Contents
1. Introduction
2. A Uniformly Moving Charge
3. A Charge in Arbitrary Motion
4. Radiation
5. The Electron Synchrotron
6. References
Introduction
Maxwellian electrodynamics is naturally relativistic; that is, it conforms to Einstein's requirement of invariance under transformations between coordinate systems moving with constant relative velocity. It has exactly the same form in any inertial system, and light travels with the same velocity c in any system. The Lorentz transformation of space and time coordinates was established for electromagnetism before relativity was formulated and it was known that this transformation applies to all physical processes. In particular, the time transformation is real, and not just a substitution that preserves the form of the equations. The formalism of relativity makes working with electromagnetism much easier. In this article, I will try to explain as clearly as I can how the electromagnetic field of a point charge in arbitrary motion can be determined at any point in space. A very important application of this theory is the radiation by moving charges.
This material is excellently explained in chapters 11, 12 and 14 of Jackson's classic text, to which the reader is referred. On first sight, the material seems very difficult and unclear, even if it has been mastered before and is now being reviewed after forgetting it. For the beginner, it may be truly frightening and formidable. The beauty of Jackson's exposition is that when one digs in and tries to understand each statement, the subject becomes continually clearer and in a few hours its elegance shines out, and it is seen to be logical and satisfying. This is the evidence of the author's comprehensive understanding achieved through labor and thought, that he communicates to you. In any case where this does not happen, the fault is more often with the author than with the reader, and particularly if the subject is a false one. Relativistic electrodynamics is one of the best-known and verified physical theories, the work of giants. Even the classical theory, as presented here for the model of a point charge, is of wide applicability. I hope I can make some points clearer in this article, and help the reader to understand.
The reader will need some preparation in relativity, but that is not difficult to acquire, even with Jackson alone. I will use the index notation for tensors with Greek indices taking the values 0,1,2,3 for the coordinates ct, x, y, z. A sum is implied over the same index in contravariant (high) and covariant (low) positions. This notation is very much like that of Euclidean tensors in 3-dimensional space, except that the distinction between contravariant and covariant is necessary because the metric tensor gαβ is diagonal with g00 = 1, g11 = g22 = g33 = -1. Changing from contravariant to covariant means only changing the sign of the space components. Then, if Aα = (A0, A) [In expressions like this, subscripts are not tensor indices, but simply labels.] Then, A · B = AαBα = A0B0 - A·B.
If xα is the coordinate four-vector (ct,r), then x · x = (ct)2 - r2 = s2 is a scalar, and hence an invariant, under Lorentz transformation. It is the invariant interval between the event X described by the vector, and the event O at the origin, t = 0 and r = 0. If s2 > 0, then a Lorentz transformation to a coordinate system where X and O occur at the same point, but at a later or earlier time, is possible. If s2 < 2 =" r2," r =" ±ct." dt =" γdτ," t =" 0." t =" γt'" t =" t'/γ." t =" γt'" r2 =" b2" 1 =" β2γ2," r =" γ[(b2" n =" (-β)[(vt" r =" (b2" ex =" Ex'," ey =" γEy'," bz =" βEy'," ex =" -γqvt/(b2" ey =" qb/(b2" bz =" βEy." v2t2 =" b2" v2t2 =" r2" v2t2 =" r2[1" e =" qr/r3γ2(1" e =" γq/r2" bz =" βEy," t =" 2b/γv." dl =" 4πI/c," x =" 0" i =" q/(2b/γv)" x0 =" ct." z =" x" k =" -(2π)-4∫d3k" ko =" ±κ,"> 0, and zo = cto <> 0, then we can close the contour below. The large semicircle contributes nothing to the integral, so the integral will be equal to 2πi time the sum of the residues at the two poles, which are exp(-iκzo)/2κ and -exp(iκzo)/2κ. This gives D(z) = θ(zo)(2π)-3∫d3k exp(ik·z)sin κzo/κ. The theta function θ(zo) is zero when its argument is negative, +1 when its argument is positive. d3k = κ2dk sinθdθdφ, so we can do the integral over the angles right away. After a little manipulation, we find D(z) = [θ(zo)/2π2]∫(0,∞)sinκr sinκzodκ. Expressing the product of the sines as the sum of the cosines of the sum and difference angles, writing them as exponentials and combining terms to extend the limits of integration from -∞ to +∞, and using the Fourier transform of the delta function, we have D(z) = [θ(zo)/4πr]δ(r - zo), where r is the distance between source and field points. The delta function forces evaluation at the retarded time t' = t - r/c. This solution has all we could desire for the effects of a disturbance from a impulse at the origin at t = 0 at the point z. The Green's function with these properties is called the retarded Green's function, or propagator. It is, of course, of general utility, and represents another theoretical triumph.
