to that of the wave. The coordinate system is oriented so that the positive zaxis coincides with the dipole moment p(l). The vector p(l) can he resolved into its components in a spherical system as illustrated in Fig. 7 2 . (22)
p(I)
=
p(l) cos e il  p ( l ) sin e i,,
435
ELrTCTRIC DIPOLE
SEC.8.51
where i,, i,, is are the unit spherical vectors designated in Fig. 8, page 52. According to (7) the Hertz vector II(O) is parallel to p(l) arid can be resolved in the same manner. Thc field vcrtors arc now calculated from the formulas (38), page 430, which in the case of a harmonic time factor become
E =VVn
(23)
+ k211,
H
=
iW€V x n.
The differential opc~rntors,cxprtwed in sphcrical coordinates by (trj), page 52, are applied t o thc v c c t o r (24) II(”) = r P cos 0 il  IT(o) sin 0 i,. If Ro = V R is a unit vector dirccted from the dipole towards the observer
and t* = t 
R )
2,
then
The general structure of a dipole field is easily determined from these expressions. Wc note in thc first place that the elect>ricvccLor lies in a meridian plane through t’hc axis of the dipole and thc magnetic vector is perpendicular to this plmc. The magnetic lines of force arc coaxial circles about’ the dipole. At zcro frequency all terms vanish witb the exception of the first in l / W in (25), which upon rcfcrcnce to (27), page 175, is seen to be identical with the field of an electrostatic dipole. Moreover, if we replace p(l) by the linear current element
i

w
I ds according
to (43), pagc 431, it is apparent fhat t’he first tcrm of (26) can he interprcted as a n extension to variable ficlds of the Biot,Savart law, (14), page 232. The terms in (25) and (26) have been arranged in invtrrsc powers of R and the ratio of the magnitudes of successive terms is k R ,
R
or 2~ .
x
I n the immediate neighborhood of t’he source the “static” and
“induction” fields in l / R 3 and l / R 2 predominate, while at distances such that R >> X, or k R 2, the term in Ci 2rm is negligible. The relation of radiation resistance to the order of excitation is shown graphically in Fig. 76. Equation (23) is valid only a t resonance, when the quantity 21/X = m is an integer. At intermediate wave lengths the radiation resistance follows a curve which oscillates above and below the resonance values indicated by Fig. 76. The exact form of this curve depends on the nature of the excitation.‘ See, for example, an interesting paper by Labus, Hochfrequenztech. u. Elektroak., 41, 17, 1933.
SEC.8.81
R A D I A T I O N DUE TO T R A V E L I N G W A V E S
445
8.8. Radiation Due to Traveling Waves.The current distribution considered in the preceding paragraph constitutes a standing wave with nodes located a t the end points. By a proper termination of the wire, a part or all of the reflected wave may be suppressed with the result that the current is propagated along the conductor as a traveling wave. Such, for example, is the case of a transmission line, where in order to secure the most efficient transfer of energy every effort is made to avoid standing waves. To calculate the radiation losses of a singlewire antenna of finite length' with no reflected current component, let us assume for the current distribution the function (24) u ( { )= IoeikE.
If, as in the previous case, the antenna is fed a t the central point, the radiation field intensities are
which, when evaluated, gives
The radiant energy flow is determincd by the function
The power dissipated in radiation is, therefore, equal to
(28)
W
=
302;
sin 0 do. (1  cos 0)' 1  cos 0, du = sin 0 dO, and reduce (28) to ~
To integrate we put u
=
klu ( 2  u ) sin2
W
=
301;
U
du.
Omitting the elementary intervening steps, one finds for W the expression
'
The problem of a line with parallel return was first treated correctly by Manneback, J. Am. Inst. Elec. Engrs., 42, 95, 1923.
446
[CHAP.V I I I
RADIATION
and for the radiation resistance
I t appears that for equal wave lengths and the same antenna the radiation resistance of a traveling wave is greater than that of a standing wave. /
/
I I \ \
I
f
t\
is
/I I
I
\
\
@
/
I
I \
t:
\ \
MarconiFranklin antennas
Idealized current distribution FIG. 77.
8.9. Suppression of Alternate Phases. The directional properties of a singlewire antenna can be accentuated by suppression of alternate current loops. This is accomplished in the Marconia Franklin type antenna by proper loading with inductance at halfwavelength in, tervals along the line. The alternate sections of opposite phase are shortened to \ such an extent as to render them negligible \ \ as radiators and the antenna is then equivFIG. 78.Evaluation of the series alcnt to a colinear set of halfwave oscilb e%mb. lators placed end to end and all in phase. m=O This is shown schematically in Fig. 77. The contribution of each oscillating segment to the radiation field at a distant point arrives with a phase advance of j3 = w d / c over its next
2
Consequently the resultant field is
lower neighbor.
(32)
447
SUPPRESSION OF ALTERNATE P H A S E S
SEC.8.91
.
E0 OOiIo 
sin 19
_
~

m=O
where n is the number of current loops. Now the quantity b ei4 is a vector in a complex plane rotated through an angle mP from the real axis. The sum indicated in (32) can, therefore, be evaluated by simple geometry. The vcctor diagram is drawn in Fig. 78, and the sum is clearly the chord of a regular polygon. Following the notation of the figure WE have
’
2
b = 2a sin 2’
2 a s i n n7P 2
=
b sin nP whence
= ___
sin P
a t an angle a
Since in the present instancc b
=:
=  (n 
P with the real axis. 1) 2
1, we obtain
(33)
For P we write w d l c = ?r cos 8, since d
x
2 cos 8, and so obtain
= 
The radiation intensity is
(T cos
sin
(35)
2a
sin 8
sin
1’’ 6
6
and the total power radiated
cos2
(36)
W
=
301;
8)
cos 6) sin2
cos 0)
(5
cos 8)
______ 
sin 0
sin2
d8.
cos 8)
448
RADIATION
[CHAP.VIII
This integral has been evaluated by BontschBruewitsch,' who obtains ns
W
(37)
= (1)n1301;
(n1)r
X  4X 0
+ 8s
(n21% ( n  3 ) r 0
0
 128 0
+. . .
A
4(n
 l)i], 0
where mr
sin2 u
0
U
du
=
600 
500 T
400 
2
300 
200 100 
0
1 (In 2may  Ci 2m7r). 2

ing segments; Fig. 79b is a plot of radiation resistance which shows t h a t a marked increase of radiating efficiency is achieved b y the suppression of alternate interfering phases. 8.10. Directional Arrays.The singlewire antenna is directive only i n the sense that the radiation is concentrated in cones of revolution about the antenna as an axis. A colinear set of halfwave oscillators excited i n phase confines the radiant energy to a thin, circular disklike region such as that of Fig.
duce a preferred direction in the equatorial plane itself, or any other form of axial asymmetry, the singlewire radiator must be replaced by a group or array of halfwave antennas suitably spaced and excited in the proper phase. In practicc the spacing between centers is usually regular, so that the array constitutes a lattice structure. There is in fact a certain parallel between the problems of xray crystallography and the design of shortwave antenna systems. From the xray diffraction pattern the physicist endeavors to locate the centers of the diffracting atomic dipoles and hence determine the structure of the crystal; the radio engineer must choose the lattice spacing and phases of the current dipoles so as t o FIG.79a. FIG.79b. FIG.79a.Radiation pattern about antenna whose current distribution is indicated by the dotted curve. FIG.796.Radiation resistance of n halfwave segments oscillating in phase. I
1
BONTSCHBRUEWITSCH, Ann. Physik, 81, 437, 1926.
SEC.
8.101
DI REC T I ON AL A R R A Y S
449
obtain a prescribed radiation pattern. I n both cases the distribution of radiation is characterized by a phase or form factor dependent upon the polar angles 0 and 4, and t’he methods which havc been developed for the analysis of crystal structure can be applied in large part to antenna design. To simplify matters we shall assume that all radiators of the array are parallel. The antennas are excited a t the central point by means of
FIQ.80.A
system of halfwave oscillators arrayed in a regular lattice structure defined by the base vectors 8 1 , 8 2 , 8 3 .
transmission lines which can be designed to radiate a negligible amount of energy. Moreover the phase velocity of propagation along these lines can be adjusted to values either less or greater than c ; consequently, the phase relations between the members of the array can be prescribed practically a t wil1.l The phase factor of a single halfwave antenna will be denoted by
PO, cos
(40)
Fo(e)
=
6
cos sin
e)
The centers of the oscillators are spaced regularly in a lattice structure whose constants are fixed by the three “base vectors” al, a2,a3,as shown ‘The technical aspects of this problem have been discussed by many recent writers. One may consult: Carter, Hansel1 and Lindenblad, Proc. Inst. Radio Engrs., 19, 1773, 1931; Wilmotte and McPetrie, J . Inst. Elec. Eng. (London),66, 949, 1928; Hund, “Phenomena in Highfrequency Systems,” McGrawHill, 1936; Hollman, “Physik und Technik der ultrakurzen Wellen,” Vol. XI, Springer, 1936.
450
RADIATION
in Fig. 80.
[CHAP.
VIII
The j t h oscillator is located in the network by a vector rj
(41)
=
+ jzaz + j3a3,
j 2 , j , are three whole numbers not excluding zero. The phase where jl, of thc j t h oscillator with respect t o the phase of the oscillator a t the origin is pi,
Pi
(42)
=j 1,l
+ jz, +
j3a3.
is the phase of any oscillat'or relative to it,s nearest neighbor on Thus the axis defined by the base vector al, and so on for the others. Thc radiation field of the j t h oscillator is, therefore, (43)
Ej
=
iGOIojFo(0)
e i k R  iwt  ikRo.r,  ipj
R
where Ro is a unit vector in the direction of the radius vector R. The resultant field int'ensity is obtained by summing over the entire array.
(44) By a proper adjustment of amplitude, spacing, and phase in (44) a radiation pattern of almost any dcsired form can be obtained. We shall consider only the most common arrangement in which all current amplitudes are the same and the base vectors al, a2, a3 define a rectangular structure whose sides are parallel t o the x, y, and zaxes. I n each row parallel to the xaxis there are nl radiators; in each row parallel to the yaxis there are 722; and in each column parallel t o the zaxis there are n3, so that the lattice is completely filled. (45)
RO ri = ~ I U sin I 0 cos 9 + jzu2sin 0 sin 4
+ j3u3cos 8
in which UI, u2, a3 are the spacings along the 2, y, zaxes respectively. The field intensity a t any point in the radiation field is
whose complex phase factors arc m
(47)
DIRECTIONAL AREAI.N
SEC.8.101
451
If we let (48)
71
=
kal sin
e cos 4 + a l l 7 3
=
yz =
kas cos 0
kaz sin
+ a3,
e sin
4
+ a2,
then (47) can be written
by (33). The complex factors 3,re eliminated from the radiation intensity and we get (50) where
. nsYs sin 2 ~
F,
=
(s =
7
sin
1, 2, 3 ) .
7s
All the radiation formulas derived above are special cases of this result. For example, the single wire of Sec. 8.7 excited in an upper mode constitutcs a linear distribution of halfwave oscillators with a1 = a2 = 0, a3 = X/2, a3 = a. Then (52)
F1= F 2 = 1,
n 3a sin  (cos 8 2 F 3 cos
6
+ 1)
cos 0)
which leads directly to (11) on page 441. If the colinear oscillators are in phase, as describcd in Sec. 8.9, wc take a1 = a2 = 0 , a3 = X/2, a p = 0, and obtain ( 3 5 ) . Considcr next a row of vertical halfwave oscillators spaced along the horizontal xaxis. We shall assume them t o be excited in phase and spaced a half wave length apart. a1 = X/2, a2 = a3 = 0, a1 = 0. Then
n o w depends o n the equatorial angle 4. The zeros occur a t those angles for which the quantity nla/2 sin e cos 4 has any one of the valucs T , 27r, . . . . In the equatorial plane e = ~ / and 2 there are radiation
which
452
RADIATION
[CHAP.
VIII
nodes along the lines determined by cos 4
(54) or
cos 4
=
=
2 24 . . . iJ nl nl n1 nl
i  7
2 i  7
nl
4
in1
FIG. 8la.Radiation pattern in the equatorial plane of two halfwave oscillators operating in phase and separated by a half wave length.
pattern in the equatorial plane of four halfwave oscillators operating in phase and with halfwavelength separation.
as nl is even or odd. The principal maximum occurs a t 4 = r / 2 , so that this array gives rise to a broadside emission of energy. Subsidiary maxima are determined by the condition aFl/a+ = 0, which in the present instance leads to the relation
(55)
nl tan
6
sin
e cos cp . (Y ) nl, so that the radiation intensity in the
)
e cos 4
=
tan  sin
When 0 = cp = r / 2 , FoFl = direction of the principal m a x i m u m i s ni times the m a x i m u m intensity of a single oscillator. On the other hand, each oscillator receives only the nlth part of the total power delivered by the source (assuming equipartition), and hence the net gain in radiation intensity in the preferred direction i s given by the factor 71.1, the number of oscillators.
DIRECTIONAL ARRAYS
SEC.8.101
453
The neighboring zeros lying on either side of the principal maximum in the equatorial plane are fixed by the relation cos 4 = f2/nl. Hence the larger nl, the narrower the heam. The beam is evidently confined to a region bounded by the angles $1 and &, where cos 42  cos q h = 4/nl,
I
FIG.82.Radiation
pattern in the equatorial plane of two halfwave oscillators 90 deg. out of phase and with quarterwave spacing.
lying on either side of 4 = 7r/2. The radiation patterns in an equatorial plane have been plotted in Figs. 81u and 81b for arrays containing two and four oscillat  r ~ ( t ~ ~ ) i .
3=1
% = I
A difficulty ariscs whcn thiJ formula is applied to colincar conductors. Consider, for cxamplc, the radiation from a single linear oscillator as in SCC.8.7. This case is the limit of two parallel wires whosc separation approaches zero. If we take the terminals of the two conductors to be coincident, t h m 1
B E L H M A Zoc. ~ N ,czt , p. 1471.
But k
=
am/l
=
1 27r/X,  d p o / e 0 = 60; hence b y (90) 2a
W
= 151;Ci
amu
2z
1
0

