First Edition, 2011
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Table of Contents Chapter 1 Kinetic Energy Chapter 2  Potential Energy Chapter 3  Conservation of Energy Chapter 4  Mechanical Work Chapter 5  Electromagnetism Chapter 6  Mass–Energy Equivalence Chapter 7  Binding Energy
Chapter 1
Kinetic Energy
The cars of a roller coaster reach their maximum kinetic energy when at the bottom of their path. When they start rising, the kinetic energy begins to be converted to gravitational potential energy. The sum of kinetic and potential energy in the system remains constant, ignoring losses to friction. The kinetic energy of an object is the energy which it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. The same amount of work is done by the body in decelerating from its current speed to a state of rest. The speed, and thus the kinetic energy of a single object is framedependent (relative): it can take any nonnegative value, by choosing a suitable inertial frame of reference. For example, a bullet passing an observer has kinetic energy in the reference frame of this observer, but the same bullet is stationary, and so has zero kinetic energy, from the point of view of an observer moving with the same velocity as the bullet. By contrast, the total kinetic energy of a system of objects cannot be reduced to zero by a suitable choice of the inertial reference frame, unless all the objects have the same velocity. In any other case the total kinetic energy has a nonzero minimum, as no inertial reference frame can be chosen in which all the objects are stationary. This minimum kinetic energy contributes to the system's invariant mass, which is independent of the reference frame. In classical mechanics, the kinetic energy of a nonrotating object of mass m traveling at a speed v is mv2/2. In relativistic mechanics, this is only a good approximation when v is much less than the speed of light.
History and etymology The adjective kinetic has its roots in the Greek word κίνησις (kinesis) meaning motion, which is the same root as in the word cinema, referring to motion pictures. The principle in classical mechanics that E ∝ mv² was first developed by Gottfried Leibniz and Johann Bernoulli, who described kinetic energy as the living force, vis viva. Willem 's Gravesande of the Netherlands provided experimental evidence of this relationship. By dropping weights from different heights into a block of clay, 's Gravesande determined that their penetration depth was proportional to the square of their impact speed. Émilie du Châtelet recognized the implications of the experiment and published an explanation. The terms kinetic energy and work in their present scientific meanings date back to the mid19th century. Early understandings of these ideas can be attributed to GaspardGustave Coriolis, who in 1829 published the paper titled Du Calcul de l'Effet des Machines outlining the mathematics of kinetic energy. William Thomson, later Lord Kelvin, is given the credit for coining the term "kinetic energy" c. 1849  1851.
Introduction Energy occurs in many forms: chemical energy, thermal energy, electromagnetic radiation, gravitational energy, electric energy, elastic energy, nuclear energy, rest energy. These can be categorized in two main classes: potential energy and kinetic energy. Kinetic energy may be best understood by examples that demonstrate how it is transformed to and from other forms of energy. For example, a cyclist uses chemical energy that was provided by food to accelerate a bicycle to a chosen speed. This speed can be maintained without further work, except to overcome airresistance and friction. The chemical energy has been converted into kinetic energy, the energy of motion, but the process is not completely efficient and produces heat within the cyclist. The kinetic energy in the moving cyclist and the bicycle can be converted to other forms. For example, the cyclist could encounter a hill just high enough to coast up, so that the bicycle comes to a complete halt at the top. The kinetic energy has now largely been converted to gravitational potential energy that can be released by freewheeling down the other side of the hill. Since the bicycle lost some of its energy to friction, it never regains all of its speed without additional pedaling. The energy is not destroyed; it has only been converted to another form by friction. Alternatively the cyclist could connect a dynamo to one of the wheels and generate some electrical energy on the descent. The bicycle would be traveling slower at the bottom of the hill than without the generator because some of the energy has been diverted into electrical energy. Another possibility would be for the cyclist to apply the brakes, in which case the kinetic energy would be dissipated through friction as heat. Like any physical quantity which is a function of velocity, the kinetic energy of an object depends on the relationship between the object and the observer's frame of reference. Thus, the kinetic energy of an object is not invariant. Spacecraft use chemical energy to launch and gain considerable kinetic energy to reach orbital velocity. This kinetic energy remains constant while in orbit because there is almost no friction in nearearth space. However it becomes apparent at reentry when some of the kinetic energy is converted to heat. Kinetic energy can be passed from one object to another. In the game of billiards, the player imposes kinetic energy on the cue ball by striking it with the cue stick. If the cue ball collides with another ball, it slows down dramatically and the ball it collided with accelerates to a speed as the kinetic energy is passed on to it. Collisions in billiards are effectively elastic collisions, in which kinetic energy is preserved. In inelastic collisions, kinetic energy is dissipated in various forms of energy, such as heat, sound, binding energy (breaking bound structures). Flywheels have been developed as a method of energy storage. This illustrates that kinetic energy is also stored in rotational motion.
Several mathematical description of kinetic energy exist that describe it in the appropriate physical situation. For objects and processes in common human experience, the formula ½mv² given by Newtonian (classical) mechanics is suitable. However, if the speed of the object is comparable to the speed of light, relativistic effects become significant and the relativistic formula is used. If the object is on the atomic or subatomic scale, quantum mechanical effects are significant and a quantum mechanical model must be employed.
Newtonian kinetic energy Kinetic energy of rigid bodies In classical mechanics, the kinetic energy of a point object (an object so small that its mass can be assumed to exist at one point), or a nonrotating rigid body, is given by the equation
where is the mass and is the speed (or the velocity) of the body. In SI units (used for most modern scientific work), mass is measured in kilograms, speed in metres per second, and the resulting kinetic energy is in joules. For example, one would calculate the kinetic energy of an 80 kg mass (about 180 lbs) traveling at 18 metres per second (about 40 mph, or 65 km/h) as Ek = (1/2) · 80 · 182 J = 12.96 kJ Since the kinetic energy increases with the square of the speed, an object doubling its speed has four times as much kinetic energy. For example, a car traveling twice as fast as another requires four times as much distance to stop, assuming a constant braking force. The kinetic energy of an object is related to its momentum by the equation:
where: is momentum is mass of the body For the translational kinetic energy, that is the kinetic energy associated with rectilinear motion, of a body with constant mass , whose center of mass is moving in a straight line with speed , as seen above is equal to
where: is the mass of the body is the speed of the center of mass of the body. The kinetic energy of any entity depends on the reference frame in which it is measured. However the total energy of an isolated system, i.e. one which energy can neither enter nor leave, does not change in whatever reference frame it is measured. Thus, the chemical energy converted to kinetic energy by a rocket engine is divided differently between the rocket ship and its exhaust stream depending upon the chosen reference frame. This is called the Oberth effect. But the total energy of the system, including kinetic energy, fuel chemical energy, heat, etc., is conserved over time, regardless of the choice of reference frame. Different observers moving with different reference frames disagree on the value of this conserved energy. The kinetic energy of such systems depends on the choice of reference frame: the reference frame that gives the minimum value of that energy is the center of momentum frame, i.e. the reference frame in which the total momentum of the system is zero. This minimum kinetic energy contributes to the invariant mass of the system as a whole. Derivation The work done accelerating a particle during the infinitesimal time interval dt is given by the dot product of force and displacement:
where we have assumed the relation p = m v. (However, also see the special relativistic derivation below.) Applying the product rule we see that:
Therefore (assuming constant mass), the following can be seen:
Since this is a total differential (that is, it only depends on the final state, not how the particle got there), we can integrate it and call the result kinetic energy:
This equation states that the kinetic energy (Ek) is equal to the integral of the dot product of the velocity (v) of a body and the infinitesimal change of the body's momentum (p). It is assumed that the body starts with no kinetic energy when it is at rest (motionless).
Rotating bodies If a rigid body is rotating about any line through the center of mass then it has rotational kinetic energy ( ) which is simply the sum of the kinetic energies of its moving parts, and is thus given by:
where: • • •
ω is the body's angular velocity r is the distance of any mass dm from that line is the body's moment of inertia, equal to
.
(In this equation the moment of inertia must be taken about an axis through the center of mass and the rotation measured by ω must be around that axis; more general equations exist for systems where the object is subject to wobble due to its eccentric shape).
Kinetic energy of systems A system of bodies may have internal kinetic energy due to the relative motion of the bodies in the system. For example, in the Solar System the planets and planetoids are orbiting the Sun. In a tank of gas, the molecules are moving in all directions. The kinetic energy of the system is the sum of the kinetic energies of the bodies it contains. A macroscopic body that is stationary (i.e. a reference frame has been chosen to correspond to the body's center of momentum) may have various kinds of internal energy at the molecular or atomic level, which may be regarded as kinetic energy, due to molecular translation, rotation, and vibration, electron translation and spin, and nuclear spin. These all contribute to the body's mass, as provided by the special theory of relativity. When discussing movements of a macroscopic body, the kinetic energy referred to is usually that of the macroscopic movement only. However all internal energies of all types contribute to body's mass, inertia, and total energy.
Frame of reference The total kinetic energy of a system depends on the inertial frame of reference: it is the sum of the total kinetic energy in a center of momentum frame and the kinetic energy the total mass would have if it were concentrated in the center of mass.
This may be simply shown: let V be the relative speed of the frame k from the center of mass frame i :
However, let
the kinetic energy in the center of mass frame,
would be simply the total momentum which is by definition zero in the center of mass frame, and let the total mass:
. Substituting, we get:
Thus the kinetic energy of a system is lowest with respect to center of momentum reference frames, i.e., frames of reference in which the center of mass is stationary (either the center of mass frame or any other center of momentum frame). In any other frame of reference there is additional kinetic energy corresponding to the total mass moving at the speed of the center of mass. The kinetic energy of the system in the center of momentum frame contributes to the invariant mass of the system, and this total mass is a quantity which is both invariant (all observers see it to be the same) and is conserved (in an isolated system, it cannot change value, no matter what happens inside the system).
Rotation in systems It sometimes is convenient to split the total kinetic energy of a body into the sum of the body's centerofmass translational kinetic energy and the energy of rotation around the center of mass (rotational energy):
where: Ek is the total kinetic energy Et is the translational kinetic energy Er is the rotational energy or angular kinetic energy in the rest frame Thus the kinetic energy of a tennis ball in flight is the kinetic energy due to its rotation, plus the kinetic energy due to its translation.
Relativistic kinetic energy of rigid bodies
In special relativity, we must change the expression for linear momentum. Using m for rest mass, v and v for the object's velocity and speed respectively, and c for the speed of light in vacuum, we assume for linear momentum that , where . Integrating by parts gives
Remembering that
, we get:
where E0 serves as an integration constant. Thus:
The constant of integration E0 is found by observing that, when , giving
and
and giving the usual formula:
If a body's speed is a significant fraction of the speed of light, it is necessary to use relativistic mechanics (the theory of relativity as developed by Albert Einstein) to calculate its kinetic energy. For a relativistic object the momentum p is equal to:
. Thus the work expended accelerating an object from rest to a relativistic speed is:
. The equation shows that the energy of an object approaches infinity as the velocity v approaches the speed of light c, thus it is impossible to accelerate an object across this boundary. The mathematical byproduct of this calculation is the massenergy equivalence formula—the body at rest must have energy content equal to:
At a low speed (v<
, So, the total energy E can be partitioned into the energy of the rest mass plus the traditional Newtonian kinetic energy at low speeds. When objects move at a speed much slower than light (e.g. in everyday phenomena on Earth), the first two terms of the series predominate. The next term in the approximation is small for low speeds, and can be found by extending the expansion into a Taylor series by one more term:
. For example, for a speed of 10 km/s (22,000 mph) the correction to the Newtonian kinetic energy is 0.0417 J/kg (on a Newtonian kinetic energy of 50 MJ/kg) and for a speed of 100 km/s it is 417 J/kg (on a Newtonian kinetic energy of 5 GJ/kg), etc. For higher speeds, the formula for the relativistic kinetic energy is derived by simply subtracting the rest mass energy from the total energy:
. The relation between kinetic energy and momentum is more complicated in this case, and is given by the equation: . This can also be expanded as a Taylor series, the first term of which is the simple expression from Newtonian mechanics. What this suggests is that the formulas for energy and momentum are not special and axiomatic, but rather concepts which emerge from the equation of mass with energy and the principles of relativity.
Quantum mechanical kinetic energy of rigid bodies In the realm of quantum mechanics, the expectation value of the electron kinetic energy, , for a system of electrons described by the wavefunction operator expectation values:
is a sum of 1electron
where me is the mass of the electron and is the Laplacian operator acting upon the coordinates of the ith electron and the summation runs over all electrons. Notice that this is the quantized version of the nonrelativistic expression for kinetic energy in terms of momentum:
The density functional formalism of quantum mechanics requires knowledge of the electron density only, i.e., it formally does not require knowledge of the wavefunction. , the exact Nelectron kinetic energy functional is Given an electron density unknown; however, for the specific case of a 1electron system, the kinetic energy can be written as
where T[ρ] is known as the von Weizsäcker kinetic energy functional.
Chapter 2
Potential Energy
In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. . The SI unit of measure for energy and work is the Joule (symbol J). The term "potential energy" was coined by the 19th century Scottish engineer and physicist William Rankine.
Overview Potential energy is energy that is stored within a system. It exists when there is a force that tends to pull an object back towards some lower energy position. This force is often called a restoring force. For example, when a spring is stretched to the left, it exerts a force to the right so as to return to its original, unstretched position. Similarly, when a mass is lifted up, the force of gravity will act so as to bring it back down. The action of stretching the spring or lifting the mass requires energy to perform. The energy that went into lifting up the mass is stored in its position in the gravitational field, while similarly, the energy it took to stretch the spring is stored in the metal. According to the law of conservation of energy, energy cannot be created or destroyed; hence this energy cannot disappear. Instead, it is stored as potential energy. If the spring is released or the mass is dropped, this stored energy will be converted into kinetic energy by the restoring force, which is elasticity in the case of the spring, and gravity in the case of the mass. Think of a roller coaster. When the coaster climbs a hill it has potential energy. At the very top of the hill is its maximum potential energy. When the car speeds down the hill potential energy turns into kinetic. Kinetic energy is greatest at the bottom. The more formal definition is that potential energy is the energy difference between the energy of an object in a given position and its energy at a reference position. There are various types of potential energy, each associated with a particular type of force. More specifically, every conservative force gives rise to potential energy. For example, the work of an elastic force is called elastic potential energy; work of the gravitational force is called gravitational potential energy; work of the Coulomb force is called electric potential energy; work of the strong nuclear force or weak nuclear force acting on the baryon charge is called nuclear potential energy; work of intermolecular
forces is called intermolecular potential energy. Chemical potential energy, such as the energy stored in fossil fuels, is the work of the Coulomb force during rearrangement of mutual positions of electrons and nuclei in atoms and molecules. Thermal energy usually has two components: the kinetic energy of random motions of particles and the potential energy of their mutual positions. As a general rule, the work done by a conservative force F will be
where ΔU is the change in the potential energy associated with that particular force. Common notations for potential energy are U, Ep, and PE.
Reference level The potential energy is a function of the state a system is in, and is defined relative to that for a particular state. This reference state is not always a real state, it may also be a limit, such as with the distances between all bodies tending to infinity, provided that the energy involved in tending to that limit is finite, such as in the case of inversesquare law forces. Any arbitrary reference state could be used, therefore it can be chosen based on convenience. Typically the potential energy of a system depends on the relative positions of its components only, so the reference state can also be expressed in terms of relative positions.
Gravitational potential energy Gravitational energy is the potential energy associated with gravitational force. If an object falls from one point to another point inside a gravitational field, the force of gravity will do positive work on the object, and the gravitational potential energy will decrease by the same amount.
The gravitational force keeps the planets in orbit around the Sun
A trebuchet uses the gravitational potential energy of the counterweight to throw projectiles over long distances For example, consider a book, placed on top of a table. When the book is raised from the floor to the table, some external force works against the gravitational force. If the book falls back to the floor, the same work will be done by the gravitational force. Thus, if the book falls off the table, this potential energy goes to accelerate the mass of the book (and is converted into kinetic energy). When the book hits the floor this kinetic energy is converted into heat and sound by the impact. The factors that affect an object's gravitational potential energy are its height relative to some reference point, its mass, and the strength of the gravitational field it is in. Thus, a book lying on a table has less gravitational potential energy than the same book on top of a taller cupboard, and less gravitational potential energy than a heavier book lying on the same table. An object at a certain height above the Moon's surface has less gravitational potential energy than at the same height above the Earth's surface because the Moon's gravity is weaker. (This follows from Newton's law of gravitation because the mass of the moon is much smaller than that of the Earth.) It is important to note that "height" in the common sense of the term cannot be used for gravitational potential energy calculations when gravity is not assumed to be a constant. The following sections provide more detail.
The strength of a gravitational field varies with location. However, when the change of distance is small in relation to the distances from the center of the source of the gravitational field, this variation in field strength is negligible and we can assume that the force of gravity on a particular object is constant. Near the surface of the Earth, for example, we assume that the acceleration due to gravity is a constant g = 9.81 m/s2 ("standard gravity"). In this case, a simple expression for gravitational potential energy can be derived using the W = Fd equation for work, and the equation
When accounting only for mass, gravity, and altitude, the equation is:
where U is the potential energy of the object relative to its being on the Earth's surface, m is the mass of the object, g is the acceleration due to gravity, and h is the altitude of the object. If m is expressed in kilograms, g in meters per second squared and h in meters then U will be calculated in joules. Hence, the potential difference is
However, if the force of gravity varies too much for this approximation to be valid, then we have to use the general, integral definition of work to determine gravitational potential energy. Now taking the arbitrary reference point where U = 0 to be when the two objects are infinite distance apart: The (now negative) gravitational potential energy of a system of masses m1 and m2 at a distance R using gravitational constant G is
for the computation of the potential energy we can integrate the gravitational force (whose magnitude is given by Newton's law of gravitation) with respect to the distance r between the two bodies from r = R to r = ∞. The total potential energy of a system of n bodies is found by summing, for all pairs of two bodies, the potential energy of the system of those two bodies. Considering the system of bodies as the combined set of small particles the bodies consist of, and applying the previous on the particle level we get the negative gravitational binding energy. This potential energy is more strongly negative than the total potential energy of the system of bodies as such since it also includes the negative gravitational
binding energy of each body. The potential energy of the system of bodies as such is the negative of the energy needed to separate the bodies from each other to infinity, while the gravitational binding energy is the energy needed to separate all particles from each other to infinity.