We can express the propagator in manifestly invariant terms by considering δ[(x - x')2]. As we know, δ[f(x)] = δ(x - x')/|f'(x')|, where x' is a root of f(x). Now δ[(x - x')2] = δ[(xo - xo')2 - |x - x'|2] = δ[(xo - xo' - r)(xo - xo' + r)] = (1/2r)[d(xo - xo' - r) + δ(xo - xo' + r)]. Since the theta functions selects only one of the delta functions in the sum, we can write D(x - x') = (1/2π)θ(xo - xo')δ[(x - x')2]. The theta function is a Lorentz invariant, since the past and future light cones are invariantly separated, and the argument of the delta function is also a Lorentz invariant. Therefore, the propagator has been expressed in a manifestly invariant form independent of a coordinate system.
The solution to our problem is now easily obtained. Omitting any solution of the homogeneous wave equation (which would be fields not due to our moving charge), we have Aα(x) = (4π/c)∫d4x'D(x - x')Jα(x'). If you substitute this expression in the inhomogeneous wave equation, you will find that the equation is satisfied. This is the promised solution.
Now we must find Jα(x'), the source current. In some coordinate system, suppose the path of the charge is r = r(t). If the point charge is q, then the charge density is ρ = qδ[x - r(t)] and the current density is ρv = qvδ[x - r(t)]. The current density Jα = (ρc, ρv) will not be expressed in a relativistically invariant way, and so our potential Aα will not be relativistically invariant. To escape from this inconvenience, we can start by parametrizing the trajectory in terms of the proper time of the charge τ instead of t. Since dτ = dt/γ, this is not hard to do by integration. Now, r(τ) specifies an invariant world line that does not depend on the coordinate system. Of course, we need a coordinate system to specify it in any particular case, but we can imagine a world line drawn in 4-dimensional space, labelled with the proper time, that is invariant, and which determines the functions r(τ) in any coordinate system.
For Jα(ct,x), we want to pick out the time t that corresponds to any point x(τ) on the world line. This can be done with a delta function δ[t - ct(τ)], where t(τ) returns the time corresponding to r(τ), and is the 0-component of the position 4-vector. Allowing a choice from any value of τ, we are led to the integral Jα = ec∫dτUα(τ)δ[x - r(τ)], where Uα = (γc, γv) is the tensor 4-velocity of the charge. The delta function is a 4-dimensional one, and r(t) is the tensor 4-position of the charge. This expression is relativistically invariant, since it contains only Lorentz tensors. If it reduces to the correct charge density and current in a particular coordinate system, then it is what we want.
Choose a coordinate system, then, and do the integral over τ using the functions that result. From the rule for the delta function of a function, integration over the time delta function sets t equal to t' and divides by γc. Then, J0 = qc(γc)(1/γc)δ[x - r(t)] = qδ[x - r(t)], exactly what is wanted. The space components of Jα also are easily seen to give the desired results. We are now confident about our invariant expression for Jα.
Now all we have to do is substitute D(x - x') and Jα in the integral for Aα to find an invariant expression for Aα. The integrals are done using the delta functions, and we find that Aα(x) = qUα(τ)/Vβ[x - r(t)]β, evaluated at τ = τo, the retarded proper time. The most important property of this expression is that it is manifestly covariant, constructed only of Lorentz tensors. The fields can now be found by differentiation, but this is not a simple process, and we shall not do it here, but only quote the results. It is simpler, in fact, to differentiate before the final integration so the retardation is easier to handle, but we won't do that either. See Jackson if you need the expressions.
The time τo comes from the condition [x - rτo]2 = 0 enforced by the space part of the 4-dimensional delta function in the integral. This means that xo - ro(τo) = |x - r(τo)| = R, the retarded distance. Further, U·(x - r) = γcR - γv·nR = γcR(1 - β·n), with the symbols meaning the same as in the section on the uniformly moving charge, and where this expression will be recognized. From the expression for Aα, we then find the components φ = [q/(1 - β·n)]ret and A = [qβ/(1 - β·n)]ret. These potentials are expressed in terms of the retarded position and time, as we defined in connection with the uniformly moving charge. They are called the Liénard-Wiechert potentials, which were found in 1898 on the basis of Maxwell's equations alone, without the help of relativity.
The electric field is E = q{(n-β)/[γ2R2(1 - β·n)3]}ret + (q/c){n x [(n - β)x(dβ/dt)]/[R(1 - β·n)3]}, and the magnetic field is B = [n x E]ret. The first term is independent of the acceleration, and falls off as R-2, so it resembles a static field. The fields of the second term fall off as R-1, so they are radiation fields that transport energy to large distances, and are smaller by a factor of c-1.