This result fails, however, since CiO is infinite. The difficulty can be avoided by calculating first the radiation of two parallel wires separated by a small but finite distance p . Upon passing to the limit p = 0, the infinite terms drop out‘ and one obtains, as in See. 8.7,
W
(91)
=
1512,(ln2may  Ci2ma).
THE KIRCHHOFFHUYGENS PRINCIPLE
8.13. Scalar Wave Functions.Lct V be any region within a homogeneous, isotropic medium bounded by a closed, regular surface s, and let +(x, y, z ) be any solution of (1)
v2*
+
k2+
=
0
which is continuous and has continuous first derivatives within V and on S. Then i t follows directly from Green’s theorem, ( 7 ) , pagc 165, that the value of a t any interior point x‘, y’, z’ can be expressed as an integral of and its normal derivative over S.
+
+
where as usual
r = d(z’  212
(3)
+ (y’  g>2 + (z’
 z)2
is the distance from the variable point x, y, z on S to the fixed interior point x’,y’, z’. Equation (2) is in fact the special case of (22), pagc 427, for which all sources are located outside S and the time factor exp (id) has been split off. 1
Details are given by Bechmann, Zoc. cit., p. 1477.
SEC.8.131
461
SCALAR WAVE FUNCTIONS
The function defined by (2) is continuous and has cont,inuous derivatives a t all interior points, but exhibits discontinuities as the point x’, y’, z’ traverses the surface S . The transition of the function f(s’,y’,
(4)
2’) = 
across S was shown to be continuous in See. 3.15; consequently, the discontinuities of (2) are identical with t,hose discussed in Scc. 3.17 in connection with the static potential. If d, y’, z‘ is a n y point, either interior or exterior, the function defined by the integral (5)
\ L’
FIQ.%.The
/
surface SI may be closed a t infinity.
is a t all interior points identical with the solution J. whosc values have been specified over S , and a t all exterior points is zero.’ As in the static case, too, the values of J. and its derivatives at all alone on S interior points are uniquely determined by the values of (Dirichlet problem), or by d J . / d n alone (Neumann problem). The values of both J. and dJ./an, therefore, cannot be specified arbitrarily over S if u and are to be identical a t interior points. The function u defined by (5) satisfies (1) and is regular within V whatever the choice of and over S , but the values assumed by u and &/an on X will in general differ from those assigned to and d J . / d n . All elements of the surface S which are infinitely remote from both the source and the point of observation contribute nothing to the value
+
+
+
The analytic properties of the Kirchhoff wave functions are discussed in detail by PoincarB, “ThBorie math6matique de la lumii?re,” Vol. 11, Chap. VII. See also the very useful treatise by Pockels, “ a e r die partielle Differentialgleiehung Au k*u = 0,”Teubner, 1891, and “Lectures on Cauchy’s Problem” by Hadamard, Yale University Press, 1923.
+
462
R A D I A TION
[CHAP.
VIII
of the integral (2). I n Fig. 86 all elements of the source are located within a finite distance 121 of a fixed origin &. The surface X is represented as an infinite surface 81 closed by a spherical surface X, of very large radius r about P as a center. Thc point of obscrvation P(z’, y’, 2’) lies within S and all elcrnents of the source are exterior t o S. If R , the distance from Q to an clement of area on Xt,is very much larger than the largest value of R1, then by (87), page 414, and ( 3 2 ) , page 406, eikRz
(6)

~ ‘v 
I22
e?!! R
2
(i)%(2n
+ 1)P,(cos TI) j,(kR1)
n=O
n0
(7) hence,
Over Sz
The terms in r I, r P 2 ,and r  3 in the expanded integrands of ( 2 ) )therefore, cancel one another, while the remaining terms are of the order r”, n > 3, and consequently contribute nothing t o the integral over Sz as r + m . Let us suppose now that the surface X represents an opaque screen separating the source from the observer. I n virtue of the theorem just proved it can be assumed that open surfaces of infinite extent are closed a t infinity. If a small opening 81 is made in the scrcen, the field will penetrate to some extent into the region occupied by the ohserwr. Thc problem is to determine the intensity and distribution of this diffracted field. Clearly if the values of and a + / d n were known over thc opening XI and over the obscrver’s side of the screen, the diffracted field could be calculated by evaluating (2). Thcse values are not known, but t o obtain an approximate solution one may assume tentatively with Kirchhoff that: ( a ) On the inner surface of the scrcen
+
SEC.8.131
SCALAR W A V E F U N C T I O N S
463
( b ) Over tjhe surface S1 of the slit or opening the field is identical with that of the unpc:rturbetl incident wave. Nearly all calculations of the diffraction patterns of slit>sand gratings made since Kirchhoff’s time have been based on these assumptions.’ It is easy t o point out fundamental analytic errors involved in this procedure. I n the first place, the assumption that $ and a$/& are zero over t’heinside of the screen implies a discontinuity about the contour C1 which bounds the opening S1,whereas Green’s theorem is valid only for functions which are continuous everywhere on the complete surface X. This difficulty cannot be obviated by the simple expedient of replacing the contour of discontinuity by a t’hin region of rapid but continuous t,ransition. If $ and a$/an are zero over any finite part of S , they arc zero a t all points of the space enclosed by AS.^ I n the eecond place, an electromagnetic field cannot in general be represent.ed by a single scalar wave function. It is characterized by a set of scalar functions which represent rectangular componcnts of the electric and magnetic vectors. Each of these scalar functions satisfies (1) and its value at an interior point z’, y’, z’ is, therefore, expressed by ( 2 ) in terms of its values over the houndary X. But these components a t a n interior poirit must not only satisfy the wave equation, they must also be solutions of the Maxwell field equations. The real problem is not the integration of a wave cquation, either scalar or vector, but of a simultaneous system of firstorder vector equations relating t’he vectors E and H. I n spite of such objections the classical Kirchhoff theory leads to satisfactory solutions of many of the diffraction problems of physical optics. This success is due primarily to thc fact that the ratio of wave length to the largest dimension of the opening in optical problems is small. As a consequence the diffracted radiation is thrown largely forward in the direction of the incident ray and the assumption of zero intcnsity on the shadow side of the screen is approximately justified. Measurcments are usually of intensity and do not take account of the polarization. As the wave length is increased, the diffraction pattern hroadens. A computation of $ from ( 2 ) now leads to intensities directly h h i n d the screen which are by no means zero, contrary to the initial assumption. Various writers have suggest’ed that this might be conOn the classical theory of diffraction see, for example, Plmck, “ I ~ h f i i h r u n gin die theorctische Optik,” Chap. IV, Hirzel, Lcipzig, 1927, or Born, “Optik,” Chap. IV, Springer, Berlin, 1933. To obtain a clearer understanding of the physical significance as well as the shortcomings of Kirchhoff’s met,hod the readcr should consult thc papcrs by Rubinowicz, Ann. I’hysik, 63, 257, 1917, and Kottler, ibid., 70, 105, 1923. Since the writing of this and the following sections a treatment of the subject Ins a p p m r d in book form: “The Mathematical Theory of Huygerrs’ Principle,” by Baker and Copson, Oxford, 1939. A direct consequence of Green’s tlieorern. See POCKELS, loc. cit., p. 212.
464
RADIATION
[CHAP.
VIII
sidered as the first of a series of successive approximations, the values of $ and a$/& over S obtained from the first calculation to be used as the boundary conditions in the second. There is, however, no proof that the process converges and the difficulties of integration make it impractical. Recent advances in the technique of generating ultrahighfrequency radio waves have stimulated interest in a number of problems previously of little practical importance. A natural consequence of this trend towards short waves is the application of methods of physical optics to the calculation of intensity and distribution of electromagnetic radiation from hollow tubes, horns, or small openings in cavity resonators. I n radio practice, however, the length of the wave is commonly of the same order as the dimensions of the opening, and the polarization of the diffracted radiation is easily observed. It is hardly to be expected that the Kirchhoff formula (2) can be relied upon under these circumstances. 8.14. Direct Integration of the Field Equations.*The problem of expressing the vectors E and H at an interior point in terms of the values of E and H over an enclosing surface has been discussed by a number of writers2 A simple and direct proof of the desired result can be obtained by applying the vector analogue of Green’s theorem to the field equations. In Sec. 4.14 it was shown that if P and Q are two vector functions of position with the proper continuity, then
(10) ~ ( Q . V X V X P  P  V X V X Q ) d v
=$s (P x v x Q  Q x B x P) encia, where S is as usual a regular surface bounding the volume V . Let us assume that the field vectors contain the time only as a factor exp (id) and write the field equations in the form
(1)
V x E  i ~ p H= J*,
(111) V * H
=
1
 p*, P
The medium is assumed to be homogeneous and isotropic, and of zero conductivity. The quantities J* and p* are fictitious densities of “magnetic current’’ and ‘(magnetic charge,” which to the best of our 8.14 and 8.15 were written in collaboration with Dr. L. J. Chu. z L o ~Phil. , Trans., (A) 197, 1901; LARMOR, Lond. Math. SOC.Proc., 1, 1, 1903; IGNATOWSKY, Ann. Physik, 23, 875, 1907, and 26,99, 1908; TONOLO, Annali d i Mat., 3, 17, 1910; MACDONALD, Proc. Lond. Math. SOC.,10, 91, 1911 and Phil. Trans., (A) 212, 295, 1912; TEDONE,Linc. Rendi., (5) 1, 286, 1917; KOTTLER,Ann. Physik, 71, 457, 1923; SCHELKUNOFF, Bell. System Tech. J . , 16, 92, 1936; BAKER and COPSON, Zoc. cit., Chap. 111. 1 Sections
SEC.8.141
465
INTEGRATION OF FIELD EQUATIONS
knowledgct have no physical existence. However, we shall have occasion shortly to assume arbitrary discontinuities in both E and H, discontinuities which are in fact physically impossible, but which would be generated by surface distributions of magnetic current or charge were they to exist. Currents and charges of both types are related by the equations of continuity,
V * J  imp
(V)
=
V J*  i m p *
0,
=
0.
The vectors E and H satisfy
V x V x E  k2E = iwpJ  V X J*, (11) V x V x H  k2H = iweJ* V X J, (12) with k2 = w2ep. I n (10) let P = E, Q = 4a, where a is a unit vector in an arbitrary direction and 4 = eikr/r. Distance r is measured from the element a t x, y, z to the point of observation a t x’, y’, z’ and is defined by (3). We have identically
+
(13)
V x Q
+

V 4 x a, V X V x Q = ak24 V(a V+), V X V X P = k2E iwpJ  V X J*.
=
+
Following the procedure of See. 4.15, i t is easily shown that a is a factor common to all the terms of (10) and, since a is arbitrary, it follows that
=
l [ i m p ( n x H)4
+ (n x E) x v@+ (n  E)v4  n x J*+] (la.
The identity
reduces this to
(16)
s
V
(iwJ4

J* X V 4
=
+
[iwp(n x
H)
+ (n x E) x V 4 + (n E)V+] du.
The exclusion of the singularity a t r = 0 was described in Sec. 4.15. A sphere of radius r1 is circumscribed about the point z’, y’, z’, its nornial directed out of V and consequently radially toward the center.
466
[CHAP.V I I I
R A D IA T IO N
and on the sphere n = ro. The area of the sphere vanishes with the radius as 4ar2,and, since
+ 
(n x E) x n (n E)n = E, (18) t)he contribution of the sphcrical surface to the righthand side (16) reduces to 4nE(z’, y’, 2 ’ ) . The value of E a t any interior point of V is, therefore, (19) E ( d , y’, z‘)

sv
4‘, 
=
&
(iwrJm

[iwp(n x H ) +
J* x v+
+
+ (n x E) x V + + (n  E)V+] da.
An obvious interchange of vectors leads to the corresponding expression for H, (20) H(d, y’,
2‘) = 4a
Ss
+ 45, 
sv
(iwrJ*m
[k(n
x E)$
+ J x v+ + P  (n x Ip)
x
V+ 

(n H)V+] da.
This last is the extension of ( 2 3 ) , page 254, to the dynamic field. If all currents and charges can be enclosed within a sphere of finite radius, the field is regular a t infinity and either side of X may be choscn as its “interior.” I n the earlier sections of this chapter the calculation of the fields of specified distributions of charge was discussed in terms of vector and scalar potentials arid of Hertzian wetors. We have now shown that E and H may be calciilated directly without the intervention of potcntials. The surface intvgrals of (19) and (20) rcymscnt the contributions of sources located outside S. If S rccedcs to infinity, it may be assumed that these contributions vanish. Discarding the densities of magnetic charges and currents, one obtains the uwf ul formulas
Since the current distribution is assumed t>obe known, the charge density can be determined from the equation of continuity. I n Sec. 9.2 it will be shown that an electromagnetic field within a bounded domain is completely determined by specification of the tangential componcnts of E or H on the surface and the initial distribution of the field throughout the enclosed volume. It follows that when a x E and n x H in (19) and (20) have been fixed, the choice of n E

467
INTEGRATION OF FIELD EQUATIONS
S w . 8.141

and n H is no longer arbitrary. The selection must be consistent with the conditions imposed on a field satisfying Maxwell’s equations. The same limitations on the choice of 1c, and d1c,/an were pointed out in Sec. 8.13. The dependence of the normal component of E upon the tangential component of H is equivalent to that of p upon J. Let i i s suppose for the moment that the charge arid current distributions in (19) are confined to a thin layer a t the surface S. As the depth of the layer diminishts, the densities niay be increased so that in the limit the volume densities arc rcplaccd in the usual way by surface densities. If the region V contains no charge or current within its interior or on its boundary S , the field a t an intcrior point is (22)
E(z’,y’,x’) = G s [ i o p ( n X H ) + + ( n X E ) x V + + ( n  E ) V + ] d a . l.I
Itj is now clear that this is cxactly the field that would be produced by a distribution of clertric currcni over S with surface density K, a distribution of magnetic current of density K*, and a surface electrir charge of density 7, where (23)
K
=
n
x
K* = n x E,
H,
6n.E.
?1 =
The values of E and H in (23) arc those ,just inside the surface S. The function E(z’, y’, z ’ ) defined by (22) is discontinuous across A!. It can be shown as above that discontinuities associated with = e c h T / r arc idcntical with those of the stationary regime, = l / r . Then by See. 3.15 the integral
+
+
(24) suffers a discontinuity on tr:tnsilion through S equal to n * AE3 = q / c , where AE, is the diffcrencc oi‘ the values outsitlr and inside. The third term of (22)’ thercforc, docs not affect the transition of the tangential component but reduces the normal component of E t o zero. Likewisc by Sec. 4.13 the discontinuity of Ez(z’, y’,
L
4‘,
2’) = 
K* X v + d a
is specified by n x AEz = K*, so that the srcond term in (22) reduces the tangential component of E t o zero without affecting thc normal coinponent. The first term in (22) is continuous arross 8, but has discontinuous derivatives. The curl of E a t z’, y’, x’ is
V’ x E(d, y’,
2’)
=
2
Jy
(n
x H) x V+ da
468
RADIATION
[CHAP.V I I I
which is of the type ( 2 5 ) . The vector E and the tangential component of its curl are zero o n the positive side of S; E i s , thwefore, zero at all external points. The same analysis applies to H. 8.16. Discontinuous Surface Distributions.The results of the preceding section hold only if the vectors E and H are continuous and have continuous first derivatives at all points of S . They cannot, therefore, be applied directly to the problem of diffraction a t a slit. T o obtain the required extension of (22) to such cases, consider the closed surface X (surfaces closed a t infinity are included) to be divided into two zones S1 and S , by a closed contour C lying on S. as in Fig. 87. The vectors E and H and their first derivatives are continuous over S1 and satisfy the field equations. The same is true over Xz. However, the components of E and H which are tangential to the surface are n o w FIG.87.C is a contour on the closed subject to a discontinuous change surface S dividing it into the two parts Si in across c from to and S2. the other. The occurrence of such discontinuities can be reconciled with the field equations only by the further assumption of a line distribution of charges or currents about the contour C. This line distribution of sources contributes to the field, and only when it is taken into account do the resultant expressions for E and H satisfy Maxwell's equations. A method of determining a contour distribution consistent with the requirements of the problem was proposed by Kottler.' It has been shown that the field a t an interior point is identical with that produced by the surface currents and charges specified in (23). A discontinuity in the tangential components of E and H in passing on the surface from zone S1 t o zone S z implies therefore an abrupt change in the surface current density. The termination of a line of current, in turn, can be accounted for according to the equation of continuity b y an accumulation of charge on the contour. Let ds bc an element of length along the contour in the positive direction as determined b y the positive normal n, Fig. 87. Let n l be a unit vector lying in thc surface, normal to both n and the contour element ds, and directed into zone (1). The line densities of electric and magnetic charge will be designated by u and u*. Then Eqs. (V), when applied to surface currents, become
UOTTLER, Ann. Physik, 71, 457, 1923.
SEC.8.151
DISCONTINUOUS SURFACE D I S T R I B U T I O N S
469
and hence by (23)
(28)
i w g = nl
 (n X 
H2
n
iwa* = nl (n X El  n

X HI) = (H2  Hl) (nl x n), x E2) =  (Ez  El) (nl x n).