Uses Gravitational potential energy has a number of practical uses, notably the generation of hydroelectricity. For example in Dinorwig, Wales, there are two lakes, one at a higher elevation than the other. At times when surplus electricity is not required (and so is comparatively cheap), water is pumped up to the higher lake, thus converting the electrical energy (running the pump) to gravitational potential energy. At times of peak demand for electricity, the water flows back down through electrical generator turbines, converting the potential energy into kinetic energy and then back into electricity. (The process is not completely efficient and much of the original energy from the surplus electricity is in fact lost to friction.) Gravitational potential energy is also used to power clocks in which falling weights operate the mechanism.
Elastic potential energy
This catapult makes use of elastic potential energy. Elastic potential energy is the potential energy of an elastic object (for example a bow or a catapult) that is deformed under tension or compression (or stressed in formal terminology). It arises as a consequence of a force that tries to restore the object to its original shape, which is most often the electromagnetic force between the atoms and molecules that constitute the object. If the stretch is released, the energy is transformed into kinetic energy.
Calculation of elastic potential energy The elastic potential energy stored in a stretched spring can be calculated by finding the work necessary to stretch the spring a distance x from its unstretched length:
an ideal spring will follow Hooke's Law:
The work done (and therefore the stored potential energy) will then be:
The equation is often used in calculations of positions of mechanical equilibrium. More involved calculations can be found at elastic potential energy.
Chemical potential energy Chemical potential energy is a form of potential energy related to the structural arrangement of atoms or molecules. This arrangement may be the result of chemical bonds within a molecule or otherwise. Chemical energy of a chemical substance can be transformed to other forms of energy by a chemical reaction. As an example, when a fuel is burned the chemical energy is converted to heat, same is the case with digestion of food metabolized in a biological organism. Green plants transform solar energy to chemical energy through the process known as photosynthesis, and electrical energy can be converted to chemical energy through electrochemical reactions. The similar term chemical potential is used by chemists to indicate the potential of a substance to undergo a chemical reaction.
Electrical potential energy
An object can have potential energy by virtue of its electric charge and several forces related to their presence. There are two main types of this kind of potential energy: electrostatic potential energy, electrodynamic potential energy (also sometimes called magnetic potential energy).
Plasma formed inside a gas filled sphere.
Electrostatic potential energy In case the electric charge of an object can be assumed to be at rest, it has potential energy due to its position relative to other charged objects. The electrostatic potential energy is the energy of an electrically charged particle (at rest) in an electric field. It is defined as the work that must be done to move it from an infinite
distance away to its present location, in the absence of any nonelectrical forces on the object. This energy is nonzero if there is another electrically charged object nearby. The simplest example is the case of two pointlike objects A1 and A2 with electrical charges q1 and q2. The work W required to move A1 from an infinite distance to a distance r away from A2 is given by:
where ε0 is the electric constant. This equation is obtained by integrating the Coulomb force between the limits of infinity and r. A related quantity called electric potential (commonly denoted with a V for voltage) is equal to the electric potential energy per unit charge.
Electrodynamic potential energy In case a charged object or its constituent charged particles are not at rest, it generates a magnetic field giving rise to yet another form of potential energy, often termed as magnetic potential energy. This kind of potential energy is a result of the phenomenon magnetism, whereby an object that is magnetic has the potential to move other similar objects. Magnetic objects are said to have some magnetic moment. Magnetic fields and their effects are best studied under electrodynamics.
Nuclear potential energy Nuclear potential energy is the potential energy of the particles inside an atomic nucleus. The nuclear particles are bound together by the strong nuclear force. Weak nuclear forces provide the potential energy for certain kinds of radioactive decay, such as beta decay. Nuclear particles like protons and neutrons are not destroyed in fission and fusion processes, but collections of them have less mass than if they were individually free, and this mass difference is liberated as heat and radiation in nuclear reactions (the heat and radiation have the missing mass, but it often escapes from the system, where it is not measured). The energy from the Sun is an example of this form of energy conversion. In the Sun, the process of hydrogen fusion converts about 4 million tonnes of solar matter per second into electromagnetic energy, which is radiated into space.
Relation between potential energy, potential and force Potential energy is closely linked with forces. If the work done moving along a path which starts and ends in the same location is zero, then the force is said to be
conservative and it is possible to define a numerical value of potential associated with every point in space. A force field can be reobtained by taking the negative of the vector gradient of the potential field. For example, gravity is a conservative force. The associated potential is the gravitational potential, often denoted by φ or V, corresponding to the energy per unit mass as a function of position. The gravitational potential energy of two particles of mass M and m separated by a distance r is
The gravitational potential (specific energy) of the two bodies is
where μ is the reduced mass. The work done against gravity by moving a infinitesimal mass from point A with U = a to point B with U = b is (b − a) and the work done going back the other way is (a − b) so that the total work done in moving from A to B and returning to A is
If the potential is redefined at A to be a + c and the potential at B to be b + c, where c is a constant (i.e. c can be any number, positive or negative, but it must be the same at A as it is at B) then the work done going from A to B is
as before. In practical terms, this means that one can set the zero of U and φ anywhere one likes. One may set it to be zero at the surface of the Earth, or may find it more convenient to set zero at infinity (as in the expressions given earlier in this section). A thing to note about conservative forces is that the work done going from A to B does not depend on the route taken. If it did then it would be pointless to define a potential at each point in space. An example of a nonconservative force is friction. With friction, the route taken does affect the amount of work done, and it makes little sense to define a potential associated with friction. All the examples above are actually force field stored energy (sometimes in disguise). For example in elastic potential energy, stretching an elastic material forces the atoms very
slightly further apart. The equilibrium between electromagnetic forces and Pauli repulsion of electrons (they are fermions obeying Fermi statistics) is slightly violated resulting in a small returning force. Scientists rarely discuss forces on an atomic scale. Often interactions are described in terms of energy rather than force. One may think of potential energy as being derived from force or think of force as being derived from potential energy (though the latter approach requires a definition of energy that is independent from force which does not currently exist). A conservative force can be expressed in the language of differential geometry as a closed form. As Euclidean space is contractible, its de Rham cohomology vanishes, so every closed form is also an exact form, and can be expressed as the gradient of a scalar field. This gives a mathematical justification of the fact that all conservative forces are gradients of a potential field.
Chapter 3
Conservation of Energy
The law of conservation of energy is an empirical law of physics. It states that the total amount of energy in an isolated system remains constant over time (is said to be conserved over time). A consequence of this law is that energy can neither be created nor destroyed: it can only be transformed from one state to another. The only thing that can happen to energy in a closed system is that it can change form: for instance chemical energy can become kinetic energy. Albert Einstein's theory of relativity shows that energy and mass are the same thing, and that neither one appears without the other. Thus in closed systems, both mass and energy are conserved separately, just as was understood in prerelativistic physics. The new feature of relativistic physics is that "matter" particles (such as those constituting atoms) could be converted to nonmatter forms of energy, such as light; or kinetic and potential energy (example: heat). However, this conversion does not affect the total mass of systems, because the latter forms of nonmatter energy still retain their mass through any such conversion. Today, conservation of “energy” refers to the conservation of the total system energy over time. This energy includes the energy associated with the rest mass of particles and all other forms of energy in the system. In addition, the invariant mass of systems of particles (the mass of the system as seen in its center of mass inertial frame, such as the frame in which it would need to be weighed) is also conserved over time for any single observer, and (unlike the total energy) is the same value for all observers. Therefore, in an isolated system, although matter (particles with rest mass) and "pure energy" (heat and light) can be converted to one another, both the total amount of energy and the total amount of mass of such systems remain constant over time, as seen by any single observer. If energy in any form is allowed to escape such systems, the mass of the system will decrease in correspondence with the loss. A consequence of the law of energy conservation is that perpetual motion machines can only work perpetually if they deliver no energy to their surroundings.
History Ancient philosophers as far back as Thales of Miletus had inklings of the conservation of which everything is made. However, there is no particular reason to identify this with what we know today as "massenergy" (for example, Thales thought it was water). In 1638, Galileo published his analysis of several situations—including the celebrated "interrupted pendulum"—which can be described (in modern language) as conservatively converting potential energy to kinetic energy and back again. It was Gottfried Wilhelm Leibniz during 1676–1689 who first attempted a mathematical formulation of the kind of energy which is connected with motion (kinetic energy). Leibniz noticed that in many mechanical systems (of several masses, mi each with velocity vi ),
was conserved so long as the masses did not interact. He called this quantity the vis viva or living force of the system. The principle represents an accurate statement of the approximate conservation of kinetic energy in situations where there is no friction. Many physicists at that time held that the conservation of momentum, which holds even in systems with friction, as defined by the momentum:
was the conserved vis viva. It was later shown that, under the proper conditions, both quantities are conserved simultaneously such as in elastic collisions. It was largely engineers such as John Smeaton, Peter Ewart, Karl Hotzmann, GustaveAdolphe Hirn and Marc Seguin who objected that conservation of momentum alone was not adequate for practical calculation and who made use of Leibniz's principle. The principle was also championed by some chemists such as William Hyde Wollaston. Academics such as John Playfair were quick to point out that kinetic energy is clearly not conserved. This is obvious to a modern analysis based on the second law of thermodynamics but in the 18th and 19th centuries, the fate of the lost energy was still unknown. Gradually it came to be suspected that the heat inevitably generated by motion under friction, was another form of vis viva. In 1783, Antoine Lavoisier and PierreSimon Laplace reviewed the two competing theories of vis viva and caloric theory. Count Rumford's 1798 observations of heat generation during the boring of cannons added more weight to the view that mechanical motion could be converted into heat, and (as importantly) that the conversion was quantitative and could be predicted (allowing for a universal conversion constant between kinetic energy and heat). Vis viva now started to be known as energy, after the term was first used in that sense by Thomas Young in 1807. The recalibration of vis viva to
which can be understood as finding the exact value for the kinetic energy to work conversion constant, was largely the result of the work of GaspardGustave Coriolis and JeanVictor Poncelet over the period 1819–1839. The former called the quantity quantité de travail (quantity of work) and the latter, travail mécanique (mechanical work), and both championed its use in engineering calculation. In a paper Über die Natur der Wärme, published in the Zeitschrift für Physik in 1837, Karl Friedrich Mohr gave one of the earliest general statements of the doctrine of the conservation of energy in the words: "besides the 54 known chemical elements there is in the physical world one agent only, and this is called Kraft [energy or work]. It may appear, according to circumstances, as motion, chemical affinity, cohesion, electricity, light and magnetism; and from any one of these forms it can be transformed into any of the others." A key stage in the development of the modern conservation principle was the demonstration of the mechanical equivalent of heat. The caloric theory maintained that heat could neither be created nor destroyed but conservation of energy entails the contrary principle that heat and mechanical work are interchangeable. The mechanical equivalence principle was first stated in its modern form by the German surgeon Julius Robert von Mayer. Mayer reached his conclusion on a voyage to the Dutch East Indies, where he found that his patients' blood was a deeper red because they were consuming less oxygen, and therefore less energy, to maintain their body temperature in the hotter climate. He had discovered that heat and mechanical work were both forms of energy, and later, after improving his knowledge of physics, he calculated a quantitative relationship between them.
Joule's apparatus for measuring the mechanical equivalent of heat. A descending weight attached to a string causes a paddle immersed in water to rotate. Meanwhile, in 1843 James Prescott Joule independently discovered the mechanical equivalent in a series of experiments. In the most famous, now called the "Joule apparatus", a descending weight attached to a string caused a paddle immersed in water to rotate. He showed that the gravitational potential energy lost by the weight in descending was equal to the thermal energy (heat) gained by the water by friction with the paddle. Over the period 1840–1843, similar work was carried out by engineer Ludwig A. Colding though it was little known outside his native Denmark. Both Joule's and Mayer's work suffered from resistance and neglect but it was Joule's that, perhaps unjustly, eventually drew the wider recognition. In 1844, William Robert Grove postulated a relationship between mechanics, heat, light, electricity and magnetism by treating them all as manifestations of a single "force" (energy in modern terms). Grove published his theories in his book The Correlation of Physical Forces. In 1847, drawing on the earlier work of Joule, Sadi Carnot and Émile
Clapeyron, Hermann von Helmholtz arrived at conclusions similar to Grove's and published his theories in his book Über die Erhaltung der Kraft (On the Conservation of Force, 1847). The general modern acceptance of the principle stems from this publication. In 1877, Peter Guthrie Tait claimed that the principle originated with Sir Isaac Newton, based on a creative reading of propositions 40 and 41 of the Philosophiae Naturalis Principia Mathematica. This is now regarded as an example of Whig history.
The first law of thermodynamics The first law of thermodynamics is an expression of the principle of conservation of energy. The law expresses that energy can be transformed, i.e. changed from one form to another, but cannot be created nor destroyed. It is usually formulated by stating that the change in the internal energy of a system is equal to the amount of heat supplied to the system, minus the amount of work performed by the system on its surroundings.
Classical statement The first explicit statement of the first law of thermodynamics was given by Rudolf Clausius in 1850: "There is a state function E, called ‘energy’, whose differential equals the work exchanged with the surroundings during an adiabatic process."
Description The first law of thermodynamics says that energy is conserved in any process involving a thermodynamic system and its surroundings. Frequently it is convenient to focus on changes in the assumed internal energy (U) and to regard them as due to a combination of heat (Q) added to the system and work done by the system (W). Taking dU as an infinitesimal (differential) change in internal energy, one writes
where δQ and δW are infinitesimal amounts of heat supplied to the system and work done by the system, respectively. Note that the minus sign in front of δW indicates that a positive amount of work done by the system leads to energy being lost from the system. When a system expands in a quasistatic process, the work done on the environment is the product of pressure (P) and volume (V) change, i.e. PdV, whereas the work done on the system is PdV. The change in internal energy of the system is:
Work and heat are due to processes which add or subtract energy, while U is a particular form of energy associated with the system. Thus the term heat for δQ means that amount of energy added as the result of heating, rather than referring to a particular form of energy. Likewise, work energy for δw means "that amount of energy lost as the result of work". Internal energy is a property of the system whereas work done and heat supplied are not. A significant result of this distinction is that a given internal energy change (dU) can be achieved by, in principle, many combinations of heat and work. Informally, the law was first formulated by Germain Hess via Hess's Law, and later by Julius Robert von Mayer
Adiabatic processes The classical statement of the first law of thermodynamics is induced from empirical evidence. It can be observed that given a system in an initial state, if work is exerted on the system in an adiabatic (i.e. thermally insulated) way, the final state is the same for a given amount of work, irrespective of how this work is performed. For instance, in Joule's experiment, the initial system is a tank of water with a paddle wheel inside. If we isolate thermally the tank and move the paddle wheel with a pulley and a weight we can relate the increase in temperature with the height descended by the mass. Now the system is returned to its initial state, isolated again, and the same amount of work is done on the tank using different devices (an electric motor, a chemical battery, a spring,...). In every case, the amount of work can be measured independently. The evidence shows that the final state of the water (in particular, its temperature) is the same in every case. It's irrelevant if the work is electrical, mechanical, chemical,... or if done suddenly or slowly, as long as it is performed in an adiabatic way. This evidence leads to the classical statement of the first law of thermodynamics For all adiabatic processes between two specified states of a closed system, the net work done is the same regardless of the nature of the closed system and the details of the process. This affirmation of path independence allows to define a state function, named internal energy, U, as the adiabatic work necessary to go from a reference state to a given one
where, following IUPAC convention we take as positive the work done on the system. To go from a state A to a state B we can take a path that goes through the reference state, since the adiabatic work is independent of the path
In particular, if no work is exerted on a thermally isolated system we have
or, in words, The internal energy of an isolated system remains constant. This is the law of conservation of energy.
Non adiabatic processes When the system evolves in a non adiabatic way, it is observed that the work exerted on the system does not coincide with the increase in its internal energy, which, being a state function, can be used for both adiabatic and nonadiabatic processes.
The difference is interpreted in terms of the heat that enters the system, so the inequality can be transformed in an equality as
This is the usual expression the first law of thermodynamics. The inclusion of an unknown term (heat) does not transform it in a tautology, since its real physical content lies in the fact that there exists a state function that can be calculated independently of heat and work.
State functional formulation The infinitesimal heat and work in the equations above are denoted by δ, rather than exact differentials denoted by d, because they do not describe the state of any system. The integral of an inexact differential depends upon the particular path taken through the space of thermodynamic parameters while the integral of an exact differential depends only upon the initial and final states. If the initial and final states are the same, then the integral of an inexact differential may or may not be zero, but the integral of an exact differential will always be zero. The path taken by a thermodynamic system through a chemical or physical change is known as a thermodynamic process. An expression of the first law can be written in terms of exact differentials by realizing that the work that a system does is, in case of a reversible process, equal to its pressure times the infinitesimal change in its volume. In other words δw = PdV where P is pressure and V is volume. Also, for a reversible process, the total amount of heat added to
a system can be expressed as δQ = TdS where T is temperature and S is entropy. Therefore, for a reversible process:
Since U, S and V are thermodynamic functions of state, the above relation holds also for nonreversible changes. The above equation is known as the fundamental thermodynamic relation. In the case where the number of particles in the system is not necessarily constant and may be of different types, the first law is written:
where dNi is the (small) number of typei particles added to the system, and μi is the amount of energy added to the system when one typei particle is added, where the energy of that particle is such that the volume and entropy of the system remains unchanged. μi is known as the chemical potential of the typei particles in the system. The statement of the first law, using exact differentials is now:
If the system has more external variables than just the volume that can change, the fundamental thermodynamic relation generalizes to:
Here the Xi are the generalized forces corresponding to the external variables xi. A useful idea from mechanics is that the energy gained by a particle is equal to the force applied to the particle multiplied by the displacement of the particle while that force is applied. Now consider the first law without the heating term: dU = − PdV. The pressure P can be viewed as a force (and in fact has units of force per unit area) while dV is the displacement (with units of distance times area). We may say, with respect to this work term, that a pressure difference forces a transfer of volume, and that the product of the two (work) is the amount of energy transferred as a result of the process. It is useful to view the TdS term in the same light: With respect to this heat term, a temperature difference forces a transfer of entropy, and the product of the two (heat) is the amount of energy transferred as a result of the process. Here, the temperature is known as a "generalized" force (rather than an actual mechanical force) and the entropy is a generalized displacement.