Let's check our previous results for a charge in uniform motion. The y component of the field will be Ey = q cosθ/γ2R2(1 - β·n)3. Multiplying top and bottom by R, and using γR(1 - β·n) = (b2 + γ2v2t2)1/2, we find our previous result at once, Ey = qγb/(b2 + γ2v2t2)3/2. Ex = -q(cosθ + β)/γ 2R2(1 - β·n)3. Again multiplying top and bottom by R, and using R cosθ + βR = vt, we find Ex = -qγvt/(b2 + γ2v2t2)3/2. This confirms our earlier procedure of Lorentz-transforming the static fields.
Radiation
When β is much less than unity, the fields are very closely E = qn/R2 + n x (n x dβ/dt)(e/cR), where n is a unit vector from the retarded position at retarded distance R. If distances are not large, R is approximately the present distance r. This is quite a different limit from the unrealistic c → ∞, where the radiation field also vanishes. In the radiation field, the electric vector lies in the plane of dv/dt = a and n and is perpendicular to the radius vector. If the angle between a and n is θ then the magnitude of the electric field is E = (qa/c2r)sinθ. The energy flux per unit solid angle in the direction of n will be dP/dΩ = r2c|E|2/4π = (q2/4πc3)a2sin2θ. The total power radiated is found by integration over 4π steradians. ∫2πsin3θdθ = 8π/3, so P = (2e2/3c3)a2. This is Larmor's result for the radiation from an accelerated charge.
Jackson shows how to find the power radiated by a relativistic electron by a clever extension of Larmor's formula. The direct use of the fields is rather laborious. The result is P = (2e2/c)γ6[(dβ/dt)2 - (β x dβ/dt)2], which Liénard found in 1898. This is a very useful formula for finding the power radiated by fast particles, as in accelerators, or in astrophysics.
The Electron Synchrotron
The first particle accelerators were linear accelerators using high-voltage DC power supplies, such as the Van de Graaff and Cockroft-Walton machines, which could attain about 1 MeV. Next came the cyclotron (1934), where the circular orbit made periodic acceleration possible with lower RF voltages on the dees, the accelerating electrodes. The angular velocity of revolution depended only on the e/m ratio of the particles, not on their speed, so a constant RF frequency could be used. Protons and other heavy particles could be accelerated to about 25 MeV. These accelerators were very useful for nuclear studies, but the energies were too low for the creation of new particles. Cyclotrons cannot accelerate electrons because of the rapid increase in mass that brings them out of step with a constant RF frequency. The betatron (1940), which could accelerate electrons to similar energies by using transformer action, was then devised. Great attention had to be paid to designing the magnetic field so that the electron orbit would remain constant in radius and focussed, but the frequency problem did not arise. Betatrons provide a pulse of fast electrons at the end of each cycle of increasing magnetic field. For more on betatrons, see The Particle Electron.
Cyclotrons could be used to accelerate heavy particles even when relativistic effects entered by changing the accelerating frequency. Such FM cyclotrons or synchrocyclotrons were indeed developed, but the frequency modulation is very troublesome. Betatrons had the disadvantage of very heavy magnetic cores, and so their size was limited. In 1947, the first electron synchrotron was constructed in California, following the ideas of McMillen (1947) in the U.S. and Veksler (1946) in the U.S.S.R., who introduced the idea of synchronous acceleration. There were some smaller experimental synchrotrons, but the electron synchrotron was born full-grown, the first one providing 300 MeV electrons, and a dozen others following soon after. These electrons are generally used to produce high-energy photons in collisions with a target, which then can cause photonuclear reactions, such as the production of π mesons.
The electron synchrotron takes advantage of two consequences of the small mass of the electron. First, the electron is easily accelerated to speeds very close to c, when their orbital period in a magnetic field will not change significantly with energy. Secondly, the electron is easily deflected by a magnetic field, even at high energies, so that accelerators can be a convenient size. A diagram of an electron synchrotron is shown at the right. The toroidal vacuum chamber, and the C-shaped magnet cores, are not shown. A magnetic field B, shown directed into the page, deflects electrons into a circular orbit of radius r in which they move anticlockwise, approximately at speed c. The kinetic energy of the electrons is T = mc2(γ - 1). The electrons are accelerated on each revolution by the electric field in a resonant cavity, shown at the left. In order for the radius to remain constant, the magnetic field B must be increased synchronously with the increase in energy of the electrons, which gives the name to the machine. At the end of an accelerating cycle, the accelerating voltage is turned off, while the magnetic field continues to increase, causing the beam to spiral in and strike the target. This happens about 60 times a second, so the beam appears continuous to the human observer.