The vector nl x n is in the direction of ds. If S2represents an opaque screen over which E2 = H2 = 0, the field a t any point on the shadow side is
which can be shown to be identical with
It remains to be shown that the fields expresscd by these integrals are in fact divergenceless and satisfy (I) and (11). Consider first the divergence of (29) a t a point z’, y’, 2’.
n
taking into account the relation V’ = V when applied to 4 or its derivatives. Now
(33)

(n x H) v + d a =
4v x H Snda  f C 4 H ds.
470
RADIA TION
[CHAP.V I I I
The line integral resulting from t.his transformation is zero when S is closed, hut otherwise just cancels the contour integral in (32). Inversely, only the presence of th,e contour integral in (29) leads to a zero divergence and waves which are transverse at great distances f r o m the opening S1. From (11) and the relation Ic2/ioe =  i w p , follows immediately the result V’ * E(x’, y’, 2’) = 0. (34) An identical proof holds for H(z’, y’, 2’). Finally, i t will be shown that (29) and (31) satisfy (I) and (11). (35)
V’ x H(z’, y’, z’)
= 4lT
L*

[(n x H1) V V +
 iwc(n
since the curl of the gradient is idcntically zero.
(n x HI) . vv+ da
(36) =
J,(n
= $c
+
=
V x H J v + da 

X El) X
n (Vvm
V+]da,
Furthermore,
1, x L x v  (Hlv+) A1
V+ HI ds  i w c
+ k2+(n x HI)
H l ) v + da
n
da
hl(n  El)v + da,
where the operator vv+acts on v+ only. The last integral takes account of t,he fact that t h e field equations arc by hypothesis sahfied on X1. Then
(37)
V‘ X H(z’, y’,
2’) =
lTpCV + H1  + 2 kl[iwdn x H1)+ ds

+ (n X El) X V + + (n  El) V+] da
= iwc
E ( d , y’,
2’).
The validity of (I) is established in the same manner.’ FOURDIMENSIONAL FORMULATION OF THE RADIATION PROBLEM
8.16. Integration of the Wave Equation.In SCC.1.21 i t was shown how by means of tensors t,he field equat’ions could be written in an exceedingly concise form. If the imaginary distance x4 = i c t be introduced as a coordinate in a fourdimensional manifold, the equations of a variable field are in fact formally identical with those which govern the static regime, and the methods which were applied to the integrat,ion of Poisson’s equation can be extended directly t o thc more general case. The fourThese formulas have bccn applied by L. J. Chu and the author to thc c:tlciilntion T h e results compare favorably with those obtained by Morse and Rubenstein, Phys. Rev., 64, 895, 1938, who solved a twodimensional slit problem rigorously by introducing coordinates of a hyperbolic cylinder. Phys. Rev., 66, 9% 1939. See also the treatment of such problems by Schelkunoff, ibid., p. 308. of diffraction by a rectangular slit in a perfectly conducting screen.
471
I N T E G R A T I O N OF T H E W A V E EQUATION
SEC.8.161
dimensional theory is more abstract than the methods described earlicr in this chapter, and consequently has not been applied to the practical problem of calculating antenna radiation. Quite apart from its formal elegance, however, the fourdimensional treatment sometimes leads in the most direct manner to very useful results. This is particularly true of the rather difficult problem of calculat>ing the field of an isolated charge moving in an arbitrary way. The discussion will be confined to the case of charges and currents in f r w space. AH in (82), page 73, the vector and scalar potentials ran be represented by a single fourvector whose rectangular components are (1)
@I
=
A,,
=
fly.
+3
A,,
=
@4
=
i C
4.
Likewise the current and chargc dcnsitics are represented by a fourvector whose romponents are (2)
J1 =
J,,
Jz
=
J,,
JX
J,,
=
Jq
= icp.
The fourpotential satisfies thr relations (3)
According to the notation of ( 3 5 ) , page 64, is a symbolic fourvector whose components are d / a m T h c x n ( 3 ) can k)e c.xyresscd concisely as (4)

nzQ,=poJ,
Q,
=
0.
Let V be a region of a fourdimcnsional spacc hounded by a threedimensional “surface” S. n is the unit outward normal to S. Then in four as in three dimensions
(5)
If u and w are two scalar functions of the four coordinates zl, xz, za, z4 which together with their first and second derivatives are continuous throughout V and on S , then (6)

Jv 17 ( u n w ) dv
=
(7) Thus, one obtains a fourdimensional analogue of Grecn’s theorem,
472
[CHAP.V I I I
RADIATlON
Let xi be the coordinates of a fixed point of observation within S, and xi those of any variable point in V or on 8. The distance from xi to xi is (9)
It can be verified by direct differentiation that the equation (10)
02w
=
0
is satisfied by
w = 1 R'
(11)
at all points exclusive of the singularity R = 0, which will be excised in the usual fashion by a sphere S1 of radius R1. If u is identified with any component @j of the fourpotential, there follows
Over SI we have
To determine the area of the hyperspherical surface S1 polar coordinates are introduced, 21 =
(14)
R cos
x2 = R sin 81 cos 02, 5 4 = R sin O1 sin O2 sin 4,
$1,
x3 = R sin O1 sin O2
COB
4,
which satisfy
x;
(15)
+ x; + xi + xi = R2.
The scale factors hi are calculated as in (70), page 48, and are found to be
(16) hl
=
1,
h2 = R,
h~ = R sin
$1,
hd = R sin
el sin 02.
The element of volume is
(17)
dv
=
hlh2h3h4dR do1 do2 d 4
=
R3 sin2 el sin O 2 d R do1 d o z d+;
hence, the area of the hypersphere is
(18) Therefore,
R3
izT
sin2 O1 sin B2 do1 do, d+
=
2dR3.
473
F I E L D OF A MOVING P O I N T C l I A l i G B
SEC.8.171
Upon replacing by  p o J , and recombining the components a fourvector, one obtains for @ a t the point x: the expression
into
To find the field vcctors oiic must compute the components of the tensor 2F a t the point 2;. By (79), page 72, ( j , k = 1, 2, 3, 4).
Let us suppose for the moment, that all sources are contained within V . Then the integral over S contributes nothing in (20) to the value of @ and in this case
where (J x R)?k = J,Rk  . J ~ R 3and ) where R is the radius vector drawn from 2 to 2’. The cross product J x R is a sixvector or antisymmetric tensor whose components are defined as in (62), page 69. In concise form,
8.17. Field of a Moving Point Charge.An isolated charge q movcs with an arbitrary velocity v in frcc space. It will be assumed that the distance from the charge t o thr observer is such that the charge can hc represented by a geometrical point. Thc motion of the charge along its trajectory is specified by cxprcwing its coordinatrs as functions of t , (24)
21 =
fi(t),
x z = f2(t).
2 3 = f3(t),
2 4
= f4(t)
=
id.
Onc must note that here 1 is riot the observer’s time but time as measurcd on the charge. Then
and by (20)
(26) The origin on the tirnc axis can be shifted a t will without loss of generality. I t will he convcnicritl t o assume that the observer’s time
im x,Plane
chargo to the ohservcr. The polcs of the irit'cgrand in (26) are locatcd a t
(28) ::
w
x4
=
ir,
2 4
=
+ir.
x., = ir
Now the data of the problcm specify the jcocoordinat~csof the trajectory for real values t which are less than t', and in the present '* x,=ir instancc less, thcrefore, than zero. Thus the fj(t) arc given only for values of 5 4 on the negativc imaginary axis. Howcver, 102
where vr is the componcnt of chargt; yelocity in the direction of thc radius vector r drawn from chargc to oljscrrer. Thc integral (26) now has thc forin
The contour is traced in the rlockwisc direction ; consequently, the integral in (32) is by Cauchy's theorem equal to  2 ~2. (ifi til
SEC.8.171
F I E L D OF A M O V I N G POINT CHARGE
475
or, in terms of vector and scalar potentials,
These arc the formulas of T i h a r d mid Wiechwt. Thc values of T and v are those associated with y at an instant preceding the observation by the interval r/c. Rather than differentiate thescb expressions to obtain the field vectors, i t is most direct to apply the s a m ’ procedure to ( 2 3 ) . This has been done by Sommerfeld‘ who obtains formulas, the derivation of which by older methods led to great complications. It is convenient to rcsolve the expressions for the field into two parts: a velocaty field which contains no terms involving the acceleration v, and an acceleration field which vanishes as v goes to zero. For the velocity field one finds
(35) where
and ro is a unit vector in the direction of r from charge to observer at the r instant t’  . These formulas can be obtained also by applying the c Lorentz transformation (111),page 79, t o the field of a static charge and noting that (111) refers to the ol3server’s time. The acceleration field is
1 SOMMERFELD, in Riemann\T.c~licr, “ Partiellen Diffcrcntialgleichungcn der 1935. See also, ABRAHAM, “Theorie der matheiriatischen Physik,” p. 786, 8th d., Elektrizitat,” Vol. 11, pp. 74J., 5th ed., 1923, and FRENKEL, “Lehrbuch der Elektrodynamik,” Vol. I, Chap. T’I.
476
RADIATION
[CHAP.VIII
Note first that both velocity and acceleration fields satisfy the condition 
H = 42.'. x E.
(38)
PO
The magnetic vector i s always perpendicular to the radius vector drawn from the eflective position of the charge, by which one means the position a t the r instant t'  . The electric vector of the velocity field is not transverse, C
but in the acceleration field
(39)
E = &H
x ro,
~
.
=
~ r
~. = 0. r
~
The velocity field decreases as I/?, while the acceleration field diminishes only as l / r . At great distances the acceleration Jield predominates; it i s purely transverse and i t alone gives rise to radiation. I n case the velocity of the charge is very much less than that of light, the equations (37) for the field at large distances become approximately
which remind one of (27), page 435, for the radiation field of a dipole. The results of this section have been based on the assumption that the motion and trajectory of thc charge zre known, just as earlier the assumption was made that the currcnt distribution can be specified. In neither case is i t exact t o treat the problem of the motion or distribution apart from t h a t of the radiation. Let us suppose, for example, that a charge is projectled into a known magnetic field. If the radiation is ignored, the mechanical force exerted on the charge is gv x B, from which the motion can be calculated by the methods of classical mechanics. I n case the velocities are large, a correction can also be made for the relativistic change in mass. Now the effect of this force is to accelerate the charge in a direction transverse t o its motion and, consequently, to introduce an energy loss through radiation. The dissipation of energy through radiation is not accounted for by the force qv x B. An additional force, the radiation reaction, which can be compared roughly to friction, must be included. The radiation reaction in turn affects the trajectory, whence it is obvious t h a t the exact solution can be found only by introducing the total effective force from the outseta very much more difficult problem. Fortunately the radiation reaction is in most cases exceedingly small, so that a satisfactory approximation for the motion can be obtained by ignoring i t entirely.
PROBLEMS
477
Problems
1. Show that the r.m.s. field intensity a t large distances from a halfwave linear oscillator is given by the formula
where W is the radiated power in watts, R the distance from the oscillator in meters, and e the angle made by the radius vector R with the axis of the oscillator. Show that the r.m.s. field intensity st large distances from an electric dipole is voIts/meter, provided the wave length is very large relativc to the length of the dipole. 2. Let $ be a solution of the equation
which together with its first and second partial derivatives is continuous within and on a closed contour C in the zyplane, 3.nd let
d(d
+
where r =  xl2 (y’  yI2, denotes the outward normal to C, and H;*)(kr) is a Hankel function. Show that, if the fixed point x‘, y’ lies outside the contour C, then u = 0, while u = $ if it lies within. The theorem holds also when H i ’ ) ( k r ) is replaced by the function N ( k r ) . This is a twodimensional analogue of the Helmholtz formula (5), page 461, derived by Weber, Math. Ann., 1, 136, 1869. 3. Discuss :he analogue of the Kirchhoff formula in two dimensions. A proof based on the Weber formula of Problem 2 has been given by Volterra, Acta Aiath., 18, 161, 1894. (Very interesting work on the propagation of waves i n a twodimensional space has recently been done by M. Iliesz. See the discussion by Baker and Copson, “ Huygens’ Principle,” Cambridge University Press, p. 54, 1939.) 4. Let z’, y’ be any fixed point within a plane twodimensional domain bounded by a closed curve C , and let \L be a solution of
which is continuous and has continuous first derivatives on C and within the enclosed area S. Show that
+
(y‘  y)2 and K O ( i k r ) is a modified Bessel function diswhere r2 = (I’ 2)’ cussed in Problem 10, Chap. VI. The normal n is drawn outward from the contour.
478
[CHAP.VIII
RADIATION
6. Lct t,he ciirrerit distribution throughout a volume V be spcclfied by thc function J(z)ePiwt, where z stands for t,hc three coordinates z, y, z. Show that the electric field intensity due to the distribution can be represented by the integral eihr
&iwt
E(z’, t )
=
__ 4rwe
Jv [ ( J .V)V
+ kzJl ;dv, 
where
2’ stands
for the three coordinates of thc point of observation, k r2
=
+
(z‘  z ) ~ (y’  y)z
+ (z’
= w d p e ,
and
z)~.
6. When the theorem of Problcrn 5 is applied to a perfect conductor one obtains eik r
[(K V)V 4 k2K]  d~ r
where K is the surface current density a t a point z, y, z on the conductor S. Let S now be the surface of a linear conductor. Rssurnc that the curvature of the wire is continuous and that a t all points the cross scction is small in comparison with the radius of curvature and with the wave length. Show that the field of such a linear conductor is given by
where V’ is applied a t the point of observation and the integrals arc extended along t h e contour C of the wire between points SI and s2. 7. Apply the formula of Problem 6 t o obtain the field of a linear oscillator of length 1 and compare with the results of Sec. 8.11. 8. In case the linear circuit of I’rohlern 6 is closed, S I = sz and the integrated term is zero. On the surface of the conductor the tangential component of E is approximately
whence the current I must be of the form
I
=
A,,
ezkms,
k,
=
2 ~ m 9 1
where 1 is the length of the circuit and m is an integer. a closed, oscillating loop is given by
Show
.
a t the field of such
The expression is exact in the limit of vanishing cross section and is approximately correct if the cross section is srnall in comparison with the wavc length and the radius of curvature at, each point of the circuit. Its application to Hertzian oscillators, such as wwe used in the c:trly days of wireless telegraphy, was discussed in an Adanis Prize essay by Mxcdonald in 1902 entitled ‘‘ Elcctric Waves.” 9. A scmiinfinite linear conductor carrying a current of frequency w / 2 ~coincides with the negative zaxis of a coordinate system. Find expressions for the components
479
PROBLEMS
of electric and magnetic field intensity and calculate the total energy crossing a plane transverse to the conductor. 10. Compute the radiation resistance of a linear halfwave oscillator by the e.m.f. method described in Sec. 8.12. 11. An isolatcd, linear, halfwave antenna radiates 50 kw. at a wave length of 15 meters. Plot the pnrallel and perpendicular components of clcctric fioltl intensity along the antenna on the assumption of a harmonic current distribution and a No. 4 wire. 12. The odd and cven functions
occur frequently in the theory of linear oscillators. Let these function are solutions of
Put F = (B), page 375,
q =
cos 0 and show that
G, and show that G satisfies the associntctf Mathieu equation (I  q2)w”
 2 ( a + I jqw‘
+ ( b  c2q2)w
=
0
for the particular case a = 1, b = a2  1, c = a. Although F(’) and Fc0) are periodic in 8 for all values of a,notc t h a t they remain finite a t the poles only for certain characteristic vnlurs. For thc csen function, these 27n I values belong to the discrete set a:) =  T ;for thc odd [unction, they form the 2 set a$’ = mr, where m is an intrgrr. Demonstrate the orthogonality of tlic functions exprcsscd by
+
m=n m z n’
the subscripts referring to characteristic values of a,and show that the functions are normal. Discuss the relation of these functions to thc radiation field of an arbitrary current distrihution on a linear antenna. 13. An electromagnetic source is located a t a point Pi, and another operating at the same frequency is located a t P 2 . Thc intrrvening mcdiurn is isotropic h u t not necessarily homogeneous. All relations betwcen field vectors nrn linear, and the time variation is harmonic. Let the field vectors due to the sourrc at PI be El and HI, t,hose due to the source a t I’2 be EParid Ha. Show that wherever these vectors are continuous and finite, they satisfy thc symmetrical relation V
*
(ELX Hz  Ez X Hi)
= 0.
This result is due to Lorentz and has been developed into a number of reciprocity theorems of fundamental importance for radio communication.
480
[CHAP.VIII
RADIATION
14. On page 434 it was shown t h a t the Hertz vector of a n electric quadrupole is
+ x:,
R 2 = x2 +xi
, pq, be the spherical components where po(&, &, f 3 ) is the density of charge. Let p ~ pe, of the vector pCz). Show that the radiation field of the quadrupole, in the region lcR 1, is given by
>>
,
r
while the radial components vanish as 1/R2. Show that the mean radiation intensity is
in the preceding prob16. Show that the components ps and p g of the vector lem are related to the components pij of the quadrupole tensor by pe
=
p+ =
+ PIZsin 26) + cos 20 (Pl3 cos 4 + sin +I,  pll) + p12 sin e cos 24 + pZ3 cos e cos 4  palcos B sin e.
4 sin 28 ( p l l cos2 + + PZZsin2 4  p33
+, sin 8 sin 24
P23
(p22
Rotate the coordinate system to coincide with the principal axes of the quadrupole tensor. Then p 1 2 = pZ3 = p31 = 0 and the above reduces to
ps = sin 20
(&I
where Qi
=
$(pi1
 Qz
+
pzz
pg
cos 24),
2
~ 3 3 ) ~
=
2 sin e sin 2+ Q 2 ,
&z = t(pzz
pd.
Show now t h a t t h e total quadrupole radiation is 
If the medium about the quadrupole is air,
+
(9: 3Q3
pe
watts.
= l/c2, and
16. A charge  e is located a t each end of a line rotating with constant angular velocity about a perpendicular axis through its center. A charge +2e is fixed at the center. The dipole moment of the configuration is zero. Calculate the components of the quadrupole moment and find the total radiation. (Van Vleck.) 17. Two fmed dipoles are located in a plane, their axes parallel but their moments directed in opposite sense. The dipoles rotate with constant angular velocity about
PROBLEMS
48 1
a parallel axis located halfway between them. Calculate the components of the quadrupole moment and find the total radiation. (Van Vleck.) 18. A positive and negative charge are bound together by quasielastic forces to form a harmonic oscillator. Show that the radiation losses can be accounted for by a frictional force which, however, is proportional to the rate of change of acceleration rather than t o the rate of change of displacement. 19. A positive point charge oscillates with very small amplitude about a fixed negative charge of equal magnitude. Show that the formulas for the field of an oscillating dipole can be obtained directly from Eqs. (35) and (37) on page 475 which apply to an accelerated point charge.
CHAPTER IX
BOUNDARY VALUE PROBLEMS It can now be assumed that the reader is familiar with the principles that govern the generation of electromagnetic waves and the manner in which they are propagated in a limitless region of homogeneous, isotropic space. The “boundlcss region” is, of course, simply an abstraction. I n point of fact the most interesting electromagnetic phenomena are those induced by surfaces of discontinuity or rapid change in the physical properties of the medium. These boundary phenomena are roughly of three types. Suppose that a wave, propagated in one medium, is incident upon a surface of discontinuity marking t h e boundary of another. I n the first case the linear dimensions of the surface measured in wave lengths are very large. A fraction of the incident energy is reJlected a t the surface and t h e remainder transmitted into the second medium. The direction of propagation is in general modified and this bending of the transmitted rays is referred to as rpfraction. The laws that govern the reflection and refraction of electromagnetic waves at surfaces of infinite extent are relatively simple. If, however, any or all the dimensions of the surface of discontinuity are of the order of the wave length, the difficulties of a mathematical discussion are vastly increased. The perturbation of t h e primary field under these circumstances is referred to as dij’hction. I n both cases t h e secondary field of induced charges and polarization is excited by a primary wave of independent origin. Both are inhomogeneous boundaryvalue problems as defined first for t h e static field on page 195. The third case referred t o above is the homogeneous problem. A conducting body is embcdded in a dielectric medium. Charge is displaced from the equilibrium distribution on the surface and then released. The resulting oscillations of charge are accompanied by oscillations in thc surrounding field. This field can in every case be represented as a supcrposition of characteristic wave functions whose form is determined by the configuration of the body and whose relative amplitudes are fixed by the initial conditions. Associated with each characteristic function is a characteristic number that determines thc frequency of that particular oscillation. The oscillations are damped, partly due to the finite conductivity of the body and partly as a rcsult of energy dissipat,ed in radiation. However, the positions of conductor and dielectric relat,ive to the surface of separation can be interchanged. Electromagnetic oscillations 482
SEC 9.11
483
B O U N D A R Y CONDITIONS
then take nlace within a dielectric cavity bounded by conducting walls. If the conductivity is infinitr, tlierc is neither heat loss nor radiation and the oscillations arc undamped a1 all frcquencies. GENERAL THEOREMS
9.1. Boundary Conditions. A convmtional proof of the conditions satisfied a t a boundary by norrnal and tangential components of the field vectors was presented in S ~ T .1.13. In Chaps. I11 arid I V the relation of these boundary conditiom to the discontinuous properties of certain integrals was cstablishcd for the stationary rcgime; on the basis of S c ~ s 8.13 . and 5.14, the same procedure c:in be clxtendzd to the variable field. We shall forego this aIial1 but s l d l give some further attention to thc important casc in which the conductil ity of one of the two media approaches infinity. At the intrrfacc. of two m d i : t thc transition of the tangential component of E and the riorrnal component of D is expresscd by (1)
n x (Ez  El)
= 0,