Similarly, a difference in chemical potential between groups of particles in the system forces a transfer of particles, and the corresponding product is the amount of energy transferred as a result of the process. For example, consider a system consisting of two phases: liquid water and water vapor. There is a generalized "force" of evaporation which drives water molecules out of the liquid. There is a generalized "force" of condensation which drives vapor molecules out of the vapor. Only when these two "forces" (or chemical potentials) are equal will there be equilibrium, and the net transfer will be zero. The two thermodynamic parameters which form a generalized forcedisplacement pair are termed "conjugate variables". The two most familiar pairs are, of course, pressurevolume, and temperatureentropy. Entropy is a function of a quantity of heat which shows the possibility of conversion of that heat into work. For a thermodynamic system with a fixed number of particles, the first law of thermodynamics may be stated as: , or equivalently, where δQ is the amount of energy added to the system by a heating process, δW is the amount of energy lost by the system due to work done by the system on its surroundings and dU is the change in the internal energy of the system. The δ's before the heat and work terms are used to indicate that they describe an increment of energy which is to be interpreted somewhat differently than the dU increment of internal energy. Work and heat are processes which add or subtract energy, while the internal energy U is a particular form of energy associated with the system. Thus the term "heat energy" for δQ means "that amount of energy added as the result of heating" rather than referring to a particular form of energy. Likewise, the term "work energy" for δW means "that amount of energy lost as the result of work". The most significant result of this distinction is the fact that one can clearly state the amount of internal energy possessed by a thermodynamic system, but one cannot tell how much energy has flowed into or out of the system as a result of its being heated or cooled, nor as the result of work being performed on or by the system. In simple terms, this means that energy cannot be created or destroyed, only converted from one form to another. For a simple compressible system, the work performed by the system may be written
where P is the pressure and dV is a small change in the volume of the system, each of which are system variables. The heat energy may be written
where T is the temperature and dS is a small change in the entropy of the system. Temperature and entropy are also system variables.
Mechanics In mechanics, conservation of energy is usually stated as
where T is kinetic and V potential energy. Actually this is the particular case of the more general conservation law
and where L is the Lagrangian function. For this particular form to be valid, the following must be true: • • •
The system is scleronomous (neither kinetic nor potential energy are explicit functions of time) The kinetic energy is a quadratic form with regard to velocities. The potential energy doesn't depend on velocities.
Noether's theorem Noether's (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918. The action of a physical system is the integral over time of a Lagrangian function (which may or may not be an integral over space of a Lagrangian density function), from which the system's behavior can be determined by the principle of least action. Noether's theorem has become a fundamental tool of modern theoretical physics and the calculus of variations. A generalization of the seminal formulations on constants of motion in Lagrangian and Hamiltonian mechanics (1788 and 1833, respectively), it does not apply to systems that cannot be modeled with a Lagrangian; for example, dissipative systems with continuous symmetries need not have a corresponding conservation law. For illustration, if a physical system behaves the same regardless of how it is oriented in space, its Lagrangian is rotationally symmetric; from this symmetry, Noether's theorem shows the angular momentum of the system must be conserved. The physical system itself need not be symmetric; a jagged asteroid tumbling in space conserves angular momentum despite its asymmetry – it is the laws of motion that are symmetric. As
another example, if a physical experiment has the same outcome regardless of place or time (having the same outcome, say, somewhere in Asia on a Tuesday or in America on a Wednesday), then its Lagrangian is symmetric under continuous translations in space and time; by Noether's theorem, these symmetries account for the conservation laws of linear momentum and energy within this system, respectively. (These examples are just for illustration; in the first one, Noether's theorem added nothing new – the results were known to follow from Lagrange's equations and from Hamilton's equations.) Noether's theorem is important, both because of the insight it gives into conservation laws, and also as a practical calculational tool. It allows researchers to determine the conserved quantities from the observed symmetries of a physical system. Conversely, it allows researchers to consider whole classes of hypothetical Lagrangians to describe a physical system. For illustration, suppose that a new field is discovered that conserves a quantity X. Using Noether's theorem, the types of Lagrangians that conserve X because of a continuous symmetry can be determined, and then their fitness judged by other criteria. There are numerous different versions of Noether's theorem, with varying degrees of generality. The original version only applied to ordinary differential equations (particles) and not partial differential equations (fields). The original versions also assume that the Lagrangian only depends upon the first derivative, while later versions generalize the theorem to Lagrangians depending on the nth derivative. There is also a quantum version of this theorem, known as the Ward–Takahashi identity. Generalizations of Noether's theorem to superspaces also exist.
Informal statement of the theorem All fine technical points aside, Noether's theorem can be stated informally If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time. A more sophisticated version of the theorem states that: To every differentiable symmetry generated by local actions, there corresponds a conserved current. The word "symmetry" in the above statement refers more precisely to the covariance of the form that a physical law takes with respect to a onedimensional Lie group of transformations satisfying certain technical criteria. The conservation law of a physical quantity is usually expressed as a continuity equation. The formal proof of the theorem uses only the condition of invariance to derive an expression for a current associated with a conserved physical quantity. The conserved quantity is called the Noether charge and the flow carrying that 'charge' is called the Noether current. The Noether current is defined up to a solenoidal vector field.
Historical context A conservation law states that some quantity X describing a system remains constant throughout its motion; expressed mathematically, the rate of change of X (its derivative with respect to time) is zero:
Such quantities are said to be conserved; they are often called constants of motion, although motion per se need not be involved, just evolution in time. For example, if the energy of a system is conserved, its energy is constant at all times, which imposes a constraint on the system's motion and may help to solve for it. Aside from the insight that such constants of motion give into the nature of a system, they are a useful calculational tool; for example, an approximate solution can be corrected by finding the nearest state that satisfies the necessary conservation laws. The earliest constants of motion discovered were momentum and energy, which were proposed in the 17th century by René Descartes and Gottfried Leibniz on the basis of collision experiments, and refined by subsequent researchers. Isaac Newton was the first to enunciate the conservation of momentum in its modern form, and showed that it was a consequence of Newton's third law; interestingly, conservation of momentum still holds even in situations when Newton's third law is incorrect. Modern physics has revealed that the conservation laws of momentum and energy are only approximately true, but their modern refinements – the conservation of fourmomentum in special relativity and the zero covariant divergence of the stressenergy tensor in general relativity – are rigorously true within the limits of those theories. The conservation of angular momentum, a generalization to rotating rigid bodies, likewise holds in modern physics. Another important conserved quantity, discovered in studies of the celestial mechanics of astronomical bodies, was the Laplace–Runge–Lenz vector. In the late 18th and early 19th centuries, physicists developed more systematic methods for discovering conserved quantities. A major advance came in 1788 with the development of Lagrangian mechanics, which is related to the principle of least action. In this approach, the state of the system can be described by any type of generalized coordinates q; the laws of motion need not be expressed in a Cartesian coordinate system, as was customary in Newtonian mechanics. The action is defined as the time integral I of a function known as the Lagrangian L
where the dot over q signifies the rate of change of the coordinates q
Hamilton's principle states that the physical path q(t) – the one truly taken by the system – is a path for which infinitesimal variations in that path cause no change in I, at least up to first order. This principle results in the Euler–Lagrange equations
Thus, if one of the coordinates, say qk, does not appear in the Lagrangian, the righthand side of the equation is zero, and the lefthand side shows that
where the conserved momentum pk is defined as the lefthand quantity in parentheses. The absence of the coordinate qk from the Lagrangian implies that the Lagrangian is unaffected by changes or transformations of qk; the Lagrangian is invariant, and is said to exhibit a kind of symmetry. This is the seed idea from which Noether's theorem was born. Several alternative methods for finding conserved quantities were developed in the 19th century, especially by William Rowan Hamilton. For example, he developed a theory of canonical transformations that allowed researchers to change coordinates so that coordinates disappeared from the Lagrangian, resulting in conserved quantities. Another approach and perhaps the most efficient for finding conserved quantities is the Hamilton– Jacobi equation.
Example 1: Conservation of energy Looking at the specific case of a Newtonian particle of mass m, coordinate x, moving under the influence of a potential V, coordinatized by time t. The action, S, is:
Consider the generator of time translations
. In other words,
. Note that x has an explicit dependence on time, whilst V does not; consequently:
so we can set
Then,
The right hand side is the energy and Noether's theorem states that (i.e. the principle of conservation of energy is a consequence of invariance under time translations). More generally, if the Lagrangian does not depend explicitly on time, the quantity
(called the Hamiltonian) is conserved.
Example 2: Conservation of center of momentum Still considering 1dimensional time, let
i.e. N Newtonian particles where the potential only depends pairwise upon the relative displacement. For , let's consider the generator of Galilean transformations (i.e. a change in the frame of reference). In other words,
Note that
This has the form of
so we can set
Then,
where is the total momentum, M is the total mass and Noether's theorem states:
is the center of mass.
Example 3: Conformal transformation Both examples 1 and 2 are over a 1dimensional manifold (time). An example involving spacetime is a conformal transformation of a massless real scalar field with a quartic potential in (3 + 1)Minkowski spacetime.
For Q, consider the generator of a spacetime rescaling. In other words,
The second term on the right hand side is due to the "conformal weight" of φ. Note that
This has the form of
(where we have performed a change of dummy indices) so set
Then,
Noether's theorem states that (as one may explicitly check by substituting the Euler–Lagrange equations into the left hand side). (Aside: If one tries to find the Ward–Takahashi analog of this equation, one runs into a problem because of anomalies.)
Applications Application of Noether's theorem allows physicists to gain powerful insights into any general theory in physics, by just analyzing the various transformations that would make the form of the laws involved invariant. For example:
•
• •
the invariance of physical systems with respect to spatial translation (in other words, that the laws of physics do not vary with locations in space) gives the law of conservation of linear momentum; invariance with respect to rotation gives the law of conservation of angular momentum; invariance with respect to time translation gives the wellknown law of conservation of energy
In quantum field theory, the analog to Noether's theorem, the Ward–Takahashi identity, yields further conservation laws, such as the conservation of electric charge from the invariance with respect to a change in the phase factor of the complex field of the charged particle and the associated gauge of the electric potential and vector potential. The Noether charge is also used in calculating the entropy of stationary black holes. The conservation of energy is a common feature in many physical theories. From a mathematical point of view it is understood as a consequence of Noether's theorem, which states every continuous symmetry of a physical theory has an associated conserved quantity; if the theory's symmetry is time invariance then the conserved quantity is called "energy". The energy conservation law is a consequence of the shift symmetry of time; energy conservation is implied by the empirical fact that the laws of physics do not change with time itself. Philosophically this can be stated as "nothing depends on time per se". In other words, if the theory is invariant under the continuous symmetry of time translation then its energy (which is canonical conjugate quantity to time) is conserved. Conversely, theories which are not invariant under shifts in time (for example, systems with time dependent potential energy) do not exhibit conservation of energy – unless we consider them to exchange energy with another, external system so that the theory of the enlarged system becomes time invariant again. Since any timevarying theory can be embedded within a timeinvariant metatheory energy, conservation can always be recovered by a suitable redefinition of what energy is. Thus conservation of energy for finite systems is valid in such modern physical theories as special relativity and quantum theory (including QED) in the flat spacetime.
Relativity With the discovery of special relativity by Albert Einstein, energy was proposed to be one component of an energymomentum 4vector. Each of the four components (one of energy and three of momentum) of this vector is separately conserved across time, in any closed system, as seen from any given inertial reference frame. Also conserved is the vector length (Minkowski norm), which is the rest mass for single particles, and the invariant mass for systems of particles (where momenta and energy are separately summed before the length is calculated). The relativistic energy of a single massive particle contains a term related to its rest mass in addition to its kinetic energy of motion. In the limit of zero kinetic energy (or equivalently in the rest frame) of a massive particle; or else in the center of momentum
frame for objects or systems which retain kinetic energy, the total energy of particle or object (including internal kinetic energy in systems) is related to its rest mass or its invariant mass via the famous equation E = mc2. Thus, the rule of conservation of energy over time in special relativity continues to hold, so long as the reference frame of the observer is unchanged. This applies to the total energy of systems, although different observers disagree as to the energy value. Also conserved, and invariant to all observers, is the invariant mass, which is the minimal system mass and energy that can be seen by any observer, and which is defined by the energy–momentum relation. In general relativity conservation of energymomentum is expressed with the aid of a stressenergymomentum pseudotensor.
Quantum theory In quantum mechanics, energy is defined as proportional to the time derivative of the wave function. Lack of commutativity of the time derivative operator with the time operator itself mathematically results in an uncertainty principle for time and energy: the longer the period of time, the more precisely energy can be defined (energy and time become a conjugate Fourier pair).
Chapter 4
Mechanical Work
In physics, mechanical work is the amount of energy transferred by a force acting through a distance. Like energy, it is a scalar quantity, with SI units of joules. The term work was first coined in 1826 by the French mathematician GaspardGustave Coriolis. According to the workenergy theorem if an external force acts upon a rigid object, causing its kinetic energy to change from Ek1 to Ek2, then the mechanical work (W) is given by:
where m is the mass of the object and v is the object's velocity. If the resultant force F on an object acts while the object is displaced a distance d, and the force and displacement act parallel to each other, the mechanical work done on the object is the dot product of the vectors F and d:
If the force and the displacement are parallel and in the same direction (θ = 0), the mechanical work is positive. If the force and the displacement are parallel but in opposite directions (i.e. antiparallel, θ = 180), the mechanical work is negative. If a force F is applied at an angle θ, only the component of the force in the same direction as the displacement (Fcosθ) does work. Thus, if the force acts perpendicular to the displacement (θ = 90 or 270), zero work is done by the force.
Units The SI unit of work is the joule (J), which is defined as the work done by a force of one newton acting over a distance of one meter. This definition is based on Sadi Carnot's 1824 definition of work as "weight lifted through a height", which is based on the fact that early steam engines were principally used to lift buckets of water, through a
gravitational height, out of flooded ore mines. The dimensionally equivalent newtonmeter (N·m) is sometimes used instead; however, it is also sometimes reserved for torque to distinguish its units from work or energy. NonSI units of work include the erg, the footpound, the footpoundal, and the literatmosphere. Heat conduction is not considered to be a form of work, since the energy is transferred into atomic vibration rather than a macroscopic displacement.
Zero work
A baseball pitcher does positive work on the ball by transferring energy into it. Work can be zero even when there is a force. The centripetal force in a uniform circular motion, for example, does zero work since the kinetic energy of the moving object doesn't change. This is because the force is always perpendicular to the motion of the object; only the component of a force parallel to the velocity vector of an object can do work on that object. Likewise when a book sits on a table, the table does no work on the book despite exerting a force equivalent to mg upwards, because no energy is transferred into or out of the book.
Mathematical calculation Force and displacement Force and displacement are both vector quantities and they are combined using the dot product to evaluate the mechanical work, a scalar quantity: (1) where is the angle between the force and the displacement vector.
In order for this formula to be valid, the force and angle must remain constant. The object's path must always remain on a single, straight line, though it may change directions while moving along the line. In situations where the force changes over time, or the path deviates from a straight line, equation (1) is not generally applicable although it is possible to divide the motion into small steps, such that the force and motion are well approximated as being constant for each step, and then to express the overall work as the sum over these steps. The general definition of mechanical work is given by the following line integral:
(2) where: is the path or curve traversed by the object; is the force vector; and is the position vector. The expression calculation of
is an inexact differential which means that the is pathdependent and cannot be differentiated to give .
Equation (2) explains how a nonzero force can do zero work. The simplest case is where the force is always perpendicular to the direction of motion, making the integrand always zero. This is what happens during circular motion. However, even if the integrand sometimes takes nonzero values, it can still integrate to zero if it is sometimes negative and sometimes positive. The possibility of a nonzero force doing zero work illustrates the difference between work and a related quantity, impulse, which is the integral of force over time. Impulse measures change in a body's momentum, a vector quantity sensitive to direction, whereas work considers only the magnitude of the velocity. For instance, as an object in uniform circular motion traverses half of a revolution, its centripetal force does no work, but it transfers a nonzero impulse.
Torque and rotation Work done by a torque can be calculated in a similar manner. A torque applied through a revolution of , expressed in radians, does work as follows:
Frame of reference
The work done by a force acting on an object depends on the inertial frame of reference, because the distance covered while applying the force does. Due to Newton's law of reciprocal actions there is a reaction force; it does work depending on the inertial frame of reference in an opposite way. The total work done is independent of the inertial frame of reference.
Chapter 5
Electromagnetism
Electromagnetism is one of the four fundamental interactions of nature. The other three are the strong interaction, the weak interaction and gravitation. Electromagnetism is the force that causes the interaction between electrically charged particles; the areas in which this happens are called electromagnetic fields. Electromagnetism is responsible for practically all the phenomena encountered in daily life, with the exception of gravity. Ordinary matter takes its form as a result of intermolecular forces between individual molecules in matter. Electromagnetism is also the force which holds electrons and protons together inside atoms, which are the building blocks of molecules. This governs the processes involved in chemistry, which arise from interactions between the electrons orbiting atoms. Electromagnetism manifests as both electric fields and magnetic fields. Both fields are simply different aspects of electromagnetism, and hence are intrinsically related. Thus, a changing electric field generates a magnetic field; conversely a changing magnetic field generates an electric field. This effect is called electromagnetic induction, and is the basis of operation for electrical generators, induction motors, and transformers. Mathematically speaking, magnetic fields and electric fields are convertible with relative motion as a four vector. Electric fields are the cause of several common phenomena, such as electric potential (such as the voltage of a battery) and electric current (such as the flow of electricity through a flashlight). Magnetic fields are the cause of the force associated with magnets. In quantum electrodynamics, electromagnetic interactions between charged particles can be calculated using the method of Feynman diagrams, in which we picture messenger particles called virtual photons being exchanged between charged particles. This method can be derived from the field picture through perturbation theory.
The theoretical implications of electromagnetism led to the development of special relativity by Albert Einstein in 1905.