The magnet windings are usually connected across a large capacitor to form a resonant circuit, so the magnet power supply need only furnish the losses, and the power factor is acceptable. At the beginning of a cycle, a large pulse of electrons is injected by a thermionic electron gun in the direction of the orbit. Most of these electrons are lost, but a sufficient number assume orbits that do not strike the vacuum chamber walls. The magnetic field B must be carefully shaped to focus the beam. In general, B bows outward like the normal fringing field at an air gap, and can be described by B = Bo(ro/r)n. If n > 0, the electron orbits will oscillate stably about the orbital plane, and if n < 1 the orbits will oscillate stably in a radial direction. As B increases, these oscillations will be damped so that the beam will become narrow and well-defined. The electrons must also form a bunch along the orbit so that they will have phase stability. An electron that receives a little too much energy on one pass will forge ahead and receive less on the next pass, and the same sort of compensation will result for an electron that falls behind. The focussing and phase stability are necessary for achieving a useful beam current, which is usually in the region of microamperes.
Electrons are injected with energies of around 100 keV. They are then accelerated to about 2 MeV by betatron action. The necessary large field within the orbit is provided by flux bars of relatively small cross section that pass between N and S poles of the magnets. Initially, much of the flux passes by this route because of the much smaller air gap, but at the end of betatron acceleration the flux bars saturate and thereafter act like an air gap, most of the flux passing through the main pole faces to steer the electrons. At 2 MeV, the electron speed has become constant enough near c that constant frequency RF acceleration can take over. With an acceleration of 10 keV per pass, and a frequency of 47.7 MHz (appropriate for r = 1 m), the electrons will gain 477 MHz in 10 ms, neglecting losses. The synchrotron principle works very well for electrons.
A large number of electron synchrotrons have been built, of which the largest seems to be the 10 GeV machine at Brookhaven National Laboratories, which has r = 100 m. The maximum magnetic field is only 3300 gauss, and the RF accelerating voltage is 10.5 MV. Note that the electrons from this machine have γ = 20,000, approximately, and they are heavier than protons! Most electron synchrotrons, however, are relatively small machines. If 10,000 gauss is taken as a convenient upper limit for the magnetic field, then for 100 MeV electrons, r is only about 33 cm. Therefore, an electron synchrotron is a convenient source of high-energy radiation.
The relation between B, r and T can be found by equating the magnetic force to the mass times acceleration, or (evB/c) = (γmv2/r), since the effective mass of the electron is γm, and it does not change in circular motion. This then gives Br = γmvc/e = βγmc2/e = βE/e ≈ E/e. Note that the expression found in some references, [T(T + 2mc2]1/2 is just βγmc2. The acceleration frequency is f = v/2πr = eB/2πγmc. If we express this in terms of r instead, using the value of Br, we find f = (c/2πr)(1 - 1/γ). If r is in metres, then f = 47.7 MHz/r. These relations are shown in the diagram of the synchrotron above.
There are losses in the synchrotron, of which the most important is the radiation loss due to the circular orbit of the electrons. Since β is at right angles to the centripetal acceleration, the formula for radiated power becomes P = (2e2/3c3)γ4|a|2, and a = v2/r, so P = (2e2c/3r2)β4γ4. Multiplying by the time per revolution, we have the energy loss per revolution due to radiation of δE = (4πe2/3r)β3γ4. In most cases, β can be set equal to 1. In evaluating these expressions, note that Gaussian units must be used, in which e = 4.803 x 10-10 esu. The result is δE (erg) = 2.125 x 106 (E4 erg/r cm). This works out to δE (MeV) = 0.0884 E4 GeV/r m. For the Brookhaven synchrotron, this is 8.84 MeV, not far below the 10.5 MV acceleration per turn.
The total power lost by radiation will be NδE/T = (I/e)δE, which works out to 106IδE W, where I is in ampere and δE in MeV. With a beam current of 1 μA, the loss in the Brookhaven machine will only be about 8.84 W, and much less in smaller machines. Nevertheless, this radiation is easily detected and quite interesting. Each electron makes a sharp pulse when it is at the point tangent to the line of sight, so the frequency spectrum is broad, extending from the orbital frequency up to the maximum energy of the electrons. The radiation is in a very narrow forward cone, that shrinks as the energy increases. It was first observed visually in 1948, looking at the radiation through a mirror, since it is not advisable to stand directly in the beam! At 60 MeV it is a red glow, then becoming white and most intense at 200 MeV, after which the maximum passes into the ultraviolet. This is synchrotron radiation, which can be seen in the Crab Nebula, among other places. This should not be confused with the radiation produced by the beam itself on impact with the target, which can be thousands of Röngen per minute. Jackson gives details on the angular distribution and spectrum of synchrotron radiation.