n (Dz  D,)
=
6,
where by previous convention n is: the unit normal directed from tho medium (1) into ( 2 ) and 6 dr1iott.s the surfacv charge l o avoid confusion with w = 2av. The flow of chcirge across or to the boundary must also satisfy the equation of continuity in caw eithci or both the conductivities are finite and not zero. a6
JI) =  at
Suppose now that the time enters only as a common factor c x p (  i ~ t ) and that apart from the boundary thc t w o media are homogeiieous and isotropic. Then (1) and (2) togither givc
12et11s s r e first uncler what circimstaiiccs thc surface charge can be zero. If 6 = 0, the determinant of (3) must vanish and h n c e U1EZ  U > € I = 0. (4) If 6 is not zero (and this is tEw iisual case), it may be vlirninated from (3) and wc obtain as: a n alternattue OoiLndary condztion o n the normal component of E :
/.~cllkiE,, /J&EI, = 0. (5) If either conductivity is infinite, (5) becomes indcterminatci; but from (3)
484
BOUNDARYVALUE PROBLEMS
and in the limit as
u1+
[CHAP.IX
m
(7) From the field equations, moreover,
so that, if by hypothesis the field intensities are bounded, both Eland H1 approach zero a t all interior points of (1) as u1 + 00. The transition of the tangential component of E is continuous across the boundary and, consequently, in the case of an infinitely conducting medium n x Ez = n x El = 0. (9) Consider now the behavior of the magnetic field a t the boundary. I n general

n (B,  B1) = 0, n x (H,  H1) = I(, (10) where the surface current density K is zero unless the conductivity of medium a t the boundary is infinite. It has just been shown that H1 vanishes if u1 becomes infinite and in this case (11)
n  B 2 = 0,
n x H, = K.
A further useful boundary condition on the magnetic field in the case of perfect conductivity can be derived as follows.' Let the boundary surface S defined by the equation, [ = fl(z, y, z ) = constant, (12) coincide with a coordinate surface in an orthogonal system of curvilinear , tangential component coordinates E, 7, { as in Sec. 1.16. If ul+ ~ i the of E2 and the tangential component of V X H,, which is proportional to it, approach zero. Therefore by (SO), page 49,
The normal coordinate is {. Since u1 is infinite, I11 is zero; hence, since the coefIicients of the unit vectors must vanish independently, we have just outside the conductor d (htH6) = 0.
The quantities h,H, and h:HE are covariant components (page 48) of the vector Hztangent to the surface. The boundary condition is, therefore: the normal derivatives of the covariant components of magnetic jield The author is indebted to E. H. Smith for the proof.
9.11
SEC.
485
B O U N D A R Y CONDITIONS
tangent to the boundary vanish as the conductivity o n one side becomes infinite. Under the head of “boundary conditions” we ~ h s l lconsider also the behavior of the jield at injinity. I n this respect the variable field differs notably from the static. If in the stationary regime all sources are located within a finite distance of the origin, the field intensities vanish a t infinity such that lirn R2E and lirn R2H are bounded as R + m , I..rhere R is the radial distance from the origin. The scalar potential satisfies V2# = 0 and lim R# is bounded as R + a. Every function C$ which is regular a t infinity and which satisfies Laplace’s equation a t all points of space (no sources) is necessarily zero. On the other hand, if variable sources are located within a finite distance of the origin, the field intensities vanish such that lim RE and lim RH are bounded as R ?. a. The scalar potential and rectangular components of the vector potential and the field intensities all satisfy llhe equation,
a t points where the source density is zero. Moreover there exist functions $ satisfying (15) throughout all space and vanishing at infinity which are not everywhere zero. A solution of Laplace’s equation is uniquely determined by the sources of the potential and the condition that it shall be regular a t infinity. These two conditions are not sufficient to determine uniquely the wave function J,, for (15) admits the possibility of convergent as well as divergent waves. This question has been investigated by Sommerfeld’ in connection with the Green’s function of (15) for spaces of infinite extent. To conditions that are analogous t o those of the static problem must be added a “radiation condition.” The problem is formulated as follows: The density g(x, y, z ) of the sourcc distribution is specified, and these sources are assumed to lie entirely within a domain of finite extent. Then is uniquely determined if: ( a ) a t all points exterior to a closed surface S (which can, if necessary, be resolved into a number of separate closed surfaces), $ satisfies
+
vz+ + k2+
(16) (b) CY*
=
9;
+ satisfies homogeneous boundary conditions over S of the type
+ /3 an  = 0, where
01
and 0 are constants or specified functions of
position;
+
( c ) vanishes in such a way t,hat lim R+ is bounded as R dition we shall again refer to as regularity a t infinity;
SOMMERFELD, Jahresber. deut. math, Ver., 21, 326, 1912.
3 00,
a con
486
BOUNDARYVALUE PROBLEMS
(d)
[CHAP.IX
9 satisfies the radiation condition
(17)
R*

which ensures that a t great distances from the source the field represents a divergent traveling wave. It has been tacitly assumed that the time enters explicitly in the factor exp( id).
The significance of (17) may be made clear by applying Green's theorem to (16) within a region bounded internally by X and externally by a surface So. Then, as in Sacs. 8.1 and 8.13,
The first two terms to the right of (18) reprercnt traveling waves diverging from the source. The third term, ho~z'cwr,expresses the sum of all waves traveling inwards from the elements of So and mnst, therefore, vanish as Xu recedes to infinity. If this third term be designated by U , we have
where d B is an clement of solid angle. The second integral in (19) is extended over the finite domain 47r and vanishes as R + rn provided is regular a t infinity as prescribed in (c) above. I n order that U shall vanish, i t is sufficient that
+
If a t grrak distanccs So is replac~cdby a sphere of radius R, (20) is identical with (17).l 9.2. Uniqueness of Solution.Lct V be a region of space bounded intcrnally by the surface S and cxternally by SO. The surface X can bc resolved, if the case demands, into a number of distinct closed surfaces Si as in Fig. 18, page 108. Thcn V is multiply connected (page 226) clnd t'he surfaces Si represent the boundaries of various foreign objects in the field. It will be assumed for t'he moment that the propert,ics of V are isotropic but the parameters p , E, and rn can be arbitrary functions of position. Now let El, HI, and E,, H2 be two solutions of t,he field equations which a t the instant t = 0 are ident'ical a t all points of V . We wish to find t'he minimum numhcr of conditions t o be imposed on the
* An alternative proof
was given in Sec. 8.13, p. 461.
SEC.
U N I Q U E N ESS OF SOL U TION
9.21
487
components of t’he field vectors a t .the boundaries S and Soin order that the two solutions shall remain idontical a t all times t > 0. In virtue of the linearity of the field eqiiations (we exclude ferromagnetic materials) the differmca jicld E = EP  El and H = H2  HI is also a solution. One may atstsimc wit,hout, loss of generality that the sources of the fields lie cntircly outside the region V , since it was shown in the preceding chapter that thti field is uniquely determined whcn the charges and currents are prescribed. Or, if sources appear within V , one must specify that the dist,rihut,ion and rate of working of the electromotive forces are in both cases the same. Then within V , by Poynting’s theorem, page 132, the difference field satisfies
In ordrr that the righthand mcmbclr of (21) shall vanish, it is only necessary that either the tangential components of El and E2,or the tangcntial components of HI and H2 11c identical for all values of t > 0; for thcn either n x E = 0 or n x H = 0 and E x H has no normal component over the boundaries. In that case we have
2 dt
.I
(5 E2 + v 2
5
The righthand member of (22) is always equal to or less than zero. The energy integral on the left is essentially positive or zero and vanishes a t t = 0. Hence (22) can only be satisfied by E = E2  El = 0, H = H2  HI = 0 for all values of t > 0, as was to be shown. A n electromagnetic field i s uniquely determined within a bounded region V at all times t > 0 by the initial valuc:; of electric and magnetic vectors throughout V , and the values of the tangcntial componmt of the electric vector (or of the magnetic vector) over the boundaries for t 2 0. If So recedes to infinity, V is cxtcmally unboundcd. To ensure t h r vanishing of the integral of a Poynting vector over an infinitely remote surface, it is only necessary to assume that the medium has a conductivity, however slight. If the field was initially established in the finite past, the difficulty may also be circumvented by the assumption that Solies beyond the zone reached a t time t b y a field propagated with a finite velocity c. The theorem just proved does not take full account of this finiteness of propagation. We have established that the valurs of E and H are uniquely determincd throughout V a t time t by a tangcntial boundary condition and the initial values everywhere in V . Physically i t is obvious, however, that this is more information than should be necessary. The field is propagated with a finite vclocity and, consequently, only those elenients of V whose disiance from the point of
488
BOUNDARY VALUE PROBLEMS
[CHAP.IX
observation is r ( t  to)c need be taken into account. The classical uniqueness proof given above has been extended in this sense by Rubinowicz.' The theorem applies also to anisotropic bodies. Electric and magnetic energies are then positive definite forms (cf. page 141), which is to say that they are positive or zero for all values of the variables. The proof remains unmodified. There is one aspect of the theorem which at first thought may be puzzling. We have seen that the field is determined by either the values of n x E or n x H on the boundary, yet in the problems to be discussed shortly we find it necessary to apply both boundary conditions,
n x (E,  El) = 0, n x (H,  HI) = 0, (23) where here E, and El denote values of field intensity on either side of the boundary. The reason for the apparent discrepancy is, of course, that n x E and n x H refer to the tangential components of the resultant field on one side of the surface. These are in general unknown and it is the object of a boundaryvalue problem to find them. Equations (23) simply specify the transition of the field across a surface of discontinuity and the two together enable us to continue analytically a given primary field from one region into another. Having thus determined the total field, the uniqueness theorem shows that there is no other possible solution. If, however, one side of a boundary is infinitely conducting, we do know the tangential component of the total field, for in this case n x E = 0, and a singleboundary condition is sufficient for the solution of the problem. 9.3. Electrodynamic Similitude.Since Newton's time the principle of similitude and the theory of models have had a most important influence on the development of applied mechanics. This is particularly true of ship and airplane design which is governed very largely by the data obtained from small models in towing tanks and wind tunnels. It is customary to express the conditions of an experiment in terms of certain dimensionless quantities such as the Reynolds number. Thus the results of a single measurement of a given model can be applied to a series of objects of identical form and differing only in scale, provided the viscosity of the fluid and the velocity of flow are varied in such a way as to keep the Reynolds number constant. Similar principles prove very helpful in the design of electromagnetic apparatus. We shall write first the field equations in a dimensionless form. In a homogeneous, isotropic conductor
l
R
~
~Physik. ~ Z., ~ 27, ~ 707, v 1926. ~ ~
~
,
SEC.9.31
ELECTRODYNAMIC SIMILITUDE
489
Now let
E (25)
E
=
H
= eE, I.c =
EoKe,
length
=
=
hH, U = UoS,
/JoKm1
time
ZOL,
=
toT,
where E, H, K,, K,, s, L, and T are the dimensionless measure numbers of the field variables in a system for which the unit quantities arc e, h, The measure numbers satisfy the equations €0, PO, lo, uo, and to.
V X E+
L
Y
l3E
V X H  PK, aT
f3H K