History of the theory Originally electricity and magnetism were thought of as two separate forces. This view changed, however, with the publication of James Clerk Maxwell's 1873 Treatise on Electricity and Magnetism in which the interactions of positive and negative charges were shown to be regulated by one force. There are four main effects resulting from these interactions, all of which have been clearly demonstrated by experiments: 1. Electric charges attract or repel one another with a force inversely proportional to the square of the distance between them: unlike charges attract, like ones repel. 2. Magnetic poles (or states of polarization at individual points) attract or repel one another in a similar way and always come in pairs: every north pole is yoked to a south pole. 3. An electric current in a wire creates a circular magnetic field around the wire, its direction depending on that of the current. 4. A current is induced in a loop of wire when it is moved towards or away from a magnetic field, or a magnet is moved towards or away from it, the direction of current depending on that of the movement. While preparing for an evening lecture on 21 April 1820, Hans Christian Ørsted made a surprising observation. As he was setting up his materials, he noticed a compass needle deflected from magnetic north when the electric current from the battery he was using was switched on and off. This deflection convinced him that magnetic fields radiate from all sides of a wire carrying an electric current, just as light and heat do, and that it confirmed a direct relationship between electricity and magnetism. At the time of discovery, Ørsted did not suggest any satisfactory explanation of the phenomenon, nor did he try to represent the phenomenon in a mathematical framework. However, three months later he began more intensive investigations. Soon thereafter he published his findings, proving that an electric current produces a magnetic field as it flows through a wire. The CGS unit of magnetic induction (oersted) is named in honor of his contributions to the field of electromagnetism. His findings resulted in intensive research throughout the scientific community in electrodynamics. They influenced French physicist AndréMarie Ampère's developments of a single mathematical form to represent the magnetic forces between currentcarrying conductors. Ørsted's discovery also represented a major step toward a unified concept of energy. This unification, which was observed by Michael Faraday, extended by James Clerk Maxwell, and partially reformulated by Oliver Heaviside and Heinrich Hertz, is one of the key accomplishments of 19th century mathematical physics. It had farreaching consequences, one of which was the understanding of the nature of light. Light and other
electromagnetic waves take the form of quantized, selfpropagating oscillatory electromagnetic field disturbances called photons. Different frequencies of oscillation give rise to the different forms of electromagnetic radiation, from radio waves at the lowest frequencies, to visible light at intermediate frequencies, to gamma rays at the highest frequencies. Ørsted was not the only person to examine the relation between electricity and magnetism. In 1802 Gian Domenico Romagnosi, an Italian legal scholar, deflected a magnetic needle by electrostatic charges. Actually, no galvanic current existed in the setup and hence no electromagnetism was present. An account of the discovery was published in 1802 in an Italian newspaper, but it was largely overlooked by the contemporary scientific community.
Overview The electromagnetic force is one of the four fundamental forces. The other fundamental forces are: the strong nuclear force (which holds quarks together, along with its residual strong force effect that holds atomic nuclei together, to form the nucleus), the weak nuclear force (which causes certain forms of radioactive decay), and the gravitational force. All other forces (e.g. friction) are ultimately derived from these fundamental forces. The electromagnetic force is the one responsible for practically all the phenomena one encounters in daily life, with the exception of gravity. Roughly speaking, all the forces involved in interactions between atoms can be traced to the electromagnetic force acting on the electrically charged protons and electrons inside the atoms. This includes the forces we experience in "pushing" or "pulling" ordinary material objects, which come from the intermolecular forces between the individual molecules in our bodies and those in the objects. It also includes all forms of chemical phenomena, which arise from interactions between electron orbitals.
Classical electrodynamics The scientist William Gilbert proposed, in his De Magnete (1600), that electricity and magnetism, while both capable of causing attraction and repulsion of objects, were distinct effects. Mariners had noticed that lightning strikes had the ability to disturb a compass needle, but the link between lightning and electricity was not confirmed until Benjamin Franklin's proposed experiments in 1752. One of the first to discover and publish a link between manmade electric current and magnetism was Romagnosi, who in 1802 noticed that connecting a wire across a voltaic pile deflected a nearby compass needle. However, the effect did not become widely known until 1820, when Ørsted performed a similar experiment. Ørsted's work influenced Ampère to produce a theory of electromagnetism that set the subject on a mathematical foundation.
An accurate theory of electromagnetism, known as classical electromagnetism, was developed by various physicists over the course of the 19th century, culminating in the work of James Clerk Maxwell, who unified the preceding developments into a single theory and discovered the electromagnetic nature of light. In classical electromagnetism, the electromagnetic field obeys a set of equations known as Maxwell's equations, and the electromagnetic force is given by the Lorentz force law. One of the peculiarities of classical electromagnetism is that it is difficult to reconcile with classical mechanics, but it is compatible with special relativity. According to Maxwell's equations, the speed of light in a vacuum is a universal constant, dependent only on the electrical permittivity and magnetic permeability of free space. This violates Galilean invariance, a longstanding cornerstone of classical mechanics. One way to reconcile the two theories is to assume the existence of a luminiferous aether through which the light propagates. However, subsequent experimental efforts failed to detect the presence of the aether. After important contributions of Hendrik Lorentz and Henri Poincaré, in 1905, Albert Einstein solved the problem with the introduction of special relativity, which replaces classical kinematics with a new theory of kinematics that is compatible with classical electromagnetism. In addition, relativity theory shows that in moving frames of reference a magnetic field transforms to a field with a nonzero electric component and vice versa; thus firmly showing that they are two sides of the same coin, and thus the term "electromagnetism".
The photoelectric effect In another paper published in that same year, Albert Einstein undermined the very foundations of classical electromagnetism. His theory of the photoelectric effect (for which he won the Nobel prize for physics) posited that light could exist in discrete particlelike quantities, which later came to be known as photons. Einstein's theory of the photoelectric effect extended the insights that appeared in the solution of the ultraviolet catastrophe presented by Max Planck in 1900. In his work, Planck showed that hot objects emit electromagnetic radiation in discrete packets, which leads to a finite total energy emitted as black body radiation. Both of these results were in direct contradiction with the classical view of light as a continuous wave, although it is now known that the photoelectric effect does not, in fact, compel one to any conclusion about light being made of "photons", as discussed in the photoelectric effect article. Planck's and Einstein's theories were progenitors of quantum mechanics, which, when formulated in 1925, necessitated the invention of a quantum theory of electromagnetism. This theory, completed in the 1940s, is known as quantum electrodynamics (or "QED"), and, in situations where perturbation theory is applicable, is one of the most accurate theories known to physics.
Units
Electromagnetic units are part of a system of electrical units based primarily upon the magnetic properties of electric currents, the fundamental SI unit being the ampere. The units are: • • • • • • • • •
ampere (current) coulomb (charge) farad (capacitance) henry (inductance) ohm (resistance) volt (electric potential) watt (power) tesla (magnetic field) weber (flux)
In the electromagnetic cgs system, electric current is a fundamental quantity defined via Ampère's law and takes the permeability as a dimensionless quantity (relative permeability) whose value in a vacuum is unity. As a consequence, the square of the speed of light appears explicitly in some of the equations interrelating quantities in this system. SI electromagnetism units Symbol
Name of Quantity
Derived Units ampere (SI base unit) coulomb
A
A (= W/V = C/s)
C
volt
V
A·s J/C = kg·m2·s−3·A−1
ohm
Ω
ohm metre watt
Ω·m kg·m3·s−3·A−2 W V·A = kg·m2·s−3 C/V = F kg−1·m−2·A2·s4 N/C = V/m kg·m·A−1·s−3
I
Electric current
Q U, ΔV, Δφ; E
ρ P
Electric charge Potential difference; Electromotive force Electric resistance; Impedance; Reactance Resistivity Electric power
C
Capacitance
E
Electric field strength volt per metre
R; Z; X
D ε χe G; Y; B
farad
Electric displacement Coulomb per field square metre Permittivity farad per metre Electric (dimensionless) susceptibility Conductance; siemens Admittance;
Unit
Base Units
V/A = kg·m2·s−3·A−2
C/m2 A·s·m−2 F/m kg−1·m−3·A2·s4 

S
Ω−1 = kg−1·m−2·s3·A2
Susceptance κ, γ, σ B
Conductivity Magnetic flux density, Magnetic induction
siemens per metre S/m kg−1·m−3·s3·A2 Wb/m2 = tesla T kg·s−2·A−1 = N·A−1·m−1 V·s = weber Wb kg·m2·s−2·A−1
Φ
Magnetic flux
H
Magnetic field strength
ampere per metre A/m A·m−1
L, M
Inductance
henry
μ
Permeability Magnetic susceptibility
henry per metre
Wb/A = V·s/A = kg·m2·s−2·A−2 H/m kg·m·s−2·A−2
(dimensionless)

χ
H

Electromagnetic phenomena With the exception of gravitation, electromagnetic phenomena as described by quantum electrodynamics (which includes as a limiting case classical electrodynamics) account for almost all physical phenomena observable to the unaided human senses, including light and other electromagnetic radiation, all of chemistry, most of mechanics (excepting gravitation), and of course magnetism and electricity. Magnetic monopoles (and "Gilbert" dipoles) are not strictly electromagnetic phenomena, since in standard electromagnetism, magnetic fields are generated not by true "magnetic charge" but by currents. There are, however, condensed matter analogs of magnetic monopoles in exotic materials (spin ice) created in the laboratory.
Chapter 6
Mass–Energy Equivalence
3metertall sculpture of Einstein's 1905 E = mc2 formula at the 2006 Walk of Ideas, Berlin, Germany In physics, mass–energy equivalence is the concept that the mass of a body is a measure of its energy content. In this concept the total internal energy E of a body at rest is equal to the product of its rest mass m and a suitable conversion factor to transform from units of mass to units of energy. If the body is not stationary relative to the observer then account must be made for relativistic effects where m is given by the relativistic mass and E the relativistic energy of the body. Albert Einstein proposed mass–energy equivalence in 1905 in one of his Annus Mirabilis papers entitled "Does the inertia of a body depend upon its energycontent?". The equivalence is described by the famous equation
where E is energy, m is mass, and c is the speed of light in a vacuum. The formula is dimensionally consistent and does not depend on any specific system of measurement units. For example, in many systems of natural units, the speed (scalar) of light is set equal to 1 ('distance'/'time'), and the formula becomes the identity E = m'('distance'2/'time'2)'; hence the term "mass–energy equivalence".
The equation E = mc2 indicates that energy always exhibits mass in whatever form the energy takes. Mass–energy equivalence also means that mass conservation becomes a restatement, or requirement, of the law of energy conservation, which is the first law of thermodynamics. Mass–energy equivalence does not imply that mass may be "converted" to energy, and indeed implies the opposite. Modern theory holds that neither mass nor energy may be destroyed, but only moved from one location to another. In physics, mass must be differentiated from matter, a more poorly defined idea in the physical sciences. Matter, when seen as certain types of particles, can be created and destroyed, but the precursors and products of such reactions retain both the original mass and energy, both of which remain unchanged (conserved) throughout the process. Letting the m in E = mc2 stand for a quantity of "matter" may lead to incorrect results, depending on which of several varying definitions of "matter" are chosen. E = mc2 has sometimes been used as an explanation for the origin of energy in nuclear processes, but mass–energy equivalence does not explain the origin of such energies. Instead, this relationship merely indicates that the large amounts of energy released in such reactions may exhibit enough mass that the massloss may be measured, when the released energy (and its mass) have been removed from the system. Einstein was not the first to propose a mass–energy relationship. However, Einstein was the first scientist to propose the E = mc2 formula and the first to interpret mass–energy equivalence as a fundamental principle that follows from the relativistic symmetries of space and time.
Conservation of mass and energy The concept of mass–energy equivalence connects the concepts of conservation of mass and conservation of energy, which continue to hold separately. The theory of relativity allows particles which have rest mass to be converted to other forms of mass which require motion, such as kinetic energy, heat, or light. However, the mass remains. Kinetic energy or light can also be converted to new kinds of particles which have rest mass, but again the energy remains. Both the total mass and the total energy inside a totally closed system remain constant over time, as seen by any single observer in a given inertial frame. In other words, energy cannot be created or destroyed, and energy, in all of its forms, has mass. Mass also cannot be created or destroyed, and in all of its forms, has energy. According to the theory of relativity, mass and energy as commonly understood, are two names for the same thing, and neither one is changed or transformed into the other. Rather, neither one appears without the other. Rather than mass being changed into energy, the view of relativity is that rest mass has been changed to a more mobile form of mass, but remains mass. In this process, neither the amount of mass nor the amount of energy changes. Thus, if energy changes type and leaves a system, it simply takes its mass with it. If either mass or energy disappears from a system, it will always be found that both have simply moved off to another place.
Fastmoving objects and systems of objects
When an object is pushed in the direction of motion, it gains momentum and energy, but when the object is already traveling near the speed of light, it cannot move much faster, no matter how much energy it absorbs. Its momentum and energy continue to increase without bounds, whereas its speed approaches a constant value—the speed of light. This implies that in relativity the momentum of an object cannot be a constant times the velocity, nor can the kinetic energy be a constant times the square of the velocity. The relativistic mass is defined as the ratio of the momentum of an object to its velocity In fact, it depends on the motion of the object. If the object is moving slowly, the relativistic mass is nearly equal to the rest mass and both are nearly equal to the usual Newtonian mass. If the object is moving quickly, the relativistic mass is greater than the rest mass by an amount equal to the mass associated with the kinetic energy of the object. As the object approaches the speed of light, the relativistic mass grows infinitely, because the kinetic energy grows infinitely and this energy is associated with mass. The relativistic mass is always equal to the total energy (rest energy plus kinetic energy) divided by c2. Because the relativistic mass is exactly proportional to the energy, relativistic mass and relativistic energy are nearly synonyms; the only difference between them is the units. If length and time are measured in natural units, the speed of light is equal to 1, and even this difference disappears. Then mass and energy have the same units and are always equal, so it is redundant to speak about relativistic mass, because it is just another name for the energy. This is why physicists usually reserve the useful short word "mass" to mean restmass. For things made up of many parts, like an atomic nucleus, planet, or star, the relativistic mass is the sum of the relativistic masses (or energies) of the parts, because energies are additive in closed systems. This is not true in systems which are open, however, if energy is subtracted. For example, if a system is bound by attractive forces and the work they do in attraction is removed from the system, mass will be lost. Such work is a form of energy which itself has mass, and thus mass is removed from the system, as it is bound. For example, the mass of an atomic nucleus is less than the total mass of the protons and neutrons that make it up, but this is only true after the energy (work) of binding has been removed in the form of a gamma ray (which in this system, carries away the mass of binding). This mass decrease is also equivalent to the energy required to break up the nucleus into individual protons and neutrons (in this case, work and mass would need to be supplied). Similarly, the mass of the solar system is slightly less than the masses of sun and planets individually. The relativistic mass of a moving object is bigger than the relativistic mass of an object that isn't moving, because a moving object has extra kinetic energy. The rest mass of an object is defined as the mass of an object when it is at rest, so that the rest mass is always the same, independent of the motion of the observer: it is the same in all inertial frames. For a system of particles going off in different directions, the invariant mass of the system is the analog of the rest mass, and is the same for all observers. It is defined as the total energy (divided by c2) in the center of mass frame (where by definition, the system
total momentum is zero). A simple example of an object with moving parts but zero total momentum, is a container of gas. In this case, the mass of the container is given by its total energy (including the kinetic energy of the gas molecules), since the system total energy and invariant mass are the same in the reference frame where the momentum is zero, and this reference frame is also the only frame in which the object can be weighed.
Applicability of the strict mass–energy equivalence formula, E = mc² As is noted above, two different definitions of mass have been used in special relativity, and also two different definitions of energy. The simple equation E = mc² is not generally applicable to all these types of mass and energy, except in the special case that the momentum is zero for the system under consideration. In such a case, which is always guaranteed when observing the system from the center of mass frame, E = mc² is true for any type of mass and energy that are chosen. Thus, for example, in the center of mass frame the total energy of an object or system is equal to its rest mass times c², a useful equality. This is the relationship used for the container of gas in the previous example. It is not true in other reference frames in which a system or object's total energy will depend on both its rest (or invariant) mass, and also its total momentum. In inertial reference frames other than the rest frame or center of mass frame, the equation E = mc² remains true if the energy is the relativistic energy and the mass the relativistic mass. It is also correct if the energy is the rest or invariant energy (also the minimum energy), and the mass is the rest or invariant mass. However, connection of the total or relativistic energy (Er) with the rest or invariant mass (m0) requires consideration of the system total momentum, in systems and reference frames where momentum has a nonzero value. The formula then required to connect the different kinds of mass and energy, is the extended version of Einstein's equation, called the relativistic energy–momentum relationship:
or
Here the (pc)2 term represents the square of the Euclidean norm (total vector length) of the various momentum vectors in the system, which reduces to the square of the simple momentum magnitude, if only a single particle is considered. Obviously this equation reduces to E = mc² when the momentum term is zero. For photons where m0 = 0, the equation reduces to Er = pc.
Meanings of the strict mass–energy equivalence formula, E = mc²
The mass–energy equivalence formula was displayed on Taipei 101 during the event of the World Year of Physics 2005. Mass–energy equivalence states that any object has a certain energy, even when it is stationary. In Newtonian mechanics, a motionless body has no kinetic energy, and it may or may not have other amounts of internal stored energy, like chemical energy or thermal energy, in addition to any potential energy it may have from its position in a field of force. In Newtonian mechanics, all of these energies are much smaller than the mass of the object times the speed of light squared. In relativity, all of the energy that moves along with an object adds up to the total mass of the body, which measures how much it resists deflection. Each potential and kinetic energy makes a proportional contribution to the mass. If a box of ideal mirrors contains light, then the photons contribute to the total mass of the box by the amount of their energy divided by c2. In relativity, removing energy is removing mass, and for an observer in the center of mass frame, the formula m = E/c² indicates how much mass is lost when energy is removed. In a chemical or nuclear reaction, the mass of the atoms that come out is less than the mass of the atoms that go in, and the difference in mass shows up as heat and
light which has the same relativistic mass as the difference (and also the same invariant mass in the center of mass frame of the system). In this case, the E in the formula is the energy released and removed, and the mass m is how much the mass decreases. In the same way, when any sort of energy is added to an isolated system, the increase in the mass is equal to the added energy divided by c². For example, when water is heated it gains about 1.11×10−17 kg of mass for every joule of heat added to the water. An object moves with different speed in different frames, depending on the motion of the observer, so the kinetic energy in both Newtonian mechanics and relativity is frame dependent. This means that the amount of relativistic energy, and therefore the amount of relativistic mass, that an object is measured to have depends on the observer. The rest mass is defined as the mass that an object has when it isn't moving (or when an inertial frame is chosen such that it is not moving). The term also applies to the invariant mass of systems when the system as a whole isn't "moving" (has no net momentum). The rest and invariant masses are the smallest possible value of the mass of the object or system. They also are conserved quantities, so long as the system is closed. Because of the way they are calculated, the effects of moving observers are subtracted, so these quantities do not change with the motion of the observer. The rest mass is almost never additive: the rest mass of an object is not the sum of the rest masses of its parts. The rest mass of an object is the total energy of all the parts, including kinetic energy, as measured by an observer that sees the center of the mass of the object to be standing still. The rest mass adds up only if the parts are standing still and don't attract or repel, so that they don't have any extra kinetic or potential energy. The other possibility is that they have a positive kinetic energy and a negative potential energy that exactly cancels.