References
J. D. Jackson, Classical Electrodynamics, 2nd ed. (New York: John Wiley & Sons, 1975). Chapters 6, 11, 12 and 14.
M. S. Livingston, High-Energy Accelerators (New York: Interscience, 1959). Chapters 2 and 3.
1. Introduction
2. A Uniformly Moving Charge
3. A Charge in Arbitrary Motion
4. Radiation
5. The Electron Synchrotron
6. References
Introduction
Maxwellian electrodynamics is naturally relativistic; that is, it conforms to Einstein's requirement of invariance under transformations between coordinate systems moving with constant relative velocity. It has exactly the same form in any inertial system, and light travels with the same velocity c in any system. The Lorentz transformation of space and time coordinates was established for electromagnetism before relativity was formulated and it was known that this transformation applies to all physical processes. In particular, the time transformation is real, and not just a substitution that preserves the form of the equations. The formalism of relativity makes working with electromagnetism much easier. In this article, I will try to explain as clearly as I can how the electromagnetic field of a point charge in arbitrary motion can be determined at any point in space. A very important application of this theory is the radiation by moving charges.
This material is excellently explained in chapters 11, 12 and 14 of Jackson's classic text, to which the reader is referred. On first sight, the material seems very difficult and unclear, even if it has been mastered before and is now being reviewed after forgetting it. For the beginner, it may be truly frightening and formidable. The beauty of Jackson's exposition is that when one digs in and tries to understand each statement, the subject becomes continually clearer and in a few hours its elegance shines out, and it is seen to be logical and satisfying. This is the evidence of the author's comprehensive understanding achieved through labor and thought, that he communicates to you. In any case where this does not happen, the fault is more often with the author than with the reader, and particularly if the subject is a false one. Relativistic electrodynamics is one of the best-known and verified physical theories, the work of giants. Even the classical theory, as presented here for the model of a point charge, is of wide applicability. I hope I can make some points clearer in this article, and help the reader to understand.
The reader will need some preparation in relativity, but that is not difficult to acquire, even with Jackson alone. I will use the index notation for tensors with Greek indices taking the values 0,1,2,3 for the coordinates ct, x, y, z. A sum is implied over the same index in contravariant (high) and covariant (low) positions. This notation is very much like that of Euclidean tensors in 3-dimensional space, except that the distinction between contravariant and covariant is necessary because the metric tensor gαβ is diagonal with g00 = 1, g11 = g22 = g33 = -1. Changing from contravariant to covariant means only changing the sign of the space components. Then, if Aα = (A0, A) [In expressions like this, subscripts are not tensor indices, but simply labels.] Then, A · B = AαBα = A0B0 - A·B.
If xα is the coordinate four-vector (ct,r), then x · x = (ct)2 - r2 = s2 is a scalar, and hence an invariant, under Lorentz transformation. It is the invariant interval between the event X described by the vector, and the event O at the origin, t = 0 and r = 0. If s2 > 0, then a Lorentz transformation to a coordinate system where X and O occur at the same point, but at a later or earlier time, is possible. If s2 < 2 =" r2," r =" ±ct." dt =" γdτ," t =" 0." t =" γt'" t =" t'/γ." t =" γt'" r2 =" b2" 1 =" β2γ2," r =" γ[(b2" n =" (-β)[(vt" r =" (b2" ex =" Ex'," ey =" γEy'," bz =" βEy'," ex =" -γqvt/(b2" ey =" qb/(b2" bz =" βEy." v2t2 =" b2" v2t2 =" r2" v2t2 =" r2[1" e =" qr/r3γ2(1" e =" γq/r2" bz =" βEy," t =" 2b/γv." dl =" 4πI/c," x =" 0" i =" q/(2b/γv)" x0 =" ct." z =" x" k =" -(2π)-4∫d3k" ko =" ±κ,"> 0, and zo = cto <> 0, then we can close the contour below. The large semicircle contributes nothing to the integral, so the integral will be equal to 2πi time the sum of the residues at the two poles, which are exp(-iκzo)/2κ and -exp(iκzo)/2κ. This gives D(z) = θ(zo)(2π)-3∫d3k exp(ik·z)sin κzo/κ. The theta function θ(zo) is zero when its argument is negative, +1 when its argument is positive. d3k = κ2dk sinθdθdφ, so we can do the integral over the angles right away. After a little manipulation, we find D(z) = [θ(zo)/2π2]∫(0,∞)sinκr sinκzodκ. Expressing the product of the sines as the sum of the cosines of the sum and difference angles, writing them as exponentials and combining terms to extend the limits of integration from -∞ to +∞, and using the Fourier transform of the delta function, we have D(z) = [θ(zo)/4πr]δ(r - zo), where r is the distance between source and field points. The delta function forces evaluation at the retarded time t' = t - r/c. This solution has all we could desire for the effects of a disturbance from a impulse at the origin at t = 0 at the point z. The Green's function with these properties is called the retarded Green's function, or propagator. It is, of course, of general utility, and represents another theoretical triumph.