~= ~
0,
~ s= E 0,
where
are dimensionless constants. Upon eliminating the common ratio e / h , one obtains
From the first of these it follows a t once that the product poco must have the dimensions of an inverse velocity squared. This is, no doubt, the most fundamental approach to the problem of units and dimcnsions.' I n order that two electromagnetic boundaryvalue problems be similar, it is necessary and sufficient that the coefficients LYK,,,, OK,, and ys be identical in both. For t o let, us take for example the period 'T of the field, and for lo any length that characterizes one of a family of bodies which differ only in scale. Thus lo may be the radius of a set of concentric spheres, or the major axis of a set of ellipsoids. The condition of similitude requires th at the two characteristic parameters C1 and 6 2 in
be invariant to a change of scale. Suppose that the characteristic length lo is halved. Then both C1 and Czremain unchanged if the permeability This is an awkward remedy I.( a t every point of the field is quadrupled. from a practical standpoint, but it is the only way in which the initial state can be simulated through the adjustment of a single parameter. If p and e are left as they were, constancy of C1 results also from halving the period, or doubling the frequency, but this alone does not take care of C,. I n order th at the halfscale model shall exactly reproduce the 1
C j . Secs. 1.8.4.8,and 4.9.
490
BOUNDARYVALUE PROBLEMS
[ C H A PI. X
iullscale conditions, i t is necessary also that the conductivity be doubled a t every point. This principle can be illustratcd by rcfercnce to the type of highfrequency radio generator that maintains standing electromagnetic waves in a cavity resonator bounded by metal walls. The frcquency of oscillation is determined essentially by the dimensions of the cavity, while the losses depend largely on the conductivity of the walls. If the dirnensions are halved, the frcqncncy will approximatcly be doubled; if the conductivity of thc walls is unchanged, it is entirely possible that the resulting increase of the rplative loss will pass the critical value, so that the halfscale apparatus fails to oscillate. REFLECTION AND REFRACTION AT A PLANE SURFACE
9.4. Snell's Laws.Two homogeneous, isotropic media have as a common boundary the plane 8, and are ot,herwisc infinite in cxtent. The unit vector n is normal t o the plane X and directed from the region
FIG.Sg.Reflertion
and refraction a t a plane surface S.
(€1, 11, a,) into thc region (€2, 12, u2). Let 0 be a fixed origin, which for convenience we locate on S. Then, if r is the position vector drawn from 0 to any point in either (1) or (2), the interfacc Xis defined by thc equation (1) n . r = 0. A plane wave, traveling in medium (2) is incident upon 8. By (27), page 272,
where EOis the complex amplitude of the incident wave and no a unit vector which fixes its direction of propagation,' as in Fig. 89. The plane defined by the pair of vectors n and n, is called the plane of incidence. 1
It is apparciit that so far :is tlic present problem is concerned it would be neater
49 1
SNELL’S LAWS
SEC.9.41
The continuation of the primary field into medium (1) is determined b y the boundary conditions a t S. To satisfy these boundary conditions, a reflected or secondary field must bc postulated in ( 2 ) . Physically i t is clear that the primary field induces an oscillatory motion of free and bound charge in the neighborhood of S , which in turn radiates a secondary field back into ( 2 ) as well as forward into (1). We shall make the tcritative assumption that both transniitted arid reflected waves are plane, and write
where i.he unit vectors nl and n, are in the directions of propagation of transmitted and reflect’ed waves respectively, and El arid E, are complex amplitudes, all as yet undetermined. By hypot’hesis, El, E,, and Eo are independent of the coordinates and, conscquent,ly, if the tangential components of the resultant field vect,ors are to he continuous across S , it is necessary that the arguments of t,hc exponential factors in (2) and (3) be identical over the surface n r = 0. But
.
r = ( n  r ) n  n x (n (4) hence, a t any point on the interface r = n
x r); x (n x r). Therefore, kznu* n x (n x r) = k2n2 n x (n x r), k2no n x (n x r) = klnl. n x (n x r),

(5)

x n) (n X r), (n, x n  nz x n) . n x r (kznox n  klnl x n) n x r
or, since no n X (n X r) = (n,
(6)

= =
0, 0.
From these two relations it follows that n, no, n,, and n2 are all coplanar. The planes of constant pkasc of both transmitted and rejlectcd waves are normal to th,e planc of in.cirlcncc. From t,hc first of (6) also
(7)
sin
02
=
sin
(T 
6,;) = sin Oo,
whence the a,ngle of inciclemc B0 is equal to the angle of reJEection From thc second of (6)
(8)
k 2 sin 60 =
Equations (7) and (8) expr
k 1
e2.
sin 81.
h d 1 ’ s ~niusof reflection and refraction.
__ from a notational point of view to let t,he incident wave travel from (1) into (2). Shortly, however, the plane S will bc rcplnced by :I closed surf:ioe S bounding a cornplete body. By previous convention n is tlircct.c:d outwnrd frnin a closcd surface and from medium (1) into ( 2 ) . The choice of n o :is above will facilitate the comparison of formulas from the present with thosc o f latcr sections.
492
[CHAP.IX
BOUNDARY 'VALU E PROBLEMS
9.6 FresneI's Equations.The boundary conditions will now be applied to determine the relation between the amplitudes Eo, El and E2. At all points on 8
+
+
n x (Ho HJ = n x HI. n x (Eo E2) = n X El, (9) By virtue of (2) and (3) the second of these two can be expressed in terms of the electric vectors. (10)
n x (no x Eo
+ nz X E2) kz
=
k n x (nl x El)2. Ccl
P2
Expansion of (10) leads to such terms as

.
n x (no X Eo) = (n Eo)no (n no)Ec. (11) The orientation of the primary vector Eois quite arbitrary but can always be resolved into a component normal to the plane of incidence and consequently tangent to S, and a second component lying in the plane of incidence. (Cf. Sec. 5.4, page 279.) The analysis is greatly simplified by a separate treatment of these two components of the incident wave. Case I. Eo Normal to the Plane of Incidence.Then


n Eo = no Eo = 0. Since the media are isotropic, the induced electric vectors of the transmitted and reflected waves must be parallel to Eo and hence also normal to the plane of incidence, so that n El = n Ez = 0. From Fig. 89 n no = cos (T  Oo) =  cos eo, (12) n nl = cos (T  0,) =  cos el, n n2 = cos O2. Upon multiplying the first of Eqs. (9) vectorially by n and making use of (11) and (12), we find that the amplitudes must satisfy


.
The relative directions of electric and magnetic vectors for this case are shown in Fig. 90. Solving (13) for El and E2 in terms of the primary amplitude Eo leads to El
=
Ez
=
(14)
(COS
e2 + cos e,)
P l ~ zcos
+
P1kz cos
e2 +
e2 cos el EO, (when n Eo = 0 ) , Plkzcos eo  P2kl cos el

P 2 ~ COS l el Eo.
These relations are not quite so simple as they appear a t first sight, for 81 is complex if either (1) or (2) is conducting and may be complex even if both media are dielectric. By (7) and (8)
SEC.
(15)
4 93
FRESNEL'S EQUATIONS
9.51
kl cos Ol = .\/k:
cos B2 = cos Bo,
 k; sin2 Bo.
The angles of reflection and refraction can be eliminated from (14) and we obtain as a n alternative form the relations
El
=
2plk2 cos eo

p l k z cos
00
+
p2
 Eo,
d k i  k i sin' Bo

(when n Eo = 0),
(16)
E2 =
p l k 2 cos Bo  pz
plkz cos Bo
mk i sin2 Bo Eo. __ 
+ p 2 d k 2 ,  k ; sin2 Bo
Complex values of the coefficients of Eo imply that the amplitudes El and E2 themselves are complcx and that the transmitted and reflected waves differ in phase from tho incident wave. /S
FIG.90.Polarization
normal to the plane of incidence.
Case 11. Eo in the Plane of Incidence.The magnetic vectors arc then normal to the plane of incidence and parallel to S.

.
.
n Ho == n HI = n H,. From (2) and (3) we have
which when substituted into (9) give cos eo H~  c~~ e2 H,
=
+ Hz
=
Ho
PlkZ ~
P'kl
HI,
cos
el H ~ ,
494
[CHAP.IX
BOUNDARYVALUE PIZORLEMS
as the conditions at the boundary. The relative directions of electric and magnetic vectors for this case are shown in Fig. '31. Solution of (18) leads to

(when n H,
or, upon eliminat,ion of 81 and
(20)
82
= 0),
by (15),

(whcn n Ho = 0), fiClpkT cos 0,,  p l k z d k :  h$ sill2 B0 H 2 =  ____ ~ ~ Ho. p & cos Oo f p l k 2 2/:kg sin2 B0 
When thc iricidcnce is normal, $0 = 0, the two caws cannot be distinguished and thc amplitudes of t,ransmitted and rcflected waves reduce to
The relations expresscd by Eqs. (14) and (19) werc first derived in a slightly less general form by Fresricl in 1823 from the dynamical properties of a hypothetical elastic ether. /S
FIG.Sl.Polariaation
parallel t o the planc of incidence.
9.6. Dielectric Media.We sh:tll study first the case in which the two conductivities u1 arid u2 arc zcro, so that both media are perfectly transparent. The permeabilities will differ by a negligible amount from po and Snell's law can now be written
Snc. 9.61
495
DIELECTRIC MEDIA
whcre v 1 and 212 are the phase velocities and n12is the relative index of refraction of the two media. If t l > € 2 it follows that n12 < 1 and there corresponds to every angle of incidence 8 0 a real angle of refraction 81. If, however, €1 < cz, as is the case when a wave emerges from a liquid or solid dielectric into air, then 81 is real only in the range for which nl2 sin eo I 1. The phenomena of total reflection occur when n12sin 0 0 > 1. We shall cxcludc this possibility for the moment and considcr the Fresnel laws for real angles between the limits 0 and a/2. When E is normal to the plane of incidence we now obtain from (14)
El
2 cos 0” sin =
7
Slll
(23)
(0,
el
+ Bo)
EO,
(n.E
=
0),
sin (01  8,) E 2 = sin . (0, e,) EO,
+
and for the components of E lying in the plane of iiicidence from (19)
nl x E l
= __
sin (0,
2 cos 0, sin 0, el) cos (0, 
+
el) no x
Eo,

(n Ho = 0),
(24)
Since the coefficients of EOin (23) and (24) are real, thc reflected and transmitted waves are either in phahe with the incident wave, or out of phase by 180 dcg. It is apparent that the phase of the transmit,ted wave is in both casos idcntical with that of the incident wave. The phase of thc reflected wave, however, will depend on the relative magnitudes of 8 0 and 61. Thus if €1 > c2, thcn O1 < eo, so that ELis opposed in direction to E” in (23) and therefore differs from i t in phase by 180 dcg. Under the same circumstances tan ( 0 0  0,) is positive, but the dcriomiriator tan (8, 01) becomes negative if Oo el > a/2 and the phase shifts accordingly. The mean energy flow is g i w n by the real part of the complex Poynting vector. I n optics this quantity is usually rclfcrrcd to as the “intensity” of light, but the term is ambiguous since i t is also applied to the amplitudcs of the fields. I n the present case
+
+
496
[CHAP.IX
B O U N D A R Y  V A L U E PROBLEMS
Now the primary encrgy which is incident per second on a unit area of the dielectric interface is not but the normal component of this flow
si
6
vector, or n Si = ~ E,2 cos 80. Likewise the energies leaving a 2 unit area of the boundary by reflection and transmission are
(26)
.
n S,
=
4% 
2
E2, cos
n St =
80,
* 2
E,2 cos
el.
According to the energy principle the normal component of energy flow across the interface must be continuous,
n
(27)
 + 3,) (Si
=
n S,,
or
(28)
6E,2 cos
80
=
6Et cos 81 + G
E ; cos
80.
It can be easily verified that the Fresnel formulas (23) and (24) satisfy this condition. The reJlection and transmission coeficients are defined by the ratios
R+T=l. I n case
E, is normal t o the planc of incidence, these coefficients are sin2 ( 8 1  80) sin2 (81 ooj'
RI= .
+
sin sin 281 sin2 (81 8,) '
TI= .
+
and in the case of En lying in the plane of incidence
If the incidence is normal, 80 (22) that
= 81 = 0,
and i t follows from (21) and
R = (32)
T =