Binding energy and the "mass defect" Whenever any type of energy is removed from a system, the mass associated with the energy is also removed, and the system therefore loses mass. This mass defect in the system may be simply calculated as Δm = ΔE/c2, but use of this formula in such circumstances has led to the false idea that mass has been "converted" to energy. This may be particularly the case when the energy (and mass) removed from the system is associated with the binding energy of the system. In such cases, the binding energy is observed as a "mass defect" or deficit in the new system and the fact that the released energy is not easily weighed may cause its mass to be neglected. The difference between the rest mass of a bound system and of the unbound parts is the binding energy of the system, if this energy has been removed after binding. For example, a water molecule weighs a little less than two free hydrogen atoms and an oxygen atom; the minuscule mass difference is the energy that is needed to split the molecule into three individual atoms (divided by c²), and which was given off as heat when the molecule formed (this heat had mass). Likewise, a stick of dynamite in theory weighs a little bit more than the fragments after the explosion, but this is true only so long as the fragments are cooled and the heat removed. In this case the mass difference is the
energy/heat that is released when the dynamite explodes, and when this heat escapes, the mass associated with it escapes, only to be deposited in the surroundings which absorb the heat (so that total mass is conserved). Such a change in mass may only happen when the system is open, and the energy and mass escapes. Thus, if a stick of dynamite is blown up in a hermetically sealed chamber, the mass of the chamber and fragments, the heat, sound, and light would still be equal to the original mass of the chamber and dynamite. If sitting on a scale, the weight and mass would not change. This would in theory also happen even with a nuclear bomb, if it could be kept in an ideal box of infinite strength, which did not rupture or pass radiation. Thus, a 21.5 kiloton (9 x 1013joule) nuclear bomb produces about one gram of heat and electromagnetic radiation, but the mass of this energy would not be detectable in an exploded bomb in an ideal box sitting on a scale; instead, the contents of the box would be heated to millions of degrees without changing total mass and weight. If then, however, a transparent window (passing only electromagnetic radiation) were opened in such an ideal box after the explosion, and a beam of Xrays and other lowerenergy light allowed to escape the box, it would eventually be found to weigh one gram less than it had before the explosion. This weightloss and massloss would happen as the box was cooled by this process, to room temperature. However, any surrounding mass which had absorbed the Xrays (and other "heat") would gain this gram of mass from the resulting heating, so the mass "loss" would represent merely its relocation. Thus, no mass (or, in the case of a nuclear bomb, no matter) would be "converted" to energy in such a process. Mass and energy, as always, would both be separately conserved.
Massless particles Massless particles have zero rest mass. Their relativistic mass is simply their relativistic energy, divided by c2, or m(relativistic) = E/c2. . The energy for photons is E = hν where h is Planck's constant and ν is the photon frequency. This frequency and thus the relativistic energy are framedependent. If an observer runs away from a photon in the direction it travels from a source, having it catch up with the observer, then when the photon catches up it will be seen as having less energy than it had at the source. The faster the observer is traveling with regard to the source when the photon catches up, the less energy the photon will have. As an observer approaches the speed of light with regard to the source, the photon looks redder and redder, by Doppler shift (the Doppler shift is the relativistic formula), and the energy of a very longwavelength photon approaches zero. This is why a photon is massless; this means that the rest mass of a photon is zero. Two photons moving in different directions cannot both be made to have arbitrarily small total energy by changing frames, or by moving toward or away from them. The reason is that in a twophoton system, the energy of one photon is decreased by chasing after it, but the energy of the other will increase with the same shift in observer motion. Two photons not moving in the same direction will exhibit an inertial frame where the combined energy is smallest, but not zero. This is called the center of mass frame or the center of
momentum frame; these terms are almost synonyms (the center of mass frame is the special case of a center of momentum frame where the center of mass is put at the origin). The most that chasing a pair of photons can accomplish to decrease their energy is to put the observer in frame where the photons have equal energy and are moving directly away from each other. In this frame, the observer is now moving in the same direction and speed as the center of mass of the two photons. The total momentum of the photons is now zero, since their momentums are equal and opposite. In this frame the two photons, as a system, have a mass equal to their total energy divided by c2. This mass is called the invariant mass of the pair of photons together. It is the smallest mass and energy the system may be seen to have, by any observer. It is only the invariant mass of a twophoton system that can be used to make a single particle with the same rest mass. If the photons are formed by the collision of a particle and an antiparticle, the invariant mass is the same as the total energy of the particle and antiparticle (their rest energy plus the kinetic energy), in the center of mass frame, where they will automatically be moving in equal and opposite directions (since they have equal momentum in this frame). If the photons are formed by the disintegration of a single particle with a welldefined rest mass, like the neutral pion, the invariant mass of the photons is equal to rest mass of the pion. In this case, the center of mass frame for the pion is just the frame where the pion is at rest, and the center of mass doesn't change after it disintegrates into two photons. After the two photons are formed, their center of mass is still moving the same way the pion did, and their total energy in this frame adds up to the mass energy of the pion. Thus, by calculating the invariant mass of pairs of photons in a particle detector, pairs can be identified that were probably produced by pion disintegration.
Consequences for nuclear physics
Task Force One, the world's first nuclearpowered task force. Enterprise, Long Beach and Bainbridge in formation in the Mediterranean, 18 June 1964. Enterprise crew members are spelling out Einstein's MassEnergy Equivalence formula E=mc² on the flight deck. Max Planck pointed out that the mass–energy equivalence formula implied that bound systems would have a mass less than the sum of their constituents, once the binding energy had been allowed to escape. However, Planck was thinking about chemical reactions, where the binding energy is too small to measure. Einstein suggested that radioactive materials such as radium would provide a test of the theory, but even though a large amount of energy is released per atom in radium, due to the halflife of the substance (1602 years), only a small fraction of radium atoms decay over experimentally measureable period of time. Once the nucleus was discovered, experimenters realized that the very high binding energies of the atomic nuclei should allow calculation of their binding energies, simply from mass differences. But it was not until the discovery of the neutron in 1932, and the measurement of the neutron mass, that this calculation could actually be performed. A little while later, the first transmutation reactions (such as 7Li + p → 2 4He) verified Einstein's formula to an accuracy of ±0.5%.
The mass–energy equivalence formula was used in the development of the atomic bomb. By measuring the mass of different atomic nuclei and subtracting from that number the total mass of the protons and neutrons as they would weigh separately, one gets the exact binding energy available in an atomic nucleus. This is used to calculate the energy released in any nuclear reaction, as the difference in the total mass of the nuclei that enter and exit the reaction.
Practical examples Einstein used the CGS system of units (centimeters, grams, seconds, dynes, and ergs), but the formula is independent of the system of units. In natural units, the speed of light is defined to equal 1, and the formula expresses an identity: E = m. In the SI system (expressing the ratio E / m in joules per kilogram using the value of c in meters per second): E / m = c2 = (299,792,458 m/s)2 = 89,875,517,873,681,764 J/kg (≈9.0 × 1016 joules per kilogram) So one gram of mass is equivalent to the following amounts of energy: 89.9 terajoules 25.0 million kilowatthours (≈25 GW·h) 21.5 billion kilocalories (≈21 Tcal) 21.5 kilotons of TNTequivalent energy (≈21 kt) 85.2 billion BTUs Any time energy is generated, the process can be evaluated from an E = mc2 perspective. For instance, the "Gadget"style bomb used in the Trinity test and the bombing of Nagasaki had an explosive yield equivalent to 21 kt of TNT. About 1 kg of the approximately 6.15 kg of plutonium in each of these bombs fissioned into lighter elements totaling almost exactly one gram less, after cooling [The heat, light, and electromagnetic radiation released in this explosion carried the missing one gram of mass.] This occurs because nuclear binding energy is released whenever elements with more than 62 nucleons fission. Another example is hydroelectric generation. The electrical energy produced by Grand Coulee Dam's turbines every 3.7 hours represents one gram of mass. This mass passes to the electrical devices which are powered by the generators (such as lights in cities), where it appears as a gram of heat and light. Turbine designers look at their equations in terms of pressure, torque, and RPM. However, Einstein's equations show that all energy has mass, and thus the electrical energy produced by a dam's generators, and the heat and light which result from it, all retain their mass, which is equivalent to the energy. The potential energy—and equivalent mass—represented by the waters of the Columbia River as it descends to the Pacific Ocean would be converted to heat due to viscous friction and the turbulence of white water rapids and waterfalls were it not for the dam and its generators. This heat would remain as mass on site at the water, were it not for the
equipment which converted some of this potential and kinetic energy into electrical energy, which can be moved from place to place (taking mass with it). Whenever energy is added to a system, the system gains mass. •
•
•
A spring's mass increases whenever it is put into compression or tension. Its added mass arises from the added potential energy stored within it, which is bound in the stretched chemical (electron) bonds linking the atoms within the spring. Raising the temperature of an object (increasing its heat energy) increases its mass. For example, consider the world's primary mass standard for the kilogram, made of platinum/iridium. If its temperature is allowed to change by 1°C, its mass will change by 1.5 picograms (1 pg = 1 × 10−12 g). A spinning ball will weigh more than a ball that is not spinning. Its increase of mass is exactly the equivalent of the mass of energy of rotation, which is itself the sum of the kinetic energies of all the moving parts of the ball. For example, the Earth itself is more massive due to its daily rotation, than it would be with no rotation. This rotational energy (2.14 x 1029 J) represents 2.38 billion metric tons of added mass.
Note that no net mass or energy is really created or lost in any of these examples and scenarios. Mass/energy simply moves from one place to another. These are some examples of the transfer of energy and mass in accordance with the principle of mass– energy conservation. Note further that in accordance with Einstein's Strong Equivalence Principle (SEP), all forms of mass and energy produce a gravitational field in the same way. So all radiated and transmitted energy retains its mass. Not only does the matter comprising Earth create gravity, but the gravitational field itself has mass, and that mass contributes to the field too. This effect is accounted for in ultraprecise laser ranging to the Moon as the Earth orbits the Sun when testing Einstein's general theory of relativity. According to E=mc2, no closed system (any system treated and observed as a whole) ever loses mass, even when rest mass is converted to energy. All types of energy contribute to mass, including potential energies. In relativity, interaction potentials are always due to local fields, not to direct nonlocal interactions, because signals can't travel faster than light. The field energy is stored in field gradients or, in some cases (for massive fields), where the field has a nonzero value. The mass associated with the potential energy is the mass–energy of the field energy. The mass associated with field energy can be detected, in principle, by gravitational experiments, by checking how the field attracts other objects gravitationally. The energy in the gravitational field itself has some differences from other energies. There are several consistent ways to define the location of the energy in a gravitational field, all of which agree on the total energy when space is mostly flat and empty. But because the gravitational field can be made to vanish locally at any point by choosing a
freefalling frame, the precise location of the energy becomes dependent on the observer's frame of reference, and thus has no exact location, even though it exists somewhere for any given observer. In the limit for low field strengths, this gravitational field energy is the familiar Newtonian gravitational potential energy.
Efficiency Although mass cannot be converted to energy, matter particles can be. Also, a certain amount of the illdefined "matter" in ordinary objects can be converted to active energy (light and heat), even though no identifiable real particles are destroyed. Such conversions happen in nuclear weapons, in which the protons and neutrons in atomic nuclei lose a small fraction of their average mass, but this massloss is not due to the destruction of any protons or neutrons (or even, in general, lighter particles like electrons). Also the mass is not destroyed, but simply removed from the system in the form of heat and light from the reaction. In nuclear reactions, typically only a small fraction of the total mass–energy of the bomb is converted into heat, light, radiation and motion, which are "active" forms which can be used. When an atom fissions, it loses only about 0.1% of its mass (which escapes from the system and does not disappear), and in a bomb or reactor not all the atoms can fission. In a fission based atomic bomb, the efficiency is only 40%, so only 40% of the fissionable atoms actually fission, and only 0.04% of the total mass appears as energy in the end. In nuclear fusion, more of the mass is released as usable energy, roughly 0.3%. But in a fusion bomb, the bomb mass is partly casing and nonreacting components, so that in practicality, no more than about 0.03% of the total mass of the entire weapon is released as usable energy (which, again, retains the "missing" mass). In theory, it should be possible to convert all of the mass in matter into heat and light (which would of course have the same mass), but none of the theoretically known methods are practical. One way to convert all matter into usable energy is to annihilate matter with antimatter. But antimatter is rare in our universe, and must be made first. Due to inefficient mechanisms of production, making antimatter always requires far more energy than would be released when it was annihilated. Since most of the mass of ordinary objects resides in protons and neutrons, in order to convert all ordinary matter to useful energy, the protons and neutrons must be converted to lighter particles. In the standard model of particle physics, the number of protons plus neutrons is nearly exactly conserved. Still, Gerard 't Hooft showed that there is a process which will convert protons and neutrons to antielectrons and neutrinos. This is the weak SU(2) instanton proposed by Belavin Polyakov Schwarz and Tyupkin. This process, can in principle convert all the mass of matter into neutrinos and usable energy, but it is normally extraordinarily slow. Later it became clear that this process will happen at a fast rate at very high temperatures, since then instantonlike configurations will be copiously produced from thermal fluctuations. The temperature required is so high that it would only have been reached shortly after the big bang.
Many extensions of the standard model contain magnetic monopoles, and in some models of grand unification, these monopoles catalyze proton decay, a process known as the Callan–Rubakov effect. This process would be an efficient mass–energy conversion at ordinary temperatures, but it requires making monopoles and antimonopoles first. The energy required to produce monopoles is believed to be enormous, but magnetic charge is conserved, so that the lightest monopole is stable. All these properties are deduced in theoretical models—magnetic monopoles have never been observed, nor have they been produced in any experiment so far. A third known method of total matter–energy conversion is using gravity, specifically black holes. Stephen Hawking theorized that black holes radiate thermally with no regard to how they are formed. So it is theoretically possible to throw matter into a black hole and use the emitted heat to generate power. According to the theory of Hawking radiation, however, the black hole used will radiate at a higher rate the smaller it is, producing usable powers at only small black hole masses, where usable may for example be something greater than the local background radiation. It is also worth noting that the ambient irradiated power would change with the mass of the black hole, increasing as the mass of the black hole decreases, or decreasing as the mass increases, at a rate where power is proportional to the inverse square of the mass. In a "practical" scenario, mass and energy could be dumped into the black hole to regulate this growth, or keep its size, and thus power output, near constant. This could result from the fact that mass and energy are lost from the hole with its thermal radiation.
Background Mass–velocity relationship In developing special relativity, Einstein found that the kinetic energy of a moving body is
with v the velocity, m0 the rest mass, and γ the gammafactor. He included the second term on the right to make sure that for small velocities, the energy would be the same as in classical mechanics:
Without this second term, there would be an additional contribution in the energy when the particle is not moving.
Einstein found that the total momentum of a moving particle is:
and it is this quantity which is conserved in collisions. The ratio of the momentum to the velocity is the relativistic mass, m.
And the relativistic mass and the relativistic kinetic energy are related by the formula:
Einstein wanted to omit the unnatural second term on the righthand side, whose only purpose is to make the energy at rest zero, and to declare that the particle has a total energy which obeys:
which is a sum of the rest energy m0c2 and the kinetic energy. This total energy is mathematically more elegant, and fits better with the momentum in relativity. But to come to this conclusion, Einstein needed to think carefully about collisions. This expression for the energy implied that matter at rest has a huge amount of energy, and it is not clear whether this energy is physically real, or just a mathematical artifact with no physical meaning. In a collision process where all the restmasses are the same at the beginning as at the end, either expression for the energy is conserved. The two expressions only differ by a constant which is the same at the beginning and at the end of the collision. Still, by analyzing the situation where particles are thrown off a heavy central particle, it is easy to see that the inertia of the central particle is reduced by the total energy emitted. This allowed Einstein to conclude that the inertia of a heavy particle is increased or diminished according to the energy it absorbs or emits.
Relativistic mass After Einstein first made his proposal, it became clear that the word mass can have two different meanings. The rest mass is what Einstein called m, but others defined the relativistic mass with an explicit index:
This mass is the ratio of momentum to velocity, and it is also the relativistic energy divided by c2 (it is not Lorentzinvariant, in contrast to m0). The equation E = mrelc2 holds for moving objects. When the velocity is small, the relativistic mass and the rest mass are almost exactly the same. •
E=mc2 either means E=m0c2 for an object at rest, or E=mrelc2 when the object is moving.
Also Einstein (following Hendrik Lorentz and Max Abraham) used velocity—and directiondependent mass concepts (longitudinal and transverse mass) in his 1905 electrodynamics paper and in another paper in 1906. However, in his first paper on E=mc2 (1905), he treated m as what would now be called the rest mass. Some claim that (in later years) he did not like the idea of "relativistic mass." When modern physicists say "mass", they are usually talking about rest mass, since if they meant "relativistic mass", they would just say "energy". Considerable debate has ensued over the use of the concept "relativistic mass" and the connection of "mass" in relativity to "mass" in Newtonian dynamics. For example, one view is that only rest mass is a viable concept and is a property of the particle; while relativistic mass is a conglomeration of particle properties and properties of spacetime. A perspective that avoids this debate, due to Kjell Vøyenli, is that the Newtonian concept of mass as a particle property and the relativistic concept of mass have to be viewed as embedded in their own theories and as having no precise connection.
Lowspeed expansion We can rewrite the expression E = γm0c2 as a Taylor series:
For speeds much smaller than the speed of light, higherorder terms in this expression get smaller and smaller because v/c is small. For low speeds we can ignore all but the first two terms:
The total energy is a sum of the rest energy and the Newtonian kinetic energy.