We can express the propagator in manifestly invariant terms by considering δ[(x - x')2]. As we know, δ[f(x)] = δ(x - x')/|f'(x')|, where x' is a root of f(x). Now δ[(x - x')2] = δ[(xo - xo')2 - |x - x'|2] = δ[(xo - xo' - r)(xo - xo' + r)] = (1/2r)[d(xo - xo' - r) + δ(xo - xo' + r)]. Since the theta functions selects only one of the delta functions in the sum, we can write D(x - x') = (1/2π)θ(xo - xo')δ[(x - x')2]. The theta function is a Lorentz invariant, since the past and future light cones are invariantly separated, and the argument of the delta function is also a Lorentz invariant. Therefore, the propagator has been expressed in a manifestly invariant form independent of a coordinate system.
The solution to our problem is now easily obtained. Omitting any solution of the homogeneous wave equation (which would be fields not due to our moving charge), we have Aα(x) = (4π/c)∫d4x'D(x - x')Jα(x'). If you substitute this expression in the inhomogeneous wave equation, you will find that the equation is satisfied. This is the promised solution.
Now we must find Jα(x'), the source current. In some coordinate system, suppose the path of the charge is r = r(t). If the point charge is q, then the charge density is ρ = qδ[x - r(t)] and the current density is ρv = qvδ[x - r(t)]. The current density Jα = (ρc, ρv) will not be expressed in a relativistically invariant way, and so our potential Aα will not be relativistically invariant. To escape from this inconvenience, we can start by parametrizing the trajectory in terms of the proper time of the charge τ instead of t. Since dτ = dt/γ, this is not hard to do by integration. Now, r(τ) specifies an invariant world line that does not depend on the coordinate system. Of course, we need a coordinate system to specify it in any particular case, but we can imagine a world line drawn in 4-dimensional space, labelled with the proper time, that is invariant, and which determines the functions r(τ) in any coordinate system.
For Jα(ct,x), we want to pick out the time t that corresponds to any point x(τ) on the world line. This can be done with a delta function δ[t - ct(τ)], where t(τ) returns the time corresponding to r(τ), and is the 0-component of the position 4-vector. Allowing a choice from any value of τ, we are led to the integral Jα = ec∫dτUα(τ)δ[x - r(τ)], where Uα = (γc, γv) is the tensor 4-velocity of the charge. The delta function is a 4-dimensional one, and r(t) is the tensor 4-position of the charge. This expression is relativistically invariant, since it contains only Lorentz tensors. If it reduces to the correct charge density and current in a particular coordinate system, then it is what we want.
Choose a coordinate system, then, and do the integral over τ using the functions that result. From the rule for the delta function of a function, integration over the time delta function sets t equal to t' and divides by γc. Then, J0 = qc(γc)(1/γc)δ[x - r(t)] = qδ[x - r(t)], exactly what is wanted. The space components of Jα also are easily seen to give the desired results. We are now confident about our invariant expression for Jα.
Now all we have to do is substitute D(x - x') and Jα in the integral for Aα to find an invariant expression for Aα. The integrals are done using the delta functions, and we find that Aα(x) = qUα(τ)/Vβ[x - r(t)]β, evaluated at τ = τo, the retarded proper time. The most important property of this expression is that it is manifestly covariant, constructed only of Lorentz tensors. The fields can now be found by differentiation, but this is not a simple process, and we shall not do it here, but only quote the results. It is simpler, in fact, to differentiate before the final integration so the retardation is easier to handle, but we won't do that either. See Jackson if you need the expressions.
The time τo comes from the condition [x - rτo]2 = 0 enforced by the space part of the 4-dimensional delta function in the integral. This means that xo - ro(τo) = |x - r(τo)| = R, the retarded distance. Further, U·(x - r) = γcR - γv·nR = γcR(1 - β·n), with the symbols meaning the same as in the section on the uniformly moving charge, and where this expression will be recognized. From the expression for Aα, we then find the components φ = [q/(1 - β·n)]ret and A = [qβ/(1 - β·n)]ret. These potentials are expressed in terms of the retarded position and time, as we defined in connection with the uniformly moving charge. They are called the Liénard-Wiechert potentials, which were found in 1898 on the basis of Maxwell's equations alone, without the help of relativity.
The electric field is E = q{(n-β)/[γ2R2(1 - β·n)3]}ret + (q/c){n x [(n - β)x(dβ/dt)]/[R(1 - β·n)3]}, and the magnetic field is B = [n x E]ret. The first term is independent of the acceleration, and falls off as R-2, so it resembles a static field. The fields of the second term fall off as R-1, so they are radiation fields that transport energy to large distances, and are smaller by a factor of c-1.