There is only one condition under which a reflection coefficient is zero. As 8 0 81 a/2, the t a n ( 8 0 81) co and in this case 211+ 0. The reflected and transmitted rays are then normal to one another
+
(nl* nz = 0) and sin 81 = sin
+
6
 00) = cos 0"; it follows from (22)
SEC.9.71
TOTAL REFLECTION
that tan
(33)
eo =
497
.$
= n21.
The angle that satisfies (33) is known as the polarizing, or Brewster angle. A wave incident upon a plane surface can be resolved, as we have seen, into two components, one polarized in a direction normal and the other parallel to the plane of incidence. The reflection coefficients of the two types differ and, consequently, the polarization of the reflected wave depends upon the angle of incidence. I n particular, if incidence occurs a t the polarizing angle, the reflected wave is polarized entirely in the direction normal to the plane of incidence. Use is sometimes made of this principle in optics to polarize natural light, although in practice it is less efficient than methods based on doublerefracting prisms. The optical indices relative to air are usually of the order of n21= 1.5; a t radio frequencies they may be very much larger with a corresponding increases increase in the polarizing angle. Thus in the case of water, nZ1 from about 1.33 to 9 a t radio frequencies and the polarizing angle from 53 to 84.6 deg., which is not far removed from grazing incidence. Irregularities of radio transmission over water can doubtless be attributed 011 certain occasions to this cause. One will note also that at the polarizing angle Ti1 is unity and th at this is the only condition under which all the energy of the primary wave can enter the second medium without loss by reflection a t the surface. 9.7. Total Reflection.We return now t o the case excluded from Sec. 9.6 of transmission from medium (2) into a medium (1) whose index of refraction is less than that of (2). The formulas of See. 9.6 are valid whatever the relative values of el and e2, but if eo is such that
n12sin
eo =
.$sin eo > 1,
they can be satisfied only by complex values of el. Physically a complex angle of refraction implies a shift of phase and the appearance of a n attenuation factor. Let us suppose, then, that sin 81 > 1. The cosine is a pure imaginary.
(34)
cos
_____
i
01
= __ 6
deZsin2 eo  €1 = inlzl/sin2 e0  nil.
1
The radical has two roots whose choice will be governed always b y the condition that the field shall never become infinite. T o simplify matters a bit, the reflecting surface S will be made to coincide with the plane 2 = 0, as in Fig. 92. All points of medium (1) correspond to negative values of 2. The phase of the transmitted wave is, therefore,
[CHAP.IX
BO U N D A K Y V AL Uh’ PROBLEMS
498

klnl r
(35)
= w
&I*,
=
6(iz
W
(z
where it is assumed that p l transmitted wave is
Et
(36)
cos fI1 ~4z sin 6,) ______ 6
1
2
= pz = po, =
e0  ng,
+ z sin
BO),
and the field intensity of the
El e / 3 i z + < a z  i w t
(x < O),
where
The field defined by (36) vanishes exponentially as x 3 confirms the choice of the positive root in (34).
which
I
Medium (1)
FIQ.92.Total
m,
reflection from a surface coinciding with the yzplane.
The amplitudes of reflected and transmitted fields are next determined from the Fresnel laws after elimination of el. From (14) we obtain
2 cos
=
cos
60
__
eo + i \/sin2 eo  nzl
EOL,
and from (19) and (3), for polarization in t,he plane of incidence,
Since the coefficients of E, are complsx, i t is apparent that the transmitted and reflected waves are no longcr in phase a t the surface with the incident wave. Thc reflection coefficient according to (29) is R = Ez @E:, and i t follows a t once from (38) and (39) that

(40)
RL
=
Rli
=
1,
TI
=
T I I= 0.
The intensity of energy flow in the reflected wave is exactly equal t o the
SEC. 9.71
TOTAL REFLECTION
499
intensity in the incident wave; there i s no average flow into the m e d i u m o j lesser refractive index. The field: intensity in medium (I), however, is by no means zero. There is, in f:Lct, an instantaneous normal component of energy flow across the surface whose time average is zero; the time average of the flow wit,hin (1) parallel to the surface (i.~.,in the direction of z) does not vanish. This lattcr component is unattenuated in the direction of propagation but falls off very rapidly as the distance from X increases due to the factor exp pix. The surfaces of constant phase in the transmitted wave are the planes z = constant, which are normal to surfaces of constant amplitude, x = constant. It is clear that Etll and Htll have components along the zaxis, which is the direction of propagation within (1). The excitation within mcdium (1) in the case of total rejlection i s a nontransverse wave (it m a y be either transverse electric or transverse magnetic, page 350) confined to the immediate neighborhood of the surface. This analysis gives no clue as to how the energy initially entered (l), for i t is based on assumptions of a steady state and of surfaces and wave fronts of infinite extent. Actua'lly, thc incident wave is boundcd in both time and space. The total reflection of a beam of light of finite cross section has been treated by Picht' who showed that the average flow normal to the surface is in this case not strictly zero. Any change causing a fluctuation in the energy flow of the transmitted wave destroys the totality of reflect.ion. Figures showing the course of the magnetic lines of force and t h e lines of energy flow in total reflection have been published by Eichenwald and by Schaefer and Gross2 Finally, let us examine the relative phases of the reflected waves. I n (38) and (39) write E,II = eiaI! E 011, Ezl = e"'l Eel, (41) [ciSd.J = lei6ll/ = 1. Then since
we have
Suppose that the incident wave is linearly polarized in a direction that is neither parallel nor normal to the plane of incidence. W7e then resolve i t into components and discover that the resultant reflected wave is PICHT, Ann. Physik, 3, 433, 1929. See also NOETHER, ibid., 11, 141, 1931. SCHAEFER a n d GROSS, Untersuchungen uber die Totalreflexion, Ann. Physik, 32, 648,1910. 2
BOU N D A R Y V A L U E PROBLEMS
500
[CHAP.IX
formed by the superposition of two harmonic oscillations a t right angles to one another and differing in phase by an amount 6 = 611  61. (44) T h e rejlected wave is elliptically polarized, since by (43) the relative phase difference 6 i s not, in general, zero. 6 tan 2
6 II  tan 61 tan 2 2 _ _ ~ 6 61 1 tan 1tan 2 2
= _

+
cos eol/sin2
eo  nZl.
sin2 O0
The phase shifts 6 1 and 611 both vanish a t the polarizing angle Oo = sin' n21, and their difference 6 is also zero a t grazing incidence, Bo = ~ / 2 .The relative phase 6 attains a maximum between these limits a t an angle found by differentiating (45) with respect to Oo and equating to zero. The maximum occurs when
which, when substituted into (45), gives
(47) This property of the totally reflected wave was used by Fresnel to produce circularly polarized light. It is first necessary that the incident wave be linearly polarized in a direction making an angle of 45 deg. with the normal to the plane of incidence. The amplitudes E z l and Ezll are then equal in magnitude. The relative index n21 and the angle of incidence Oo are next adjusted so that 6 = n/2, or tan 6/2 = 1. According to (47) this condition can be satisfied only if 1  ngl 2 2n21, or nzl < 0.414, or n12 > 2.41. I n the visible spectrum such a minimum value of the index of refraction is larger than occurs in any common transparent substance. To overcome this difficulty, Fresnel caused the ray of light to be totally reflected twice between the inner surfaces of a glass parallelepiped of proper angle. I n the radio spectrum, on the other hand, the index of refraction may assume very much larger values. Thus, in the case of a surface formed by water and air, n12= 9, nzl = 0.11. The condition tan 6/2 = 1 is then satisfied by either of the angles 00 = 6.5 deg. or 00 = 44.6 deg. The latter figure has been confirmed by measurements at a wave length of 250 cm.1 9.8. Refraction in a Conducting Medium.The phenomena of reflection and refraction are modified to a striking degree by the presence of a BERGMANN, Die Erzeugung zirkular polarisierter elektrischer Wellen durch einmalige Totalreflexion, Physik. Z., 33, 582, 1932.
501
R E F R A C T I O N I N A CONDUCTING M E D I U M
SEC. 9.81
conductivity in either medium. The laws of Snell and Fresnel are still valid in a purely formal way but, as in the case of total reflection, the complex range of the angle 81 leads to a very different physical interpretation. Let us suppose that medium ( 2 ) is still a perfect dielectric but that the refracting medium (1) is now conducting. The propagation constants are defined b y
where a 1 and 8, are cxpreswd in terms of page 276. By Snell’s law we have
61,
p ~ and ,
UI
by (48) and (49),
(49) which it is convenient to writc as sin
(50)
01
=
( a  i b ) sin 80.
The complex cosine is then _____
cos 81 = 4 1  (u2  b2  2abi) sin2 B0
(51)
= peir.
The magnitude p and phase y are found by squaring (51) and equating real and imaginary parts on either side. p2
(52)
p2
cos 2y = p2(2 cos2 y  1) = 1  (a2  b2) sin2 Bo, sin 2y = 2p2 sin y cos y = 2ab sin2 BO.
The phase of the refracted wave is, as in (35) and Fig. 92,
+ i@l)( 8, + z sin 8,) = cos y  p1 sin y)  i z p ( p 1 cos y + sin y) + ~ ( c t a+l sin B0 + iz(ap1 bal) sin BO. From (49) and (50) it is readily seen that (aal + bP1) sin O0 = a2 sin o,, (53)
klnl.r
= (a1
J
CQS
zp(aI
a1
bP1)

and (up1  bad = 0. Within the conducting medium the transmitted wave is represented, therefore, by (54)
Et = El e p r + i (  g z + n z z s i n B o  ~ o t )
where (55)
p q
= p(P1 = p(a1
7
(x < 01,
+
cos y a1 sin y) cos y  P 1 sin y).
Note that the surfaces of constant ampZitude are the planes p z = constant, the surfaces of constant phase are the planes gz a2 s i n t10 z = constant, and these two families do not, in general, coincide. Within (1) the field is represented by a system of inhomogeneous plane waves, as in the case of
+
502
[CHAP.IX
BOUNDARYVALUE PROBLEMS
total reflection. The planes of const)ant amplitude are parallel to the reflecting surface S; the direction of propagation is determined by the normal to thr plancs of const,ant phase. Thc angle II, made by this wave normal with the normal to the boundary plane (in the present instance the negative xaxis) is the true angle of refraction and is defined by (56) or (57)
cos +b =
x cos J/
+ x sin # = constant,
Q
sin +b
~
fl+CY;Sin2e,’
=
CQ
sin Bo ayit sin2
+
eo
The relation of the planes of constant amplitude to the planes of constant phase is illustrated in Fig. 93. I X
Medium
////,,? /
Medium (1)
FIG.93.Refraction
a t a plane, conducting surface.
A modified Sncll’s law for real angles is expressed by (57).
The quantity n(&)is a real index of refraction which now depends on the angle of incidencc, a notable deviation from the law of refraction in nonabsorbing media. The phase vclocity, defined as the velocity of propagation of the planes of constant phase, is (59) Not only does this velocity depend on the angle of incidence, but as in the case of total reflection there are also components of field intensity in the direction of propagation. The field within the conductor is not strictly transverse.
SEC.
9.81
REFRACTION I N A CONDUCTING
nmnirAu
503
The calculation of p , q, and n in terms of the constants of the media and the angle of incidence is a tedious but elementary task. From ( 5 2 ) and (55), together with the definition of a and b implicit in (49), one obtains the following rclations :
p2(eo) = +[a;
(60)
+ + a; sin2 Oo + 4 4 0 3 3 + (a!  ~9
 a; sin2
00)2],
P:
 a; sin2
@I,
+ a; sin’ + f i a $ j _+_ _ _~p:
 a; sin2
0,)2].
qz(Oo) = +[a; /3;  a; sin2 Bo
+ 64zE+cLy’;
a;nZ(Bo)= +[a; 
p:
00
(01;
From these a subsidiary set of relations, known as Kettclcr’s equations, can easily be derived.
The junctional relation of n(B0) to thc angle of incidence and the constants a1 and 01 expressed by (60) has hecn confirmed by mcasurements in the visible region of the light spectrum.’ No direct connection exists, however, bctwecn the observed values of a1 and 81 at optical frequencies and the static or quasistatic values of the parameters €1, p1, and ul. I n fact a1 and can assume values a t optical frequencies which at radio frequencics are possible only in densely ionized media. Thus Shea found for copper aI/az = 0.48, a value less than u n i t y , and P l / a z = 2.61, instead of the exceedingly large value one might anticipatc for such a good conductor. I n this case the apparent phase velocity within the metal is greater than that of light. The anomalous behavior of these parameters a t optical frequencies gives rise to some very interesting phenomena in the domain of metal optics, which lie beyond the limits we have imposed upon the prcsent t heory.2 Although there appear to be no experimental data available in support of Eqs. (60) a t radio frequencies, there is every reason t o believe them exact. We shall discuss only the case in which the conduction current in the medium is very much greater than the displacement current. Let 7 = u / w e . Then the assumption is that q:>> 1, and under these circumstances it will be recalled (page 277) t h a t ~SHEA U’ied. , Ann., 47, 271, 1892; W~LSEY, Phys. Rev., 8, 391, 1916. 2 An excellent account of the optical problems of reflection and refraction is given by Konig in his chapter on the electrornagnetic theory of light in the “Handbuch der Physik,” Vol. XX, pp. 197253, Sprlngcr, 1928.
504
BOUNDARYVALUE PROBLEMS
[CHAP.
JX
but: since 7; is assumed to be very much larger than the maximum value of sin2 B0, we obtain the approximate formulas
hence, n 4
w
as a 1 4
00
or o + 0. ,4t the same time
and $ 4 0 . As the conductivity increases or the frequency decreases, the planes of constant phase align themselves parallel to the planes of constant amplitude and the propagation i s into the conductor in a direction normal to the surface. In the case of copper, u1 = 5.82 X lo' mhos/meter, and it is obvious that $ differs from zero by an imperceptible amount. Whatever the angle of incidence, the transmitted wave travels in the direction of the normal. The factor
measures the depth of penetration. It is characteristic of all skineffect phenomena and gives the distance within the conductor of a point at which the amplitude of the electric vector is equal to l / e = 0.3679 of its value at the surface. In this distance the phase lags 180 deg. Since the value of 6 measures the effectiveness of a material for shielding purposes, it is interesting to know its order of magnitude. The table below gives the values of 6 in the case of copper for several frequencies. They are obtained from the approximate formula 6 = 6 . 6 ~ cm. 4 Y
cycles/sec. 60 103 106
6, cm. 0.85 0.21 0.007
The depth of penetration is decreased by an increase in permeability, but this is usually offset by the poor conductivity of many highly permeable magnetic materials.
SEC.9.91
REFLECTION AT A CONDUCTING SURFACE
505
The angle 1c. may be approximately zero for materials of much lower conductivity than the metals. In the case of sea water, u1
= 3 mhos/meter,
el =
81co,
and we find
At v = lo6 we see that n = 164, J. < 0.35 deg., and 6 = 29 cm. The approximation is just valid a t v = los, for then vf = 45 >> 1, and in this case n = 16.4, J. < 3.5 deg., 6 = 2.9 cm. 9.9. Reflection at a Conducting Surface.We shall examine next the phase and amplitude of the wave reflected a t the plane interface of a dielectric and a conductor. By Snell’s law
kl cos el
(68)
=
l/lcq  k%sin2 oO,
and from (60) and (61) it follows that pp =
(69)
WP1,
q2  p 2 = a:  p:  a; sin2 O0, q2 p 2 = [4a4p4 (a?  p:  a: sin2 &)2]+.
+
+
It can easily be verified from these relations that
where
Upon substituting (70) into the Fresnel equation (14) for the reflected component of the electric vector polarized normal to the plane of incidence, one obtains (72)
Ezl =
pla2
cos
60
pla2
cos
00
 pa 4 
+
pLp
ia
eT
4 EOl.
dFW e
The fraction must be rationalized to give amplitude and phase. (73) Ezi = p l e  i 6 J  E,,l,
Then
and after a relatively simple calculation we find
(74) The other component of polarization is found in the same way but the computation is considerably more tedious. According to (20),
506
[CHAP.IX
BOUNDARYVALUE PROBLEMS
the second Fresnel equation is pz
(75) H ~ L =
cos
00
+
(a? _ _ P?~ 
v'FW + p l a 2 d=i
2a1~1i)
+
/*la2
c e2
Hal.
t!D
p2 cos 00 (a:  py 2a1pli) e3 The lengthy process of rationalizing this expression may be skipped over and the result set down at once. We abbreviate n2 x E ~ I=I pile26iI no x EGII, (76) and find [ 2 p 2 a l ~cos 1 eo  pla2p12 = [ p 2 ( a ?  PT) cos eo  P1a2q12 [pLz(a?  E ) cos 8 0 p1azq12 [2p2a1P1 cos 8 0 p1azp1"
+
+
+
+
If the conductor is nonmagnetic, so that p 1 = p2, the expression for the amplitude pi1 factors into the form deduced by Pfeifferl in the course of his optical studies :  (q  a2 cos &J2 p 2 (q  a2 sin 0 0 t a n Bo)2 p2 _____ (78) 1'1  ( q a 2 cos eo)2 p2 ( q a2 sin o0 t a n o 0 ) 2 p f As in the case of total reflection, the two components of polarization are reflected from an absorbing surfacc according to different laws. Consequently a n inciderit uiaw which zs linearly polarized, but whose direction of polarization i s neither normal nor parallel to the plane of incidence, will be rejected with elliptic polarization. The polarization is determined by the ratio
+ +
+
+ +
+
(79) Only in the case of nonmagnetic materials do the expressions for p and 6 reduce to a relatively simple form. If p 1 = p 2 ) one finds after another laborious calculation that p2
O,, t,an B o ) 2 + p 2 + a2 sin eo tan eo)2 + ~  _2a2 _ 11_sin Oo  _t.an_ 80_ ~ . a: sin2 Bo tan2 Oo  (q2 + p ' )
 a2 sin = (q ___
~
(q
(80)
tan 6
=
p2'
The reflection coefficients are again defined as the ratio of the energy flows in the incident and reflected waves. Thus (81) RL = p i 2 , I n the case of normal incidence R , (82)
a:= k (pza1