The classical energy equation ignores both the m0c2 part, and the highspeed corrections. This is appropriate, because all the highorder corrections are small. Since only changes in energy affect the behavior of objects, whether we include the m0c2 part makes no difference, since it is constant. For the same reason, it is possible to subtract the rest energy from the total energy in relativity. By considering the emission of energy in different frames, Einstein could show that the rest energy has a real physical meaning. The higherorder terms are extra correction to Newtonian mechanics which become important at higher speeds. The Newtonian equation is only a lowspeed approximation, but an extraordinarily good one. All of the calculations used in putting astronauts on the moon, for example, could have been done using Newton's equations without any of the higherorder corrections.
History While Einstein was the first to have correctly deduced the mass–energy equivalence formula, he was not the first to have related energy with mass. But nearly all previous authors thought that the energy which contributes to mass comes only from electromagnetic fields.
Newton: Matter and light In 1717 Isaac Newton speculated that light particles and matter particles were interconvertible in "Query 30" of the Opticks, where he asks: Are not the gross bodies and light convertible into one another, and may not bodies receive much of their activity from the particles of light which enter their composition?
Electromagnetic rest mass There were many attempts in the 19th and the beginning of the 20th century—like those of J. J. Thomson (1881), Oliver Heaviside (1888), and George Frederick Charles Searle (1897)—to understand how the mass of a charged object depends on the electrostatic field. Because the electromagnetic field carries part of the momentum of a moving charge, it was also suspected that the mass of an electron would vary with velocity near the speed of light. Searle calculated that it is impossible for a charged object to supersede the velocity of light because this would require an infinite amount of energy. Following Thomson and Searle (1896), Wilhelm Wien (1900), Max Abraham (1902), and Hendrik Lorentz (1904) argued that this relation applies to the complete mass of bodies, because all inertial mass is electromagnetic in origin. The formula of the mass–energyrelation given by them was m = (4 / 3)E / c2. Wien went on by stating, that if it is assumed that gravitation is an electromagnetic effect too, then there has to be a strict proportionality between (electromagnetic) inertial mass and (electromagnetic) gravitational mass. This interpretation is in the now discredited electromagnetic worldview, and the formulas that they discovered always included a factor of 4/3 in the
proportionality. For example, the formulas given by Lorentz in 1904 for the prerelativistic longitudinal and transverse masses were (in modern notation):
, where
In July 1905 (published 1906), nearly at the same time when Einstein found the simple relation from relativity, Poincaré was able to explain the reason that the electromagnetic mass calculations always had a factor of 4/3. In order for a particle consisting of positive or negative charge to be stable, there must be some sort of attractive force of nonelectrical nature which keeps it together. If the mass–energy of this force field is included in a way which is consistent with relativity theory, the attractive contribution adds an amount − (1 / 3)E / c2 to the energy of the bodies, and this explains the discrepancy between the pure electromagnetic theory and relativity.
Inertia of energy and radiation James Clerk Maxwell (1874) and Adolfo Bartoli (1876) found out that the existence of tensions in the ether like the radiation pressure follows from the electromagnetic theory. However, Lorentz (1895) recognized that this led to a conflict between the action/reaction principle and Lorentz's ether theory. Poincaré In 1900 Henri Poincaré studied this conflict and tried to determine whether the center of gravity still moves with a uniform velocity when electromagnetic fields are included. He noticed that the action/reaction principle does not hold for matter alone, but that the electromagnetic field has its own momentum. The electromagnetic field energy behaves like a fictitious fluid ("fluide fictif") with a mass density of E / c2 (in other words m = E/c2). If the center of mass frame is defined by both the mass of matter and the mass of the fictitious fluid, and if the fictitious fluid is indestructible—it is neither created or destroyed—then the motion of the center of mass frame remains uniform. But electromagnetic energy can be converted into other forms of energy. So Poincaré assumed that there exists a nonelectric energy fluid at each point of space, into which electromagnetic energy can be transformed and which also carries a mass proportional to the energy. In this way, the motion of the center of mass remains uniform. Poincaré said that one should not be too surprised by these assumptions, since they are only mathematical fictions.
But Poincaré's resolution led to a paradox when changing frames: if a Hertzian oscillator radiates in a certain direction, it will suffer a recoil from the inertia of the fictitious fluid. In the framework of Lorentz ether theory Poincaré performed a Lorentz boost to the frame of the moving source. He noted that energy conservation holds in both frames, but that the law of conservation of momentum is violated. This would allow a perpetuum mobile, a notion which he abhorred. The laws of nature would have to be different in the frames of reference, and the relativity principle would not hold. Poincaré's paradox was resolved by Einstein's insight that a body losing energy as radiation or heat was losing a mass of the amount m = E / c2. The Hertzian oscillator loses mass in the emission process, and momentum is conserved in any frame. Einstein noted in 1906 that Poincaré's solution to the center of mass problem and his own were mathematically equivalent (see below). Poincaré came back to this topic in "Science and Hypothesis" (1902) and "The Value of Science" (1905). This time he rejected the possibility that energy carries mass: "... [the recoil] is contrary to the principle of Newton since our projectile here has no mass, it is not matter, it is energy". He also discussed two other unexplained effects: (1) nonconservation of mass implied by Lorentz's variable mass γm, Abraham's theory of variable mass and Kaufmann's experiments on the mass of fast moving electrons and (2) the nonconservation of energy in the radium experiments of Madame Curie. Abraham and Hasenöhrl Following Poincaré, Max Abraham in 1902 introduced the term "electromagnetic momentum" to maintain the action/reaction principle. Poincaré's result was verified by him, whereby the field density of momentum per cm3 is E / c2 and E / c per cm2. In 1904, Friedrich Hasenöhrl specifically associated inertia with radiation in a paper, which was according to his own words very similar to some papers of Abraham. Hasenöhrl suggested that part of the mass of a body (which he called apparent mass) can be thought of as radiation bouncing around a cavity. The apparent mass of radiation depends on the temperature (because every heated body emits radiation) and is proportional to its energy, and he first concluded that m = (8 / 3)E / c2. However, in 1905 Hasenöhrl published a summary of a letter, which was written by Abraham to him. Abraham concluded that Hasenöhrl's formula of the apparent mass of radiation is not correct, and on the basis of his definition of electromagnetic momentum and longitudinal electromagnetic mass Abraham changed it to m = (4 / 3)E / c2, the same value for the electromagnetic mass for a body at rest. Hasenöhrl recalculated his own derivation and verified Abraham's result. He also noticed the similarity between the apparent mass and the electromagnetic mass. However, Hasenöhrl stated that this energy–apparentmass relation only holds as long a body radiates, i.e. if the temperature of a body is greater than 0 K. However, Hasenöhrl did not include the pressure of the radiation on the cavity shell. If he had included the shell pressure and inertia as it would be included in the theory of relativity, the factor would have been equal to 1 or m = E / c2. This calculation assumes
that the shell properties are consistent with relativity, otherwise the mechanical properties of the shell including the mass and tension would not have the same transformation laws as those for the radiation. Nobel Prizewinner and Hitler advisor Philipp Lenard claimed that the mass–energy equivalence formula needed to be credited to Hasenöhrl to make it an Aryan creation.
Einstein: Mass–energy equivalence Albert Einstein did not formulate exactly the formula E = mc2 in his 1905 Annus Mirabilis paper "Does the Inertia of a Body Depend Upon Its Energy Content?"; rather, the paper states that if a body gives off the energy L in the form of radiation, its mass diminishes by L/c2. (Here, "radiation" means electromagnetic radiation, or light, and mass means the ordinary Newtonian mass of a slowmoving object.) This formulation relates only a change Δm in mass to a change L in energy without requiring the absolute relationship. Objects with zero mass presumably have zero energy, so the extension that all mass is proportional to energy is obvious from this result. In 1905, even the hypothesis that changes in energy are accompanied by changes in mass was untested. Not until the discovery of the first type of antimatter (the positron in 1932) was it found that all of the mass of pairs of resting particles could be converted to radiation. First correct derivation (1905) The correctness of Einstein's 1905 derivation of E=mc2 was criticized by Max Planck (1907), and also by Herbert Ives (1952), and also in a recent book (2008) by Hans Ohanian. Einstein considered a body at rest with mass M. If the body is examined in a frame moving with nonrelativistic velocity v, it is no longer at rest and in the moving frame it has momentum P = Mv. Einstein supposed the body emits two pulses of light to the left and to the right, each carrying an equal amount of energy E/2. In its rest frame, the object remains at rest after the emission since the two beams are equal in strength and carry opposite momentum. But if the same process is considered in a frame moving with velocity v to the left, the pulse moving to the left will be redshifted while the pulse moving to the right will be blue shifted. The blue light carries more momentum than the red light, so that the momentum of the light in the moving frame is not balanced: the light is carrying some net momentum to the right. The object hasn't changed its velocity before or after the emission. Yet in this frame it has lost some rightmomentum to the light. The only way it could have lost momentum is by losing mass. This also solves Poincaré's radiation paradox, discussed above.
The velocity is small, so the rightmoving light is blueshifted by an amount equal to the nonrelativistic Doppler shift factor 1 − v/c. The momentum of the light is its energy divided by c, and it is increased by a factor of v/c. So the rightmoving light is carrying an extra momentum ΔP given by:
The leftmoving light carries a little less momentum, by the same amount ΔP. So the total rightmomentum in the light is twice ΔP. This is the rightmomentum that the object lost.
The momentum of the object in the moving frame after the emission is reduced by this amount:
So the change in the object's mass is equal to the total energy lost divided by c2. Since any emission of energy can be carried out by a two step process, where first the energy is emitted as light and then the light is converted to some other form of energy, any emission of energy is accompanied by a loss of mass. Similarly, by considering absorption, a gain in energy is accompanied by a gain in mass. Einstein concludes that the mass of a body is a measure of its energy content. Relativistic centerofmass theorem – 1906 Like Poincaré, Einstein concluded in 1906 that the inertia of electromagnetic energy is a necessary condition for the centerofmass theorem to hold. On this occasion, Einstein referred to Poincaré's 1900 paper and wrote: Although the merely formal considerations, which we will need for the proof, are already mostly contained in a work by H. Poincaré2, for the sake of clarity I will not rely on that work. In Einstein's more physical, as opposed to formal or mathematical, point of view, there was no need for fictitious masses. He could avoid the perpetuum mobile problem, because on the basis of the mass–energy equivalence he could show that the transport of inertia which accompanies the emission and absorption of radiation solves the problem. Poincaré's rejection of the principle of action–reaction can be avoided through Einstein's E = mc2, because mass conservation appears as a special case of the energy conservation law.
Others During the nineteenth century there were several speculative attempts to show that mass and energy were proportional in various ether theories. In 1873 Nikolay Umov pointed out a relation between mass and energy for ether in the form of Е = kmc2, where 0.5 ≤ k ≤ 1. . The writings of Samuel Tolver Preston, and a 1903 paper by Olinto De Pretto, presented a mass–energy relation. De Pretto's paper received recent press coverage when Umberto Bartocci discovered that there were only three degrees of separation linking De Pretto to Einstein, leading Bartocci to conclude that Einstein was probably aware of De Pretto's work. Preston and De Pretto, following Le Sage, imagined that the universe was filled with an ether of tiny particles which are always moving at speed c. Each of these particles have a kinetic energy of mc2 up to a small numerical factor. The nonrelativistic kinetic energy formula did not always include the traditional factor of 1/2, since Leibniz introduced kinetic energy without it, and the 1/2 is largely conventional in prerelativistic physics. By assuming that every particle has a mass which is the sum of the masses of the ether particles, the authors would conclude that all matter contains an amount of kinetic energy either given by E = mc2 or 2E = mc2 depending on the convention. A particle ether was usually considered unacceptably speculative science at the time, and since these authors didn't formulate relativity, their reasoning is completely different from that of Einstein, who used relativity to change frames. Independently, Gustave Le Bon in 1905 speculated that atoms could release large amounts of latent energy, reasoning from an allencompassing qualitative philosophy of physics.
Radioactivity and nuclear energy It was quickly noted after the discovery of radioactivity in 1897, that the total energy due to radioactive processes is about one million times greater than that involved in any known molecular change. However, it raised the question where this energy is coming from. After eliminating the idea of absorption and emission of some sort of Lesagian ether particles, the existence of a huge amount of latent energy, stored within matter, was proposed by Ernest Rutherford and Frederick Soddy in 1903. Rutherford also suggested that this internal energy is stored within normal matter as well. He went on to speculate in 1904: If it were ever found possible to control at will the rate of disintegration of the radioelements, an enormous amount of energy could be obtained from a small quantity of matter. Einstein's equation is in no way an explanation of the large energies released in radioactive decay (this comes from the powerful nuclear forces involved; forces that were still unknown in 1905). In any case, the enormous energy released from radioactive decay (which had been measured by Rutherford) was much more easily measured than the (still
small) change in the gross mass of materials, as a result. Einstein's equation, by theory, can give these energies by measuring mass differences before and after reactions, but in practice, these mass differences in 1905 were still too small to be measured in bulk. Prior to this, the ease of measuring radioactive decay energies with a calorimeter was thought possibly likely to allow measurement of changes in mass difference, as a check on Einstein's equation itself. Einstein mentions in his 1905 paper that mass–energy equivalence might perhaps be tested with radioactive decay, which releases enough energy (the quantitative amount known roughly by 1905) to possibly be "weighed," when missing from the system (having been given off as heat). However, radioactivity seemed to proceed at its own unalterable (and quite slow, for radioactives known then) pace, and even when simple nuclear reactions became possible using proton bombardment, the idea that these great amounts of usable energy could be liberated at will with any practicality, proved difficult to substantiate. It had been used as the basis of much speculation, causing Rutherford himself to later reject his ideas of 1904; he was reported in 1933 to have said that: "Anyone who expects a source of power from the transformation of the atom is talking moonshine."
The popular connection between Einstein, E = mc2, and the atomic bomb was prominently indicated on the cover of Time magazine in July 1946 by the writing of the equation on the mushroom cloud itself. This situation changed dramatically in 1932 with the discovery of the neutron and its mass, allowing mass differences for single nuclides and their reactions to be calculated directly, and compared with the sum of masses for the particles that made up their composition. In 1933, the energy released from the reaction of lithium7 plus protons giving rise to 2 alpha particles (as noted above by Rutherford), allowed Einstein's equation to be tested to an error of ± 0.5%. However, scientists still did not see such reactions as a source of power. After the very public demonstration of huge energies released from nuclear fission after the atomic bombings of Hiroshima and Nagasaki in 1945, the equation E = mc2 became directly linked in the public eye with the power and peril of nuclear weapons. The equation was featured as early as page 2 of the Smyth Report, the official 1945 release by the US government on the development of the atomic bomb, and by 1946 the equation was linked closely enough with Einstein's work that the cover of Time magazine prominently featured a picture of Einstein next to an image of a mushroom cloud emblazoned with the equation. Einstein himself had only a minor role in the Manhattan Project: he had cosigned a letter to the U.S. President in 1939 urging funding for research into atomic energy, warning that an atomic bomb was theoretically possible. The letter persuaded Roosevelt to devote a significant portion of the wartime budget to atomic research. Without a security clearance, Einstein's only scientific contribution was an analysis of an isotope separation method based on the rate of molecular diffusion through pores, a nowobsolete process that was then competitive and contributed a fraction of the enriched uranium used in the project. While E = mc2 is useful for understanding the amount of energy potentially released in a fission reaction, it was not strictly necessary to develop the weapon, once the fission process was known, and its energy measured at 200 MeV (which was directly possible, using a quantitative Geiger counter, at that time). As the physicist and Manhattan Project participant Robert Serber put it: "Somehow the popular notion took hold long ago that Einstein's theory of relativity, in particular his famous equation E = mc2, plays some essential role in the theory of fission. Albert Einstein had a part in alerting the United States government to the possibility of building an atomic bomb, but his theory of relativity is not required in discussing fission. The theory of fission is what physicists call a nonrelativistic theory, meaning that relativistic effects are too small to affect the dynamics of the fission process significantly." However the association between E = mc2 and nuclear energy has since stuck, and because of this association, and its simple expression of the ideas of Albert Einstein himself, it has become "the world's most famous equation". While Serber's view of the strict lack of need to use mass–energy equivalence in designing the atomic bomb is correct, it does not take into account the pivotal role which this relationship played in making the fundamental leap to the initial hypothesis that large
atoms were energetically allowed to split into approximately equal parts (before this energy was in fact measured). In late 1938, while on the winter walk on which they solved the meaning of Hahn's experimental results and introduced the idea that would be called atomic fission, Lise Meitner and Otto Robert Frisch made direct use of Einstein's equation to help them understand the quantitative energetics of the reaction which overcame the "surface tensionlike" forces holding the nucleus together, and allowed the fission fragments to separate to a configuration from which their charges could force them into an energetic "fission". To do this, they made use of "packing fraction", or nuclear binding energy values for elements, which Meitner had memorized. These, together with use of E = mc2 allowed them to realize on the spot that the basic fission process was energetically possible: ...We walked up and down in the snow, I on skis and she on foot. ...and gradually the idea took shape... explained by Bohr's idea that the nucleus is like a liquid drop; such a drop might elongate and divide itself... We knew there were strong forces that would resist, ..just as surface tension. But nuclei differed from ordinary drops. At this point we both sat down on a tree trunk and started to calculate on scraps of paper. ...the Uranium nucleus might indeed be a ginger kid, ready to divide itself... But, ...when the two drops separated they would be driven apart by electrical repulsion, about 200 MeV in all. Fortunately Lise Meitner remembered how to compute the masses of nuclei... and worked out that the two nuclei formed... would be lighter by about onefifth the mass of a proton. Now whenever mass disappears energy is created, according to Einstein's formula E = mc2, and... the mass was just equivalent to 200 MeV; it all fitted!
Chapter 7
Binding Energy
Binding Energy is the mechanical energy required to disassemble a whole into separate parts. A bound system typically has a lower potential energy than its constituent parts; this is what keeps the system together often this means that energy is released upon the creation of a bound state. The usual convention is that this corresponds to a positive binding energy. In general, binding energy represents the mechanical work which must be done against the forces which hold an object together, disassembling the object into component parts separated by sufficient distance that further separation requires negligible additional work. At the atomic level the atomic binding energy of the atom derives from electromagnetic interaction and is the energy required to disassemble an atom into free electrons and a nucleus. Electron binding energy is a measure of the energy required to free electrons from their atomic orbits. This is more commonly known as ionization energy . At the nuclear level, binding energy is also equivalent to the energy liberated when a nucleus is created from other nucleons or nuclei . This nuclear binding energy (binding energy of nucleons into a nuclide) is derived from the strong nuclear force and is the energy required to disassemble a nucleus into the same number of free unbound neutrons and protons it is composed of, so that the nucleons are far/distant enough from each other so that the strong nuclear force can no longer cause the particles to interact . In astrophysics, gravitational binding energy of a celestial body is the energy required to expand the material to infinity. This quantity is not to be confused with the gravitational potential energy, which is the energy required to separate two bodies, such as a celestial body and a satellite, to infinite distance, keeping each intact (the latter energy is lower).