Let's check our previous results for a charge in uniform motion. The y component of the field will be Ey = q cosθ/γ2R2(1 - β·n)3. Multiplying top and bottom by R, and using γR(1 - β·n) = (b2 + γ2v2t2)1/2, we find our previous result at once, Ey = qγb/(b2 + γ2v2t2)3/2. Ex = -q(cosθ + β)/γ 2R2(1 - β·n)3. Again multiplying top and bottom by R, and using R cosθ + βR = vt, we find Ex = -qγvt/(b2 + γ2v2t2)3/2. This confirms our earlier procedure of Lorentz-transforming the static fields.
Radiation
When β is much less than unity, the fields are very closely E = qn/R2 + n x (n x dβ/dt)(e/cR), where n is a unit vector from the retarded position at retarded distance R. If distances are not large, R is approximately the present distance r. This is quite a different limit from the unrealistic c → ∞, where the radiation field also vanishes. In the radiation field, the electric vector lies in the plane of dv/dt = a and n and is perpendicular to the radius vector. If the angle between a and n is θ then the magnitude of the electric field is E = (qa/c2r)sinθ. The energy flux per unit solid angle in the direction of n will be dP/dΩ = r2c|E|2/4π = (q2/4πc3)a2sin2θ. The total power radiated is found by integration over 4π steradians. ∫2πsin3θdθ = 8π/3, so P = (2e2/3c3)a2. This is Larmor's result for the radiation from an accelerated charge.
Jackson shows how to find the power radiated by a relativistic electron by a clever extension of Larmor's formula. The direct use of the fields is rather laborious. The result is P = (2e2/c)γ6[(dβ/dt)2 - (β x dβ/dt)2], which Liénard found in 1898. This is a very useful formula for finding the power radiated by fast particles, as in accelerators, or in astrophysics.
The Electron Synchrotron
The first particle accelerators were linear accelerators using high-voltage DC power supplies, such as the Van de Graaff and Cockroft-Walton machines, which could attain about 1 MeV. Next came the cyclotron (1934), where the circular orbit made periodic acceleration possible with lower RF voltages on the dees, the accelerating electrodes. The angular velocity of revolution depended only on the e/m ratio of the particles, not on their speed, so a constant RF frequency could be used. Protons and other heavy particles could be accelerated to about 25 MeV. These accelerators were very useful for nuclear studies, but the energies were too low for the creation of new particles. Cyclotrons cannot accelerate electrons because of the rapid increase in mass that brings them out of step with a constant RF frequency. The betatron (1940), which could accelerate electrons to similar energies by using transformer action, was then devised. Great attention had to be paid to designing the magnetic field so that the electron orbit would remain constant in radius and focussed, but the frequency problem did not arise. Betatrons provide a pulse of fast electrons at the end of each cycle of increasing magnetic field. For more on betatrons, see The Particle Electron.
Cyclotrons could be used to accelerate heavy particles even when relativistic effects entered by changing the accelerating frequency. Such FM cyclotrons or synchrocyclotrons were indeed developed, but the frequency modulation is very troublesome. Betatrons had the disadvantage of very heavy magnetic cores, and so their size was limited. In 1947, the first electron synchrotron was constructed in California, following the ideas of McMillen (1947) in the U.S. and Veksler (1946) in the U.S.S.R., who introduced the idea of synchronous acceleration. There were some smaller experimental synchrotrons, but the electron synchrotron was born full-grown, the first one providing 300 MeV electrons, and a dozen others following soon after. These electrons are generally used to produce high-energy photons in collisions with a target, which then can cause photonuclear reactions, such as the production of π mesons.
The electron synchrotron takes advantage of two consequences of the small mass of the electron. First, the electron is easily accelerated to speeds very close to c, when their orbital period in a magnetic field will not change significantly with energy. Secondly, the electron is easily deflected by a magnetic field, even at high energies, so that accelerators can be a convenient size. A diagram of an electron synchrotron is shown at the right. The toroidal vacuum chamber, and the C-shaped magnet cores, are not shown. A magnetic field B, shown directed into the page, deflects electrons into a circular orbit of radius r in which they move anticlockwise, approximately at speed c. The kinetic energy of the electrons is T = mc2(γ - 1). The electrons are accelerated on each revolution by the electric field in a resonant cavity, shown at the left. In order for the radius to remain constant, the magnetic field B must be increased synchronously with the increase in energy of the electrons, which gives the name to the machine. At the end of an accelerating cycle, the accelerating voltage is turned off, while the magnetic field continues to increase, causing the beam to spiral in and strike the target. This happens about 60 times a second, so the beam appears continuous to the human observer.
The magnet windings are usually connected across a large capacitor to form a resonant circuit, so the magnet power supply need only furnish the losses, and the power factor is acceptable. At the beginning of a cycle, a large pulse of electrons is injected by a thermionic electron gun in the direction of the orbit. Most of these electrons are lost, but a sufficient number assume orbits that do not strike the vacuum chamber walls. The magnetic field B must be carefully shaped to focus the beam. In general, B bows outward like the normal fringing field at an air gap, and can be described by B = Bo(ro/r)n. If n > 0, the electron orbits will oscillate stably about the orbital plane, and if n < 1 the orbits will oscillate stably in a radial direction. As B increases, these oscillations will be damped so that the beam will become narrow and well-defined. The electrons must also form a bunch along the orbit so that they will have phase stability. An electron that receives a little too much energy on one pass will forge ahead and receive less on the next pass, and the same sort of compensation will result for an electron that falls behind. The focussing and phase stability are necessary for achieving a useful beam current, which is usually in the region of microamperes.
Electrons are injected with energies of around 100 keV. They are then accelerated to about 2 MeV by betatron action. The necessary large field within the orbit is provided by flux bars of relatively small cross section that pass between N and S poles of the magnets. Initially, much of the flux passes by this route because of the much smaller air gap, but at the end of betatron acceleration the flux bars saturate and thereafter act like an air gap, most of the flux passing through the main pole faces to steer the electrons. At 2 MeV, the electron speed has become constant enough near c that constant frequency RF acceleration can take over. With an acceleration of 10 keV per pass, and a frequency of 47.7 MHz (appropriate for r = 1 m), the electrons will gain 477 MHz in 10 ms, neglecting losses. The synchrotron principle works very well for electrons.
A large number of electron synchrotrons have been built, of which the largest seems to be the 10 GeV machine at Brookhaven National Laboratories, which has r = 100 m. The maximum magnetic field is only 3300 gauss, and the RF accelerating voltage is 10.5 MV. Note that the electrons from this machine have γ = 20,000, approximately, and they are heavier than protons! Most electron synchrotrons, however, are relatively small machines. If 10,000 gauss is taken as a convenient upper limit for the magnetic field, then for 100 MeV electrons, r is only about 33 cm. Therefore, an electron synchrotron is a convenient source of high-energy radiation.
The relation between B, r and T can be found by equating the magnetic force to the mass times acceleration, or (evB/c) = (γmv2/r), since the effective mass of the electron is γm, and it does not change in circular motion. This then gives Br = γmvc/e = βγmc2/e = βE/e ≈ E/e. Note that the expression found in some references, [T(T + 2mc2]1/2 is just βγmc2. The acceleration frequency is f = v/2πr = eB/2πγmc. If we express this in terms of r instead, using the value of Br, we find f = (c/2πr)(1 - 1/γ). If r is in metres, then f = 47.7 MHz/r. These relations are shown in the diagram of the synchrotron above.
There are losses in the synchrotron, of which the most important is the radiation loss due to the circular orbit of the electrons. Since β is at right angles to the centripetal acceleration, the formula for radiated power becomes P = (2e2/3c3)γ4|a|2, and a = v2/r, so P = (2e2c/3r2)β4γ4. Multiplying by the time per revolution, we have the energy loss per revolution due to radiation of δE = (4πe2/3r)β3γ4. In most cases, β can be set equal to 1. In evaluating these expressions, note that Gaussian units must be used, in which e = 4.803 x 10-10 esu. The result is δE (erg) = 2.125 x 106 (E4 erg/r cm). This works out to δE (MeV) = 0.0884 E4 GeV/r m. For the Brookhaven synchrotron, this is 8.84 MeV, not far below the 10.5 MV acceleration per turn.
The total power lost by radiation will be NδE/T = (I/e)δE, which works out to 106IδE W, where I is in ampere and δE in MeV. With a beam current of 1 μA, the loss in the Brookhaven machine will only be about 8.84 W, and much less in smaller machines. Nevertheless, this radiation is easily detected and quite interesting. Each electron makes a sharp pulse when it is at the point tangent to the line of sight, so the frequency spectrum is broad, extending from the orbital frequency up to the maximum energy of the electrons. The radiation is in a very narrow forward cone, that shrinks as the energy increases. It was first observed visually in 1948, looking at the radiation through a mirror, since it is not advisable to stand directly in the beam! At 60 MeV it is a red glow, then becoming white and most intense at 200 MeV, after which the maximum passes into the ultraviolet. This is synchrotron radiation, which can be seen in the Crab Nebula, among other places. This should not be confused with the radiation produced by the beam itself on impact with the target, which can be thousands of Röngen per minute. Jackson gives details on the angular distribution and spectrum of synchrotron radiation.
References
J. D. Jackson, Classical Electrodynamics, 2nd ed. (New York: John Wiley & Sons, 1975). Chapters 6, 11, 12 and 14.
M. S. Livingston, High-Energy Accelerators (New York: Interscience, 1959). Chapters 2 and 3.
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