+
RII =
PII'.
= ail =
a,
Fla2)2
Pla2)2
+PX. + P X
PFEIFFER,Diss. Giessen, 1912; K ~ N I GZoc. , cit., p. 242. Cf. also WILSEY, zoc. cit., p. 393.
SEC.9.91
REFLECTION A T A CONDUCTING SURFACE
507
The degree of polarization is commonly measured by the ratio
where g2 = E2,II/E:l, the ratio of the incident intensities. When one considers that the quantities q and p are functions of the angle of incidence as well as of a11 the parameters of the medium [Eqs. (60)], the complexity of what appeared a t first to be the simplest of problemsthe reflection of a plane wave from a plane, absorbing surfaceis truly amazing. The formulas nre far too involved to be understood from a casual inspection, and the nature of the reflection phenomena becomes apparcnt only whcn one treats certain limiting cases. Of these, one of the most important for electrical theory is that in which Case
I.
$ >> 1. *
7;"=
lhen
€1
as was shown on page 504, and
(I 
2
cos 0")?
p 1 2 'u 
( + p1ff9 cos 1
(85)
: pza1
0")'
+1 +1
Since p1/p2 = pJp0 = ~,,,l i s the magnetic permeability of the conductor and e z / e o = ~~1 is the specific inductive capacity of the dielectric, we may write
(86)
5 =
E! =
.Pzff 1
Egi
=
2.11 X 104
where v is the frequency. I n the case of all metallic conductors j2(,)1,
 2n T =  e1A2a3X 0.054. 3
HANSEN,J . Applied l'hys., 9, 654, 1938.
Next Page
SEC.9>25]
E X P A N S I O N OF T H E DIFFRACTED FIELD
563
Next we define an inductance L by the formula
F = +LP = $LIT. (43) If the current I in (43) is arbitrarily taken to be that which crosses the equator, the equivalent inductance is by (38), (42), and (43) L =0 . 0 7 7 ~ ~ ~ henrys. Thus far it has been assumed that the conductivity of the walls is infinite, in which case EO vanishes a t R = a and the attenuation is zero. If, however, the walls are metallic, the tangential component of E a t the surface may be obtained from the approximation (47), page 534. A t R=a (44)
There is now a radial flow of energy into the metal whose mean density is
The energy lost per unit time is obtained by integrating (46) over the surface of the sphere.
where 6
=
d+
is the skin depth defined on page 504. An equiva
wlIP2cz
lent series resistance 6i is expressed by (48) whence
w = a72 = +@IT,
(49) The parameter Q , defined as the ratio of inductive reacta,nctt to series resistance, is often used as a measure of circuit efficiency. Tho Q of the cavity oscillating a t its fundamental frequency is
DIFFRACTION OF A PLANE WAVE BY A SPHERE
9.26. Expansion of the Diffracted Field.A periodic wave incident upon a material body of any description gives rise to a forced oscillation of free and bound charges synchronous with the applied field. These constrained movements of charge set up in turn a secondary field both
Previous Page
564
BOUNDARYVALUE PROBLEMS
[CHAP.I X
inside and outside the body. The resultant field at any point is then the vector sum of the primary and secondary fields. In general the forced oscillations fail to match the conditions prevailing at the instant the primary field was first established. To ensure fulfillment of the boundary conditions at all times, a transient term must be added, constructed from the natural modes with suitable amplitudes. Such transient oscillations, however, are quickly damped by absorption and radiation losses, leaving only the steadystate, synchronous term. The simplest problem of this class and at the same time one of great practical interest is that of a plane wave falling upon a sphere. As in the preceding section, we shall suppose the sphere of radius a and propagation constant kl to be embedded in an infinite, homogeneous medium k2. A plane wave, whose electric vector is linearly polarized in the xdirection, is propagated in the direction of the positive zaxis. The expansion of this incident field in vector spherical wave functions has been given in (36), page 419.
where Eo is the amplitude and
the prime denoting differentiation with respect to the argument k2R. The induced secondary field must be constructed in two parts, the one applying to the interior of the sphere and the other valid at all external points with the necessary regularity at infinity. By analogy with the problem of plane boundaries discussed in Secs. 9.4 to 9.9, these two parts will be referred to as transmitted and reflected waves, although such terms are strictly appropriate only when the wave length is very much smaller than the radius of the sphere. Let us write
SEZ.9.251
EXPANSION OF THE DIFFRACTED FIELD
valid when R
> a, and
which hold when R
< a.
565
The functions mL3) and ni3) are obtained by *I”
replacing j,(k2R) by hA1)(k2R)in (2) and (3). kl replaces k , in (5). The boundary conditions at h! = a are il
+
E,) = i, X E,, x (Hi t HT)= il x Ht.
il X (Ei
(6)
These lead to two pairs of inhomogeneous equations for the expansion coefficients. a5n(NP)  a3L1’(p) = j n b ) , (7) IL2ah“pjnWP)l’  P l a 3 P h?(P)l’ = P l b jn(PIJ’7 PZNb;jn(Np)  P ~ V K ( P=)~ i j n ( ~ ) , (8) b f i N p j , ( N ~ ) l ’  N b i b h2’(~)1’= NLPjn(p)I’, where again
kl = Nk2, p = kza, kla = N p . This system is now solved for the coefficients of the external field (9)
If the conductivity or inductive capacity of the sphere is large relative to the ambient medium, and at the same time the radius a is not too small, then lNpl >> 1 and (10) and (11) can be greatly simplified by use of the asymptotic expressions
In this case (13)
Since h;l)(p) form (1 4)
= jn(p)
a:
+ i n,(p), N
these coefficients can also be put in the
ieiy sin yn,
b,’
N
i e i ~ hsin y’ n,
566
BOUNDARY VALU E PROBLEMS
[CHAP.IX
where
To sum up, we have found that the primary wave excites certain partial oscillations in the sphere. These are not the natural modes discussed in Sec. 9.22, for all are synchronous with the applied ficld. These partial oscillations and their associated fields, however, are of electric and magnetic type for the same reasons that were set forth in Sec. 9.22. The coefficients a, are the amplitudes of the oscillations of magnetic type, while the 6 , are the amplitudes of electric oscillations. Whenever the impressed frequency w approaches a characteristic frequency wns of the free oscillations, resonance phenomena will occur. Now the characteristic frequencies of the magnetic oscillations have been shown to satisfy (lo), Sec. 9.22. But this is just the condition which makes the denominator of a;, in (10) above, vanish. Likewise the natural frequencies of the electric oscillations satisfy (19) in Sec. 0.22, which is the condition that the denominator of b; shall vanish. Note, however, that the wns are always complex, whereas the frequency w of the forcing field is real. Consequently the denominators of a; and b; can be reduced to minimum values, but never quite to zero, so that the catastrophe of infinite amplitudes is safely avoided. In Fig. 112 the electric and magnetic lines of force are shown for the first, four partial waves of electric type. The drawings are reproduced from the original paper by Mie.l The incident wave is linearly polarized with its electric vector parallel to the zaxis. A t very great distances from the sphere the radial component of the secondary field vanishes as 1/R2, while the tangential components Ere and E,, diminish as 1/R. In this radiation zone the field is transverse to the direction of propagation. Moreover, the components EO , and E , are perpendicular to each other and differ, in general, in phase. T h e secondary radiation f r o m the sphere i s elliptically polarized. There are two exceptional directions. When (b = 0, we note that E,, = 0, and when (b = 9/2 we have E,o = 0. Consequently, when viewed along the x or the yaxis, the secondary radiation is linearly polarized. Inversely, if the primary wave is unpolarized, as in the case of natural light, the secondary radiation exhibits partial polarization dependent upon the direction of observation. This effect has been studied in connection with the scattering of light by suspensions of colloidal particles. The most complete numerical investigation of the problem thus far in the visible spectrum has been made by Blumer.2 MIE, Ann. Physik, 26, 377, 1908. 2 B ~ n2.~Physik, ~ ~ 32, , 119, 1925; 38, 304, 1926; 38, 920, 1926; 39, 195, 1926.
SEC.9.251
EXPANSION OF T H E DIFFRACTED FIELD
First Mode

S e c o n d Mode
T h i r d Mode
F o u r t h Mode Magnetic Lines of Force Electric Lines of Force of force corresponding to the first four modes of electric type.
FIG.ll2.Lines
567
568
BOUNDARYVALUE PROBLEMS
[CHAP.IX
9.26. Total Radiation.The resultant field at any point outside the sphere is the sum of the primary and secondary fields.
+
+
E = Ei E,, H = Hi H,. (16) The radial component of the total complex flow vector is SE = +(EBB+  EJJe), (17) which can be resolved by virtue of (16) into three groups: (18) 8;
=
a(EieAi+ EiJie) f * ( E r Z r +  Ev&frO) +(EtBgr+ EreHi,  EiJ:,s  EJTie).
+
+
Let us draw about the diffracting sphere a concentric spherical surface of radius R. The real part of 8; integrated over this sphere is equal to the net flow of energy across its surface. To simplify matters we shall assume the external medium (2) to be a perfect dielectric, so that k , = a&.. If the diffracting sphere is also nonconducting, the net flow across any surface enclosing the sphere must be zero. If, however, energy is converted into heat within the sphere, the net flow is equal to the amount absorbed and is directed inwards. We shall call the total energy absorbed by the sphere W,,
W , = Re
(19)
cr
SER2 sin 0 d$ do.
The first term on the righthand side of (18) measures the flow of energy in the incident wave. When integrated over a closed surface this gives zero so long as uz = 0. The second term obviously measures the outward flow of the secondary or ficattered energy from the diffracting sphere, and the tot,al scattered energy is
To maintain the energy balance the third term of (18) must be equal in magnitude to the sum of the absorbed and scattered energies. (21)
Wt = W,
+ W,
=
1 Re 2
Liz'
+ Er&+
(Eisg,.+
 E+Brs
 Er+Rie)R2sin 0 d8 d+. W t measures, therefore, the total energy derived from the primary wave and dissipated as heat and scattered radiation. To calculate W , and Wt we allow R to grow very large and introduce the asymptotic values
SEC.
TOTAL RADIATION
9.261
569
into the field functions (2) and (3). The integrals can then be evaluated with the help of Eq. (20), page 417,
ifn #m,
and the relation
One finds for the scattered cnorgy
for the sum of absorbed and scattered energies,
The mean energy flow of the incident wave per unit area is
The scattering cross section of the sphere is defined as the ratio of the total scattered energy per second to the energy density of the incident wave. m
Likewise one may define the cross section Qt by
(29)
If the conductivity or inductive capacity of the sphere is so large t h a t lNp\ >> 1, then thc approximations (14) and ( 1 5 ) can be introduced and the cross sections reduce to
3 m
(30)
Q8 =
(2n
n=l m
+ l)(sin2 Y~ + sin2d),
570
BOUNDARYVALUE PROBLEMS
[CHAP.IX
If the sphere is nonabsorbing, y n and y: are real, in which case it is obvious that Q3 =  Q t . 9.27. Limiting Cases.Although in appearance, the preceding formulas are exceedingly simple, the numerical calculation of the coefficients usually presents a serious task. There are two limiting cases which can be handled with ease. If IpI = (klal >> 1, the functions jn(p) and hA1)(p)may be replaced by their asymptotic values. The quantity p is essentially the ratio of radius to wave length, so that this is the case of a sphere whose radius is very much larger than the wave length. Results obtained by this method should approach those deduced from the KirchhoffHuygens principle of Secs. 8.13 to 8.15. If, on the other hand, the wave length is very much larger than the radius of the sphere, so that IpI > 1. If the asymptotic representations (12) and (22) are substituted into (10) and ( l l ) , one obtains for the coefficients a;:% .ne
p1
sin x  p 2 N cos x tan y ' PI  i p 2 N tan y
What is apparent here is also true in the general case: the expansion coefficients are oscillating functions of p and the order n. Small changes in either p or n may give rise to wide variations in the values of the coefficients. The absolute magnitudes of the coefficients oscillate between the limits zero and one. If in (32) we replace n by n 1, it will be observed that a',+l 'v T h e amplitude of an electric oscillation of order n i s approximately of the same magnitude as the amplitude of the magnetic oscillation of next higher order. The asymptotic expressions used in the derivations of (32) and (33) are valid only so long as the order n is very much less than the arguments Ip[ and ( N p l . Asymptotic formulas for the Bessel functions which hold for all orders have been found by Debye, who has also shown that the number of terms to be retained in the series expansions is just equal to the number 1pI.l Methods of summing the coefficients have been discussed by Jobst.2 Case 11. Ipl >
+
1 a4 El = , hl aul
1 a+ E O = ,
h2 au2
a n d that these potential and stream functions satisfy
a++ + .a+= o a2
at
’
az
a*+  p e a24 a22
at2
+ e  =
at
=
0,
592
BOUNDARYVALUE PROBLEMS
[CHAP.I X
where p and B pertain to the external dielectric. Let q be the charge per unit length, Z the current on a particular conductor, and C a closed contour linking this conductor only and lying in a plane z = constant. Show that
the differentiation being in the direction of the outward normal. The theory applies to any number of parallel conductors of constant but arbitrary cross section provided the sum of the charges q in any transverse plane is zero. Otherwise the functions $ and $ are irregular (logarithmic) at infinity. I n the case of two parallel conductors a and b we have q. =  q h . The capacity and inductance per unit length are defined by C = Q L , 4 a  +b
exactly as in the static regime.
L=$a  $b, 1,
Prove that in m.k.s. units p€ =
LC
and show that current and charge satisfy
18. The theory of propagation along a system of parallel, cylindrical conductors discussed in Problem 17 was extended by Abraham t o the case of finite conductivity. The solution is approximate but valid and useful in most cases of technical importance. One assumes ( a ) t h a t the waves are transverse magnetic. We have seen in Sec. 9.15 that, apart from the case of an axially symmetric field, this is not strictly correct, but the error involved is, in general, extremely small. Assume, next, that the field is harmonic and take for the scalar potential
Show that a t any point in the conductor or in the dielectric the components of the field are given by
El
=
1 a$ , 11,
aul
E',
=
1 a$ ,
hz au2
593
PROBLEMS
Calculate the zcomponent of the complex Poynting vector S*and integrate this over the entire transverse zyplane. Let the value of this integral be W*. Show t h a t l$T*
=
2f
 x magnetic energy in dielectric wpee
27 +X magnetic energy in conductor, WPi
~ aW*
=
Joule heat
az
+ 2iw
(electric energy
 magnetic
energy),
where the subscript e refers to the external or dielectric medium, i to the conductor, and all quantities are per unit length of line. Now make the following assumptions: (b) I n the dielectric the longitudinal component r E , is very small relative to the transverse field, and consequently in this medium the components E l , E2, H I , H z have the values computed in thc case of perfect conductivity. Furthermore, the values of capacity C and externul inductance L, are unchanged, with ~~e = L,C as in Problem 17. The contribution of the longitudinal component to the electric energy in the dielectric is negligible. (c) In the metal the intensity of thc transverse electric components is negligible with respect to the longitudinal component, and the electric energy is negligible with respect to the magnetic energy. Consider a pair of conductors carrying equal currents in opposite directions. Let R be twice the alternatingcurrent resistance per unit length per conductor, Li the internal inductance per unit length of the double circuit. Both these quantities are functions of the frequency and depend upon the proximity of the conductors as well as upon their cross sections. They are obtained approximately by the method outlined in Problem 13. Remembering that the energies involved are time averages, show that
aw* az
__ = i ( y  .t)TV'+.
The second term in W* can be neglected and one obtains for the propagation factor y2 =
w*pee
[ + 2) + 5.. (1
19. According t o Problem 18 the propagation factor for waves along a pair of parallel, cylindrical conductors is given by
Let y
=
01
+ i@
and L = Li
+ L,, the total inductance per unit length.
Show thac
594
[CHAP. IX
BOUNDARYVALUE PROBLEMS
20. A Lecher wire system consists of two long parallel wires of circular cross section. The distance between centers is 2a and the radius of each wire is b. Show that the capacitance and external selfinductances are
y) 7) T€
=
(a
=a ’” In
L,
+
farads/meter,
henrys/meter.
+
>>
If a b, the effect of one wire on the current distribution in the other can be neglected and the resistance and internal selfinductance per meter per conductor are given approximately by the formulas at the end of Sec. 9.17. T o obtain an idea of the orders of magnitude assume the wires to be 2 mm. in radius (about a No. 6 gauge) and of copper, spaced 10 cm. Calculate C , L,, Li, R for frequencies of 108, 105, and lo2 cycles/sec. and compute the attenuation in decibels per meter. Discuss the effect of a change in spacing or radius on the various parameters. 21. When leakage through the dielectric is considered in the preceding problems, one obtains Eq. (27), page 550, for the general equation of propagation of current along a pair of parallel conductors. a21
a21
a22
at2
  LC
a1  (U:+ RC)  RGI at
=O.
This equation was first studied in detail by Lord Kelvin in connection with submarine cables. By the methods of the Fourier or Laplace transform discussed in Chap. V, obtain a general solution in terms of initial conditions imposed on current and voltage. Show that by an appropriate choice of parameters the velocity and attenuation can be made independent of frequency so that a signal is propagated without distortion. 22. The feasibility of a n Atlantic cable was first demonstrated in a theoretical paper published by Lord Kelvin in 1855. For the case in question the selfinductance and leakage could be neglected with respect to the series resistance and the potential with respect to ground was governed by
According to Kelvin the time necessary to produce a given potential a t a distance z from the origin is proportional to C R multiplied by the square of the distance. Verify this result. 23. The problem of eliminating “cross talk” between pairs of telephone lines by means of cylindrical shields is of great technical importance. To reduce the problem to a form which can be handled satisfactorily i t is usually assumed that variation along the line can be neglected, so that the field is essentially twodimensional. This proves justifiable as long as the wave length is very much greater than the distances between wires and between wires and shield. Show first t h a t on this assumption the field equations resolve into two groups:
595
PROBLEMS
Interference caused by a field of type (I) is commonly called "electrostatic" cross talk; that associated with a field of type (11) is known as "electromagnetic" cross talk. The terms are unfortunate and misleading but firmly entrenched in engineering literature. Group (11) represents the field of one or more alternatingcurrent filaments. Show that the field due to a single filament Z is
whose radial impedance is
zI 
E, He
N$')(kr) __. k Hil)(kr)
imp
A current solenoid or vortex line produces a field of type (I),but such a field is also generated by pairs of charged filaments of opposite sign, whence the term "electrostatic" interference. 24. A transmission line consists of two parallel wires of radius b whose separation between centers is 2a. The inductance and capacitance per unit length are known from the results of Problem 20. So far as t h e external field is concerned, the system is equivalent t o a pair of current filaments of strength Z and opposite sign, and a pair of line charges of strength q per unit length and opposite sign. A study of the problem in bipolar coordinates shows that the location of these filaments does not coincide exactly with the centers of the wires, but the effect of the deviation on the field can be ignored except perhaps in the immediate neighborhood of the wire itself. Let the central line between conductors coincide with the zaxis of a cylindrical coordinate system. At distances from this axis which are very much less than the wave length the field is governed by the equations of Problem 23. Show t h a t at points whose distance r from the central axis a, the field of a current pair or doublet is
E,
*
>>
aZH$')(kr) cos e e  i w t , 2 i k a1 Z i   H!')(kr) sin e e'wt, 2 r i k aI H,q =   [H$')(kr)  krH!"(kr)] cos 8 e'ot, 2 7 7
=

and find expressions for the field of a corresponding charge doublet.
396
BOUNDARYVALUE PROBLEMS
[CHAP.IX
26. Extend the results of Problem 24 with the help of the addition formula ( l l ) , Sec. 6.11, and show t h a t any pair of eccentric filaments can be replaced by a system of line sources on the central axis emitting cylindrical waves of different orders. 26. The transmission line described in Problem 24 is enclosed by a coaxial copper shield of inner radius c and thickness 6 . Assume c a and obtain expressions for the field penetrating the shield. Discuss the results in terms of the frequency and dimensions of the system. Considrr fields of both electric and magnetic type. (On this and the preceding problems see the discussion and bibliography by Schelkunoff, Re21 System Tech. Jour., 13, 532, 1934.) 27. A hollow tube of rectangular cross section is bounded by thick metal walls. The inner dimensions are x = a, y = b. Assuming first perfect conductivity, obtain expressions for the allowed modes of the transverse electric field ( H waves) and the corresponding transverse magnetic modes ( E waves). The components of the field can be written down directly from the results of Problem 1 , Chap. VI. If the tangential components of E are to vanish a t the boundaries, t h e fields must be periodic in the x and ydirections and hence hl = mn/a, h, = nr/b. Correspondingly, the various orders of E wave are designated by E,,,,; for a n H wave of order m, n, one writes Hm,,,. Find expressions for the phase velocity, group velocity, wave length within the tube, critical frequency, and critical wave length. Sketch the lines of electric and magnetic field intensity for the first few orders for both E and I€ waves. The Ho.1 wave has the simplest structure, the lowest critical frequency, and the lowest attenuation in the case of tubes of finite conductivity. For these reasons i t is the mode commonly used in practice. Assuming the walls to be metal of finite conductivity, derive a n approximate ,:xpression for the attenuation of the Ho.1 wave in decibels per meter. Show that there are optimum values of the ratio a/b and the frequency leading t o a minimum attenuation. 28. Find the lowest natural frequency for electromagnetic oscillations in a cavity whose form is t h a t of a right circular cylinder of radius a and length 2. The walls are metal of conductivity u. Calculate the equivalent inductance, resistance, and the parameter Q for this mode. Follow the procedure of Sec. 9.24. 29. Discuss the propagation of transverse electric and transverse magnetic waves in a perfectly conducting hollow tube of elliptic cross section. 30. The axis of a n infinitely long, circular cylinder of radius a coincides with the zaxis of a coordinate system. The propagation factor of the cylinder is kl, that of the external medium is 122. A plane wave whose direction is that of the positive zaxis is incident upon the cylinder. Derive expressions for the diffracted field and the scattering cross section. Obtain approximate formulas for the case a / h 1. Discuss the two cases: a. Electric vector of the incident wave parallel to zaxis; b. Magnetic vector of the incident wave parallel t o zaxis. The theory has been confirmed experimentally by Schaefer, Ann. Physik, 31, 455, 1910; Zeit. Physik, 13, 166, 1923. 31. A plane, linearly polarized wave is scattered b y a spherical body of radius a. The material of the sphere has a propagation factor k l ; t h a t of the external medium, which is assumed to be nonconducting, is kz. Let R be the radius vector from the center of the sphere to a point of observation. The plane containing R and the axis of propagation is called the plane of vision. The plane containing the direction of polarization of the incident wave and the axis of propagation is the plane of oscillation. Let + b e the angle between the plane of vision and the plane of oscillation, and e the
>>
a, the intensity of the scattered Rayleigh radiation is given by
8 SO
 =
I
1
as k:  k: (2T)4__ X4Rz kf 2k:
+
2
(cos2 0 cos2 +
+ sin2 +),
where 8 is the mean value of the scattered intensity and 8,t h a t of the incident wave. Discuss the polarization of the scattered radiation. 32. Linearly polarized light is scattered by very small, nonconducting, spherical particles. Show t h a t in a direction parallel to the electric vector of the incident light the intensity of the scattered radiation varies inverscly as the eighth power of the wave length. Polarized white light scattered in this direction appears as a purer blue than when viewed a t other angles and the effect is referred to as Tyndall’s “residual blue.” 33. Discuss the attenuation of ultrahighfrequency radio waves due to scattering in rain and fog from the following data, taken from Humphreys, “Physics of the Air.”
Fog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Light rain., . . . . . . . . . . . . . . . . . . . . . . . Moderate raiii.. . . . . . . . . . . . . . . . . . . . Excessive rain. . . . . . . . . . . . . . . . . . . . .
Radius, meters
Drops /meter 3
5 X 2 X lo‘ 5 X 103
4,000 500 450
107
Express the decrease in intensity in decibels per kilometer. 34. Calculate the force exerted by a plane, linearly polarized wave on a dielectric sphere, assuming the wave length to be very large relative to the radius of the sphere. (Debye, Ann. Physik, 30, 57, 1909.) 36. A plane, linearly polarized electromagnetic wave is incident upon a perfectly conducting sphere. Assuming the wave length to be very large relative to the radius of the sphere, calculate the total force exerted on the sphere. (Debye, Ann. Physik, 30, 57, 1909.) Estimate the change in the result in case the sphere were of copper. 36. A spherical cavity 17.5 cm. in radius, such as was described in Sec. 9.24, is bounded by thick copper walls. Calculate the frequency and logarithmic decrement of the lowest possible mode of oscillation. Repeat for a similar sphere of half the radius. 37. An oscillating electric dipole is located at a point whose distance from the center of a sphere is b. The radius of the sphere is a, with a < b, and the dipole is oriented in the radial direction. The propagation factor of the sphere is kl;t h a t of the external medium is ks. Show that the Hertz vector of a field satisfying the boundary conditions and behaving as eikaR/ik,R at the dipole is
when r
> a, and
598 when r
BOUNDARYVALUE PROBLEMS
[CHAP.
IX
< a. where
and where R is the distance from the dipole to the observer, r the distance from the center to the observer, and e the polar angle measured from the dipole axis. The Hertz vector is radial. 38. The axis of a n infinite, perfectly conducting, right circular cone coincides with the negative zaxis of a coordinate system. The vertex of the cone is at the origin and its generators make an angle e = eo with the positive zaxis. The external space is dielectric. Investigate the natural modes of propagation on the cone. (Macdonald, " Electric Waves," Cambridge University Press, 1902.) 39. Discuss the radiation from the open end of a semiinfinite, straight wire carrying an oscillating current by allowing eo x in the results of Problem 38. Compare this with Problem 9, Chap. VIII. 40. I n Problem 9, Chap. VII, expressions were found for a field with rotational symmetry in spheroidal coordinates. Apply this theory t o a perfectly conducting prolate spheroid whose eccentricity differs only slightly from unity. Calculate the frequency of the fundamental mode of oscillation and carry the approximations far enough to give a n expression for the damping due t o radiation. (Abraham, Math. Ann., 62, 81, 1899.) 41. A circular loop of wire 15 meters in radius carries a 60cycle alternating current of 10 amp. The loop lies on a flat surface of earth whose constants are mho/meter. c/eO = 4, u = a. Calculate the electric field intensity in volts per meter at a point on the surface of the earth whose distance from the center of the loop is r. b. What is the effective resistance at the driving point of the loop? Neglect losses in the wire and assume that all dissipation is due to the finite conductivity of the earth. (Note: At this low frequency the loop is equivalent to a vertical magnetic dipole. The field can be calculated by the methods of Secs. 9.28 to 9.31, and the losses determined from the mean vertical energy flow a t the surface of the earth.) 42. Consider a system of conductors and dielectrics characterized by the parameters t, p, u. The media are isotropic but not necessarily homogeneous. Sharp boundary surfaces may be replaced by layers of rapid transition. There are two kinds of impressed forces, the one FI causing dielectric polarization, the other Fz producing conduction currents. Thus
D
=
c(E
+ Fi),
J = u(E
+ FZ).
FIand FZare continuous functions of position. These forces are started during an infinitely short interval At, ending at the instant t = 0, and henceforth remain constant. Let
be the work done on the system by the impressed forces from the initial instant to the time t , and
PROBLEMS
599
the total generation of heat at the final rate, where J is the final value of current density. Then if U and T are the electric and magnetic energies of the field in the final state, show that
A
=
W
+ 2(U  T ) .
The theorem was stated first by Heaviside, “Electrical Papers,” Vol. 11, page 412, and proved by Lorentz, Proc. Natl. Acad. Sci., 8, 333, 1922. A similar theorem for electrical circuits is expressed in Pierce, “Electric Oscillations,” page 40, 1st ed., 1920.
APPENDIX I A. NUMERICAL VALUES OF FUNDAMENTAL CONSTANTS
Permittivity of free space
€0,
1 8.854 X 1012 = __ X 36x Permeability of free space P O , po = 4~ x 107 = 1.257 x
farad/meter.
=
€0
henrylmeter.
1
c = __ = 2.998 X 1 0 8 . v 3 X
z/iz
1/60 = l/po =
1.129 x 1011 7.958 X lo5
lo8 nieters/sec.
meterslfarad. meters/henry.
,,/: I
=
L& 27r
ohms.
= 59.95 'V 60
&
= 2.654
Propagation constant k k2
376.7
= epw2
= a
x
ohms. mho.
103
+ iP,
+ i w p u = epuw2(1 + iq), 1.796 X 10"
cl
U
N
1.8 X 10*O, VKe
VKe
where cl is the conductivity in mhos per meter; v, t,he frequency; and the specific inductive capacity e / e O .
k where
If
K~ =
q
= a
+ ip
N
K,
l/zwlra,
specific magnetic permeability p / p o .