In bound systems, if the binding energy is removed from the system, it must be subtracted from the mass of the unbound system, simply because this energy has mass, and if subtracted from the system at the time it is bound, will result in removal of mass from the system . System mass is not conserved in this process because the system is not closed during the binding process.
Mass deficit Classically a bound system is at a lower energy level than its unbound constituents, its mass must be less than the total mass of its unbound constituents. For systems with low binding energies, this "lost" mass after binding may be fractionally small. For systems with high binding energies, however, the missing mass may be an easily measurable fraction. Since all forms of energy have mass, the question of where the missing mass of the binding energy goes, is of interest. The answer is that this mass is lost from a system which is not closed. It transforms to heat, light, higher energy states of the nucleus/atom or other forms of energy, but these types of energy also have mass, and it is necessary that they be removed from the system before its mass may decrease. The "mass deficit" from binding energy is therefore removed mass that corresponds with removed energy, according to Einstein's equation E = mc2. Once the system cools to normal temperatures and returns to ground states in terms of energy levels, there is less mass remaining in the system than there was when it first combined and was at high energy. Mass measurements are almost always made at low temperatures with systems in ground states, and this difference between the mass of a system and the sum of the masses of its isolated parts is called a mass deficit. Thus, if binding energy mass is transformed into heat, the system must be cooled (the heat removed) before the massdeficit appears in the cooled system. In that case, the removed heat represents exactly the mass "deficit", and the heat itself retains the mass which was lost. As an illustration, consider two objects attracting each other in space through their gravitational field. The attraction force accelerates the objects and they gain some speed toward each other converting the potential (gravity) energy into kinetic (movement) energy. When either the particles 1) pass through each other without interaction or 2) elastically repel during the collision, the gained kinetic energy (related to speed), starts to revert into potential form driving the collided particles apart. The decelerating particles will return to the initial distance and beyond into infinity or stop and repeat the collision (oscillation takes place). This shows that the system, which loses no energy, does not combine (bind) into a solid object, parts of which oscillate at short distances. Therefore, in order to bind the particles, the kinetic energy gained due to the attraction must be dissipated (by resistive force). Complex objects in collision ordinarily undergo inelastic collision, transforming some kinetic energy into internal energy (heat content, which is atomic movement), which is further radiated in the form of photons—the light and heat. Once the energy to escape the gravity is dissipated in the collision, the parts will oscillate at closer, possibly atomic, distance, thus looking like one solid object. This lost energy, necessary to overcome the potential barrier in order to separate the objects, is the binding
energy. If this binding energy were retained in the system as heat, its mass would not decrease. However, binding energy lost from the system (as heat radiation) would itself have mass, and directly represent of the "mass deficit" of the cold, bound system. Closely analogous considerations apply in chemical and nuclear considerations. Exothermic chemical reactions in closed systems do not change mass, but (in theory) become less massive once the heat of reaction is removed. This mass change is too small to measure with standard equipment. In nuclear reactions, however, the fraction of mass that may be removed as light or heat, i.e., binding energy, is often a much larger fraction of the system mass. It may thus be measured directly as a mass difference between rest masses of reactants and products. This is because nuclear forces are comparatively stronger than Coulombic forces associated with the interactions between electrons and protons that generate heat in chemistry.
Mass defect In simple words, the definition of mass defect can be stated as follows: Definition: The difference between the unbound system calculated mass and experimentally measured mass of nucleus is called mass defect. It is denoted by Δm. It can be calculated as follows: Mass defect = (unbound system calculated mass)  (measured mass of nucleus) i.e, (sum of masses of protons and neutrons)  (measured mass of nucleus) In nuclear reactions, the energy that must be radiated or otherwise removed as binding energy may be in the form of electromagnetic waves, such as gamma radiation, or as heat. Again, however, no mass deficit can in theory appear until this radiation has been emitted and is no longer part of the system. The energy given off during either nuclear fusion or nuclear fission is the difference between the binding energies of the fuel and the fusion or fission products. In practice, this energy may also be calculated from the substantial mass differences between the fuel and products, once evolved heat and radiation have been removed. When the nucleons are grouped together to form a nucleus, they lose a small amount of mass, i.e., there is mass defect. This mass defect is released as (often radiant) energy according to the relation E = mc2; thus binding energy = mass defect × c2. This energy is a measure of the forces that hold the nucleons together, and it represents energy which must be supplied from the environment if the nucleus is to be broken up. It is known as binding energy, and the mass defect is a measure of the binding energy because it simply represents the mass of the energy which has been lost to the environment after binding.
Mass excess
It is observed experimentally that the mass of the nucleus is smaller than the number of nucleons each counted with a mass of 1 a.m.u.. This difference is called mass excess. The difference between the actual mass of the nucleus measured in atomic mass units and the number of nucleons is called mass excess i.e. Mass excess = M  A = Excessenergy / c2 with : M equals the actual mass of the nucleus, in u. and : A equals the mass number. This mass excess is a practical value calculated from experimentally measured nucleon masses and stored in nuclear databases. For middleweight nuclides this value is negative in contrast to the mass defect which is never negative for any nuclide.
Nuclear binding energy A general and simple description of nuclear binding energy is the energy required to break apart, split, or break down, the nucleus of the atom into its component parts (nucleons), i.e. neutrons and protons. If the binding energy for the products is higher when light nuclei fuse, or when heavy nuclei split, either of these processes will result in a release of the "extra" binding energy, and this energy is referred to as nuclear energy. It is also loosely called nuclear power. The mass of the atom's nucleus is always less than the sum of the individual masses of the constituent protons and neutrons. This notable difference is a measure of the nuclear binding energy, which is a result of forces that hold the nucleus together. Because these forces result in the removal of energy when the nucleus is formed, and this energy has mass, mass is removed and is "missing" in the resulting nucleus. This missing mass is known as the mass defect and represents the evolved energy when the nucleus is bound. This energy may be removed as photons (gamma rays) or as the mass or kinetic energy of a number of different ejected particles. Total mass is conserved throughout the process, but the "mass defect" mass has merely moved off to a different place. The nuclear binding energies and forces are on the order of a million times greater than the electron binding energies of light atoms like hydrogen. Determining the relevant nuclear binding energy encompasses three steps of calculation, which involves the creation of mass defect by removing the mass as released energy. The mass defect of a nucleus represents the mass of the energy of binding of the nucleus, and is the difference between the mass of a nucleus and the sum of the masses of the nucleons of which it is composed.
Introduction
In order to obtain a reasonable understanding of nuclear binding energy, it is useful to first consider some of the basic principles involved in nuclear physics.
Nuclear energy An absorption or release of nuclear energy occurs in nuclear reactions or radioactive decay; the former are called endothermic reactions and the latter exothermic reactions. Energy is consumed or liberated because of differences in the nuclear binding energy between the incoming and outgoing products of the nuclear transmutation. It is useful to consider the bestknown classes of exothermic nuclear transmutations: fission and fusion. Nuclear energy may be liberated by atomic fission, when heavy atomic nuclei like uranium and plutonium are broken apart into lighter nuclei. The energy from fission is used to generate electric power in hundreds of locations worldwide. Nuclear energy is also released during atomic fusion, when light nuclei like hydrogen are combined to form heavier nuclei such as helium. The Sun and other stars use nuclear fusion to generate thermal energy which is later radiated from the surface, a type of stellar nucleosynthesis. In any exothermic nuclear process, nuclear mass may ultimately be converted to thermal energy, which is given off as heat and in doing so, carries away the mass with it. In order to quantify the energy released or absorbed in any nuclear transmutation, one must know the nuclear binding energies of the species involved.
The nuclear force Electrons and nuclei are kept together by electric attraction (negative attracts positive). Furthermore, electrons are sometimes shared by neighboring atoms or transferred to them (by processes of quantum physics), and this link between atoms creates many chemical compounds. But something else is needed to hold nuclei together, since all protons carried positive charges and repelled each other. Electric forces are definitely not the glue that holds nuclei together, they act in the wrong direction. It has been established that binding neutrons to nuclei clearly requires a nonelectrical attraction. All that suggests a different kind of force, a nuclear force, that is holding nuclei together. That force must be stronger than the electric repulsion at short distances, but weaker far away, or else different nuclei might tend to clump together. In other words, it had to be a shortrange force. A simplified visual analogy is the force between two small magnets. These are very difficult to separate when stuck together, but once pulled a short distance apart, the force between them drops almost to zero. Another way of communicating this process is: unlike gravity or electrical forces, the nuclear force is effective only at very short distances. At greater distances, the protons repel each other because they are positively charged, and charges of the same kind repel.
For that reason, the protons forming the nuclei of ordinary hydrogen; for instance, in a balloon filled with hydrogen; do not combine to form helium (a process which also would require some to combine with electrons and become neutrons). They cannot get close enough for the nuclear force, which attracts them to each other, to become important. Only under conditions of extreme pressure and temperature (for example, within the core of the sun), can such a process take place.
Physics of nuclei Nature contains nuclei of many different sizes. In hydrogen they contain just one proton, in heavy hydrogen ("deuterium") a proton and a neutron; in helium, two protons and two neutrons, and in carbon, nitrogen and oxygen  6, 7 and 8 of each particle, respectively. The weight of all these nuclei has been measured, and an interesting fact was noted: a helium nucleus weighed less than the sum of the weights of its components. The same held even more for carbon, nitrogen and oxygen  the carbon nucleus, for instance, was found to be slightly lighter than three helium nuclei. This describes the mass defect. Mass defect The reason for this "mass defect" has to do with Einstein's famous formula E=mc2, expressing the equivalence of energy and mass. By this formula, adding energy also increases mass (both weight and inertia), removing energy, decreases it. If a combination of particles contains extra energy; for instance, in a molecule of the explosive TNT—weighing it will reveal some extra mass (compared to its end products—an immeasurably small difference, for TNT). If on the other time we need invest energy to separate it into its components, the weight will be less than that of the components. The latter is the case with nuclei such as helium: to break them up into protons and neutrons, we would have to invest energy. On the other hand, if a process existed going in the opposite direction, by which hydrogen atoms could be combined to form helium, a lot of energy would be released. Namely, E=mc2 per nucleus, where m is the difference between the mass of the helium nucleus and the mass of four protons (plus 2 electrons, absorbed to create the neutrons of helium). As we go on to elements heavier than oxygen, the energy which can be gained by assembling them from lighter elements decreases, up to iron. For nuclei heavier than iron, one actually gains energy by breaking them up into 2 fragments. That, of course, is how energy is extracted by breaking up uranium nuclei in nuclear power reactors. The reason the trend reverses after iron is the growing positive charge of the nuclei. The electric force may be weaker than the nuclear force, but its range is greater: in an iron nucleus, each proton repels 25 other protons, while (one may argue) the nuclear force only binds close neighbors.
As nuclei grow bigger still, this disruptive effect becomes steadily more significant. By the time plutonium is reached (94 protons), nuclei can no longer accommodate their large positive charge, but emit their excess protons quite rapidly in the process of alpha radioactivity—the emission of helium nuclei, each containing two protons and two neutrons. (Helium nuclei are an especially stable combination.) This process becomes so rapid that still heavier nuclei are not found naturally on Earth. Solar binding energy A simplified explanation for nuclear fusion process, is as follows: Five billion years ago, the new sun having formed when gravity pulled together a vast cloud of gas and dust, from which the Earth and other planets also arose. The gravitational pull released energy and heated the early Sun, much in the way Helmholtz had proposed. Heat is the motion of atoms and molecules: the higher the temperature, the greater is their velocity and the more violent are their collisions. When the temperature at the center of the newlyformed Sun became great enough for collisions between nuclei to overcome their electric repulsion, nuclei began to stick together and protons were combined into helium, with some protons changing in the process to neutrons (plus positrons, positive electrons, which combine with electrons and are destroyed). This released nuclear energy and kept up the high temperature of the Sun's core, and the heat also kept the gas pressure high, keeping the Sun puffed up and stopping gravity from pulling it together any more. That, in greatly simplified terms, is the "nuclear fusion" process which still takes place inside the Sun. Different nuclear reactions may predominate at different stages of the Sun's existence, including the protonproton reaction and the carbonnitrogen cycle which involves heavier nuclei, but whose final product is still the combination of protons to form helium. A branch of physics, the study of "controlled nuclear fusion," has tried since the 1950s to derive useful power from "nuclear fusion" reactions which combine small nuclei into bigger ones, i.e., power to heat boilers, whose steam could turn turbines and produce electricity. Unfortunately, no earthly laboratory can match one feature of the solar powerhouse; the great mass of the Sun, whose weight keeps the hot plasma compressed and confines the "nuclear furnace" to the Sun's core. Instead, physicists use strong magnetic fields to confine the plasma, and for fuel they use heavy forms of hydrogen, which "burn" more easily. Still, magnetic traps can be rather unstable, and any plasma hot enough and dense enough to undergo nuclear fusion tends to slip out of them after a short time. Even with ingenious tricks, the confinement in most cases lasts only a small fraction of a second. Combining nuclei
Other small nuclei can similarly combine into bigger ones and release energy, but in combining such nuclei, the amount of energy released is much smaller. The reason is that while the process gains energy from letting the nuclear attraction do its work, it has to invest energy to force together positively charged protons, which also repel each other with their electric charge. Once iron is reached—a nucleus with 26 protons—this process no longer gains energy. In even heavier nuclei, we find energy is lost, not gained by adding protons. Overcoming the electric repulsion (which affects all protons in the nucleus) requires more energy than what is released by the nuclear attraction (effective mainly between close neighbors). Energy could actually by gained, however, by breaking apart nuclei heavier than iron. In the biggest nuclei (elements heavier than lead), the electric repulsion is so strong that some of them spontaneously eject positive fragments—usually nuclei of helium, which form very stable combinations ("alpha particles"). This spontaneous breakup is one of the forms of radioactivity found in nuclei. Nuclei heavier than uranium break up too quickly to be found in nature, although they can be produced artificially. As a broad trend, the heavier they are, the faster is their spontaneous decay. In summary, then: iron nuclei are the most stable ones, and the best sources of energy are therefore nuclei as far removed from iron as possible. One can combine the lightest ones—nuclei of hydrogen (protons)to form nuclei of helium, and that is how the Sun gets its energy. Or else one can break up the heaviest ones—nuclei of uranium—into smaller fragments, and that is what nuclear power companies do.
Nuclear Binding Energy A carbon nucleus of 12C (for instance) contains 6 protons and 6 neutrons. The protons are all positively charged and repel each other: they nevertheless stick together, showing the existence of another force—a nuclear attraction, the "nuclear force" which overcomes electric repulsion at very close range. Hardly any effect of this force is observed outside the nucleus, so it must have a much steeper dependence on distance—it is a short range force. The same force is also found to pull neutrons together, or neutrons and protons. The energy of the nucleus is negative with regard to the energy of the particles pulled apart to infinite distance (just like the energy of planets of the solar system), for one must invest energy to tear a nucleus apart into its individual protons and neutrons. "Mass spectrometers" have actually measured the masses of nuclei, which are always less than the sum of the masses of protons and neutrons which form them, and the difference—by Einstein's famous E = mc2 gives the binding energy of the nucleus. Nuclear Fusion
The binding energy of helium is appreciable, and seems to be the energy source of the Sun and of most stars. The sun is composed of 74% by mass hydrogen, and element whose nucleus is a single proton; energy is released when 4 protons combine into a helium nucleus, a process in which two of them are also converted to neutrons. The conversion of protons to neutrons is the result of another nuclear force, known as "the weak force" (the word "nuclear" is assumed here). The weak force also has a short range but is much weaker than the strong force. The weak force tries to make the number of neutrons and protons into the most energetically stable configuration. For nuclei containing less than 40 particles, these numbers are usually about equal. Protons and neutrons are closely related and are sometimes collectively known as nucleons. As the number of particles increases toward a maximum of about 209, the number of neutrons to main stability begins to outstrip the number of protons, until the ratio of neutrons to protons is about 2 to 3. The protons of hydrogen combine to helium only if they have enough velocity to overcome each other's repulsion and get within range of the strong nuclear attraction, which means they must form a very hot gas. Hydrogen hot enough for combining to helium requires an enormous pressure to keep it confined, but suitable conditions exist in the central regions of the Sun ("core"), where such pressure is provided by the enormous weight of the layers above the core, created by the Sun's strong gravity. The process of combining protons to form helium is an example of nuclear fusion. Our oceans have plenty of hydrogen, and helium does not harm the environment, so it would be desirable if physicists could harness nuclear fusion for more than the short time of an thermonuclear bomb] explosion, in order to provide the world with energy. Experiments in that direction have so far have only partially succeeded. Sufficiently hot hydrogen will also be ionized, and to confine it, very strong magnetic fields have been used, because charged particles (like those trapped in the Earth's radiation belt) are guided by magnetic field lines. Fusion experiments also rely on heavy hydrogen which fuses more easily, and gas densities have been kept moderate. In spite of all such tricks, though fusion energy has been released, so far more energy is consumed by the apparatus then is yielded by the process. The Curve of Binding Energy In the main isotopes of light nuclei, such as carbon, nitrogen and oxygen, the most stable combination of neutrons and of protons is indeed equal (this continues to element 20, calcium). However, as we move to heavier nuclei, the disruptive energy of protons increases, since they are confined to a tiny volume and repel each other. The energy of the strong force holding the nucleus together also increases, but at a slower rate, as if inside the nucleus, only nucleons close to each other are tightly bound, not ones more widely separated. The net binding energy of a nucleus is that of the nuclear attraction, minus the disruptive energy of the electric force. As nuclei get heavier than helium, their net binding energy
per nucleon (deduced from the difference in mass between the nucleus and the sum of masses of component nucleons) grows more and more slowly, reaching its peak at iron. As nucleons are added, the total nuclear binding energy always increases—but the total disruptive energy of electric forces (positive protons repelling other protons) also increases, and past iron, the second increase outweighs the first. One may say 56Fe is the most efficiently bound nucleus. To reduce the disruptive energy, the weak interaction allows the number of neutrons to exceed that of protons—for instance, in the main isotope of iron, 26 protons but 30 neutrons. Of course, isotopes also exist where the number of neutrons differs, but if these are too far from stability, after some time nucleons convert to a more stable isotope by beta emission radioactivity—protons turn into neutrons by dragging in an electron (or emitting a positron, the positive counterpart of the electron), or neutrons become protons by emitting electrons. Among the heaviest nuclei, containing 210 or more nucleons, electric forces may be so destabilizing that entire chunks of the nucleus get ejected, usually in combinations of 2 protons and 2 neutrons ("alpha particles," actually fast helium nuclei) which are extremely stable. The curve of binding energy (drawing) plots binding energy per nucleon against atomic mass. It has its main peak at iron and then slowly decreases again, and also a narrow isolated peak at helium, which as noted is very stable. The heaviest nuclei in nature, uranium 238U , are unstable, but having a lifetime of 4.5 billion years, close to the age of the Earth, they are still relatively abundant; they (and other nuclei heavier than iron) may have formed in a supernova explosion (reference #8) preceding the formation of the solar system. The most common isotope of thorium, 232Th, also undergoes α particle emission, and its halflife (time over which half a number of atoms decays) is even longer, by several times. In each of these, radioactive decay produces "daughter isotopes" which are also unstable, starting a chain of decays which ends in some stable isotope of lead.
Determining nuclear binding energy Calculation can be employed to determine the relevant nuclear binding energy involved. This process is three general steps: • • •
Determining the Mass defect Conversion of mass defect into energy Expressing nuclear binding energy as energy per mole of atoms, or as energy per nucleon.
Conversion of mass defect into energy Mass defect is called the difference between the mass of a nucleus and the sum of the masses of the nucleons of which it is composed. The mass defect is determined by calculating three ascertained quantities.
These are: the actual mass of the nucleus, the composition of the nucleus (number of protons and of neutrons), and the masses of a proton and of a neutron. This is then followed by converting the mass defect into energy. This quantity is the nuclear binding energy, however it must be expressed as energy per mole of atoms or as energy per nucleon.
Fission and fusion Nuclear energy is released by the splitting (fission) or merging together (fusion) of the nuclei of atom(s). The conversion of nuclear massenergy to a form of energy which can remove some mass when the energy is removed, is consistent with the massenergy equivalence formula ΔE = Δmc², in which ΔE = energy release, Δm = mass defect, and c = the speed of light in a vacuum (a physical constant). When this equation is used in this way, the mass "changes" only because it is removed from the system, not because it is converted or destroyed (the removed binding energy retains and accounts for the missing mass, which is a conserved quantity). Nuclear energy was first discovered by French physicist Henri Becquerel in 1896, when he found that photographic plates stored in the dark near uranium were blackened like Xray plates, which had been just recently discovered at the time 1894. Nuclear chemistry can be used as a form of alchemy to turn lead into gold or change any atom to any other atom (albeit through many steps). Radionuclide (radioisotope) production often involves irradiation of another isotope (or more precisely a nuclide), with alpha particles, beta particles, or gamma rays. Nickel62 has the highest binding energy per nucleon of any isotope. If an atom of lower average binding energy is changed into an atom of higher average binding energy, energy is given off. The chart shows that fusion of hydrogen, the combination to form heavier atoms, releases energy, as does fission of uranium, the breaking up of a larger nucleus into smaller parts. Stability varies between isotopes: the isotope U235 is much less stable than the more common U238.
Nuclear energy is released by three exoenergetic (or exothermic) processes: •
• •
Radioactive decay, where a neutron or proton in the radioactive nucleus decays spontaneously by emitting either particles, electromagnetic radiation (gamma rays), neutrinos (or all of them) Fusion, two atomic nuclei fuse together to form a heavier nucleus Fission, the breaking of a heavy nucleus into two (or more rarely three) lighter nuclei
Practice: Binding energy for atoms The amount of energy required to break the nucleus of an atom into its isolated nucleons is called nuclear binding energy. The measured mass deficits of isotopes are always listed as mass deficits of the neutral atoms of that isotope, and mostly in MeV. As a consequence, the listed mass deficits are not a measure for the stability or binding energy of isolated nuclei, but for the whole atoms. This has very practical reasons, because it is very hard to totally ionize heavy elements, i.e. strip them of all of their electrons. This practice is useful for other reasons, too: Stripping all the electrons from a heavy unstable nucleus (thus producing a bare nucleus) will change the lifetime of the nucleus, indicating that the nucleus cannot be treated independently (Experiments at the heavy ion accelerator GSI). This is also evident from phenomena like electron capture. Theoretically, in orbital models of heavy atoms, the electron orbits partially inside the nucleus (it doesn't orbit in a strict sense, but has a nonvanishing probability of being located inside the nucleus).
Of course, a nuclear decay happens to the nucleus, meaning that properties ascribed to the nucleus will change in the event. But for the following considerations and examples, you should keep in mind that "mass deficit" as a measure for "binding energy", and as listed in nuclear data tables, means "mass deficit of the neutral atom" and is a measure for stability of the whole atom.
Nuclear binding energy curve
Binding energy per nucleon of common isotopes. In the periodic table of elements, the series of light elements from hydrogen up to sodium is observed to exhibit generally increasing binding energy per nucleon as the atomic mass increases. This increase is generated by increasing forces per nucleon in the nucleus, as each additional nucleon is attracted by all of the other nucleons, and thus more tightly bound to the whole. The region of increasing binding energy is followed by a region of relative stability (saturation) in the sequence from magnesium through xenon. In this region, the nucleus has become large enough that nuclear forces no longer completely extend efficiently across its width. Attractive nuclear forces in this region, as atomic mass increases, are nearly balanced by repellent electromagnetic forces between protons, as the atomic number increases.
Finally, in elements heavier than xenon, there is a decrease in binding energy per nucleon as atomic number increases. In this region of nuclear size, electromagnetic repulsive forces are beginning to gain against the strong nuclear force. At the peak of binding energy, nickel62 is the most tightly bound nucleus (per nucleon), followed by iron58 and iron56. This is the approximate basic reason why iron and nickel are very common metals in planetary cores, since they are produced profusely as end products in supernovae and in the final stages of silicon burning in stars. However, it is not binding energy per defined nucleon (as defined above) which controls which exact nuclei are made, because within stars, neutrons are free to convert to protons to release even more energy, per generic nucleon, if the result is a stable nucleus with a larger fraction of protons. Thus, iron56 has the most binding energy of any group of 56 nucleons (because of its relatively larger fraction of protons), even while having less binding energy per nucleon than nickel62, if this binding energy is computed by comparing Ni62 with its disassembly products of 28 protons and 34 neutrons. In fact, it has been argued that photodisintegration of 62Ni to form 56Fe may be energetically possible in an extremely hot star core, due to this beta decay conversion of neutrons to protons. It is generally believed that iron56 is more common than nickel isotopes in the universe for mechanistic reasons, because its unstable progenitor nickel56 is copiously made by staged buildup of 14 helium nuclei inside supernovas, where it has no time to decay to iron before being released into the interstellar medium in a matter of a few minutes as a star explodes. However, nickel56 then decays to iron56 within a few weeks. The gamma ray light curve of such a process has been observed to happen in type IIa supernovae, such as SN1987a. In a star, there are no good ways to create nickel62 by alphaaddition processes, or else there would presumably be more of this highly stable nuclide in the universe.
Measuring the binding energy The existence of a maximum in binding energy in mediumsized nuclei is a consequence of the tradeoff in the effects of two opposing forces which have different range characteristics. The attractive nuclear force (strong nuclear force), which binds protons and neutrons equally to each other, has a limited range due to a rapid exponential decrease in this force with distance. However, the repelling electromagnetic force, which acts between protons to force nuclei apart, falls off with distance much more slowly (as the inverse square of distance). For nuclei larger than about four nucleons in diameter, the additional repelling force of additional protons more than offsets any binding energy which results between further added nucleons as a result of additional strong force interactions; such nuclei become less and less tightly bound as their size increases, though most of them are still stable. Finally, nuclei containing more than 209 nucleons (larger than about 6 nucleons in diameter) are all too large to be stable, and are subject to spontaneous decay to smaller nuclei.
Nuclear fusion produces energy by combining the very lightest elements into more tightly bound elements (such as hydrogen into helium), and nuclear fission produces energy by splitting the heaviest elements (such as uranium and plutonium) into more tightly bound elements (such as barium and krypton). Both processes produce energy, because middlesized nuclei are the most tightly bound of all. As seen above in the example of deuterium, nuclear binding energies are large enough that they may be easily measured as fractional mass deficits, according to the equivalence of mass and energy. The atomic binding energy is simply the amount of energy (and mass) released, when a collection of free nucleons are joined together to form a nucleus. Nuclear binding energy can be easily computed from the easily measurable difference in mass of a nucleus, and the sum of the masses of the number of free neutrons and protons that make up the nucleus. Once this mass difference, called the mass defect or mass deficiency, is known, Einstein's massenergy equivalence formula E = mc² can be used to compute the binding energy of any nucleus. (As a historical note, early nuclear physicists used to refer to computing this value as a "packing fraction" calculation.) For example, the atomic mass unit (1 u) is defined to be 1/12 of the mass of a 12C atom— but the atomic mass of a 1H atom (which is a proton plus electron) is 1.007825 u, so each nucleon in 12C has lost, on average, about 0.8% of its mass in the form of binding energy.
Semiempirical formula for nuclear binding energy For a nucleus with A nucleons, including Z protons and N neutrons, a semiemipirical formula for the binding energy (B.E.) per nucleon (A) is:
where the binding energy is in MeV for the following numerical values of the constants: a = 14.0; b = 13.0; c = 0.585; d = 19.3; e = 33. The first term is called the saturation contribution and ensures that the B.E. per nucleon is a surface tension is the same for all nuclei to a first approximation. The term effect and is proportional to the number of nucleons that are situated on the nuclear surface; it is largest for light nuclei. The term is the Coulomb electrostatic repulsion; this becomes more important as Z increases. The symmetry correction term takes into account the fact that in the absence of other effects the most stable arrangement has equal numbers of protons and neutrons; this is because the np interaction in a nucleus is stronger than either the nn or pp interaction. The pairing term
is purely empirical; it is + for eveneven nuclei and  for oddodd nuclei.
Electron binding energy
The term ionization energy (EI) of an atom or molecule is the minimum energy required to remove (to infinity) an electron from the atom or molecule isolated in free space and in its ground electronic state. This quantity was formerly called ionization potential, and was at one stage measured in volts. The name "ionization energy" is now strongly preferred. In atomic physics the ionization energy is measured using the unit "electronvolt" (eV). In chemistry, the value is usually given in kJ/mol (or formerly kcal/mol). This value is strictly the "molar ionization energy" and corresponds to the energy required to remove (to infinity) one mole of electrons from one mole of gaseous atoms or molecules. However it is often just called "ionization energy" . More generally, the nth ionization energy is the energy required to strip off the nth electron after the first n − 1 electrons have been removed. It is considered a measure of the "reluctance" of an atom or ion to surrender an electron, or the "strength" by which the electron is bound; the greater the ionization energy, the more difficult it is to remove an electron. The ionization energy is, thus, an indicator of the reactivity of an element. Elements with a low ionization energy tend to be reducing agents and form cations, which in turn combine with anions to form salts. The term "ionization energy" is sometimes used as a name for the work needed to remove (to infinity) the topmost electron from an atom or molecule adsorbed onto a surface. However, due to interactions with the surface, this value differs from the ionization energy of the atom or molecule in question when it is in free space. So, in the case of surfaceadsorbed atoms and molecules, it may be better to use the more general term "electron binding energy", in order to avoid confusion. Both these names are also sometimes used to describe the work needed to remove an electron from a "lower" orbital (i.e., not the topmost orbital) to infinity, both for free and for adsorbed atoms and molecules; in such cases it is necessary to specify the orbital from which the electron has been removed. Electron binding energy (BE), more accurately, is the energy required to release an electron from its atomic or molecular orbital when adsorbed to a surface rather than a free atom. Binding energy values are normally reported as positive values with units of "eV". The binding energies of 1s electrons are roughly proportional to (Z1)² (Moseley's law).
Values and trends Generally the (n+1)th ionization energy is larger than the nth ionization energy. Always, the next ionization energy involves removing an electron from an orbital closer to the nucleus. Electrons in the closer orbitals experience greater forces of electrostatic attraction; thus, their removal requires increasingly more energy. Some values for elements of the third period are given in the following table: Successive molar ionization energies in kJ/mol (96.485 kJ/mol = 1 eV/particle) Element First Second Third Fourth Fifth Sixth Seventh 496 4,560 Na 738 1,450 7,730 Mg 577 1,816 2,881 11,600 Al 786 1,577 3,228 4,354 16,100 Si 1,060 1,890 2,905 4,950 6,270 21,200 P 999.6 2,260 3,375 4,565 6,950 8,490 27,107 S
Cl Ar
1,256 1,520
2,295 3,850 2,665 3,945
5,160 6,560 9,360 11,000 5,770 7,230 8,780 12,000
Large jumps in the successive molar ionization energies occur when passing noble gas configurations. For example, as can be seen in the table above, the first two molar ionization energies of magnesium (stripping the two 3s electrons from a magnesium atom) are much smaller than the third, which requires stripping off a 2p electron from the very stable neon configuration of Mg2+.
Periodic trend for ionization energy. Each period begins at a minimum for the alkali metals, and ends at a maximum for the noble gases. Ionization energy is also a periodic trend within the periodic table organization. Moving left to right within a period or upward within a group, the ionization energy generally increases. As the atomic radius decreases, it becomes harder to remove an electron that is closer to a more positively charged nucleus.
Electrostatic explanation Atomic ionization energy can be predicted by an analysis using electrostatic potential and the Bohr model of the atom, as follows. Consider an electron of charge e and an atomic nucleus with charge +Ze, where Z is the number of protons in the nucleus. According to the Bohr model, if the electron were to approach and bind with the atom, it would come to rest at a certain radius a. The electrostatic potential V at distance a from the ionic nucleus, referenced to a point infinitely far away, is:
Since the electron is negatively charged, it is drawn inwards by this positive electrostatic potential. The energy required for the electron to "climb out" and leave the atom is:
This analysis is incomplete, as it leaves the distance a as an unknown variable. It can be made more rigorous by assigning to each electron of every chemical element a characteristic distance, chosen so that this relation agrees with experimental data. It is possible to expand this model considerably by taking a semiclassical approach, in which momentum is quantized. This approach works very well for the hydrogen atom, which only has one electron. The magnitude of the angular momentum for a circular orbit is:
The total energy of the atom is the sum of the kinetic and potential energies, that is:
Velocity can be eliminated from the kinetic energy term by setting the Coulomb attraction equal to the centripetal force, giving:
Solving the angular momentum for v and substituting this into the expression for kinetic energy, we have:
This establishes the dependence of the radius on n. That is:
Now the energy can be found in terms of Z, e, and r. Using the new value for the kinetic energy in the total energy equation above, it is found that:
At its smallest value, n is equal to 1 and r is the Bohr radius a0. Now, the equation for the energy can be established in terms of the Bohr radius. Doing so gives the result:
Quantummechanical explanation According to the more complete theory of quantum mechanics, the location of an electron is best described as a probability distribution. The energy can be calculated by integrating over this cloud. The cloud's underlying mathematical representation is the wavefunction which is built from Slater determinants consisting of molecular spin orbitals. These are related by Pauli's exclusion principle to the antisymmetrized products of the atomic or molecular orbitals. In general, calculating the nth ionization energy requires calculating the energies of Z − n + 1 and Z − n electron systems. Calculating these energies exactly is not possible except for the simplest systems (i.e. hydrogen), primarily because of difficulties in integrating the correlation terms. Therefore, approximation methods are rountinely employed, with different methods varying in complexity (computational time) and in accuracy compared to empirical data. This has become a wellstudied problem and is routinely done in computational chemistry. At the lowest level of approximation, the ionization energy is provided by Koopmans' theorem.
Vertical ionization energy and adiabatic ionization energy The geometry of a molecular ion may be different from the neutral molecule. The measured ionization energy can refer to the vertical ionization energy, in which case the ion has the same geometry as the neutral molecule, or to the adiabatic ionization energy, in which case the ion has its lowest energy, relaxed geometry. This is illustrated in the figure. For a diatomic molecule the only geometry change possible is the bond length. The figure shows an ion with a slightly longer bond length than the neutral molecule. The harmonic potential energy surfaces are shown in green (neutral) and red (ion) with vibrational energy levels. The vertical ionization energy is always greater than the adiabatic ionization energy.
Gravitational binding energy The gravitational binding energy of an object consisting of loose material, held together by gravity alone, is the amount of energy required to pull all of the material apart, to infinity. It is also the amount of energy that is liberated (usually in the form of heat) during the accretion of such an object from material falling from infinity. The gravitational binding energy of a system is equal to the negative of the total gravitational potential energy, considering the system as a set of small particles. For a system consisting of a celestial body and a satellite, the gravitational binding energy will
have a larger absolute value than the potential energy of the satellite with respect to the celestial body, because for the latter quantity, only the separation of the two components is taken into account, keeping each intact. For a spherical mass of uniform density, the gravitational binding energy U is given by the formula
where G is the gravitational constant, M is the mass of the sphere, and r is its radius. This is 80% greater than the energy required to separate to infinity the two hemispheres of the spherical mass. Assuming that the Earth is a uniform sphere (which is not correct, but is close enough to get an orderofmagnitude estimate) with M = 5.97×1024kg and r = 6.37×106m, U is 2.24×1032J. This is roughly equal to one week of the Sun's total energy output. It is 37.5 MJ/kg, 60% of the absolute value of the potential energy per kilogram at the surface. According to the virial theorem, the gravitational binding energy of a star is about two times its internal thermal energy.
Derivation for a uniform sphere The gravitational binding energy of a sphere is found by imagining that it is pulled apart by successively moving spherical shells to infinity, the outermost first, and finding the total energy needed for that. If we assume a constant density ρ then the masses of a shell and the sphere inside it are: and The required energy for a shell is the negative of the gravitational potential energy:
Integrating over all shells we get:
Remembering that ρ is simply equal to the mass of the whole divided by its volume for objects with uniform density we get:
And finally, plugging this in to our result we get: