Classical Electrodynamics

1 4 Radiation by Moving Charge s It is well known that accelerated charges emit electromagnetic radiation. In Chapter 9 we discussed examples of radiation by macroscopic time-varying charge and current densities, which are fundamentally charges in motion . We will return to such problems in Chapter 16 where multipole radiation is treated in a systematic way . But there is a class of radiation phenomena where the source is a moving point charge or a small number of such charges. In these problems it is useful to develop the formalism in such a way that the radiation intensity and polarization are related directly to properties of the charge's trajectory and motion . Of particular interest are the total radiation emitted, the angular distribution of radiation, and its frequency spectrum. For nonrelativistic motion the radiation is described by the well-known Larmor result (see Section 14.2) . But for relativistic particles a number of unusual and interesting effects appear . It is these relativistic aspects which we wish to emphasize . In the present chapter a number of general results are derived and applied to examples of charges undergoing prescribed motions, especially in external force fields . Chapter 15 deals with radiation emitted in atomic or nuclear collisions. 14.1 Lienard-Wiechert Potentials and Fields for a Point Charg e In Chapter 6 it was shown that for localized charge and current distributions without boundary surfaces the scalar and vector potentials can be written as f 1JM(x', 1) 6 (t' + R A``(X' t) - R - t) d3x' dt' (14. 1) C where R = (x - x'), and the delta function provides the retarded behavior 464 [Sect. 14.1 ] Radiation by Moving Charges 465 demanded by causality . For a point charge e with velocity c p(t) at the point r(t) the charge-current density i s Jµ(x, t) = ec#u S[x -- r(t)] (14.2) where ~,_ (P, i) . With this source density the spatial integration in (14 .1) can be done immediate ly, yielding t' Au(x, t) = e~ ~u~ ) b t' + Rat ~ - t dt' (14.3) R(t' ) I where now R(t') = Ix - r(t')I . Although (14 .3) is a convenient form to utilize in calculating the fields, the integral over dt' can be performed, provided we recall from Section 1 .2 that when the argument of the delta function is a function of the variable of integration the standard results are modified as follows : fg(x) 6U(x) - cx] dx = (14 .4) dfJdx f(x) = a The function f (t') = t' + [R(t')Ic] has a derivative = 1 - n•~3 (14 .5) dt' K +dRc where cp is the instantaneous velocity of the particle, and n =RJR is a unit vector directed from the position of the charge to the observation point. With (14.5) in (14.4) and (14.3) the potentials of the point charge, called the Lienard-Wiechert potentials, are r 1 (D(x, t) = e 1 L KR rot (14.6) A(x,t) =e [!Rl rPt The square bracket with subscript ret means that the quantity in brackets is to be evaluated at the retarded time, t' = t - [R(t') f c]. We note that, for nonrelativistic motion, ►c -1 1 . Then the potentials (14 .6) reduce to the well-known nonrelativistic results _ To determine the fields E and B from the potentials A . it is possible to perform the specified differential operations directly on (14.6). But this is a more tedious procedure than working with the form (14.3) . We note that in (14 .3) the only dependence on the spatial coordinates x of the observation point is through R . Hence the gradient operation is equivalent to O D OR a =n a (14.7) 7R 7R 466 Classical Electrodynamics Consequently the electric and magnetic fields can be written a s KR c / cR ~ c R (14 .8) 6 ( +~ - t ~ 1 B (X, t) = e (n x ) + -61 t' +- _ R t d t` f ~ R2 cR C The primes on the delta functions mean differentiation with respect to their arguments. If the variable of integration is changed to f (t') = t' -1- jR(t')Ic], we can integrate by parts on the derivative of the delta function . Then we find readily n 1 In ~31 E(x, t) -= e C KR2 + cK dt d ' ~ KR ~ ret (14 .9) [' xn + 1 d B(x, t) = e ('KXR ret ►cR2 Ca Cat' I It is convenient to perform first the differentiation of the unit vector n. It is evident from Fig. 14.1 that the rate of change of n with time is the negative of the ratio of the perpendicular component of v to R . Thus do n x (n x (14.10) cd t' R When we perform the differentiation of n wherever it appears explicitly, we obtain n n d 1 ~i 1 d P E(x, t) = e [rc'R' + c rc d t ~ rcR~ x2R 2 crc dt' `rcR~ ret [ (_~ + B(x, t) = e 1 d (A)) x n (14.11) rc2R 2 Crt Cat' ►cR I ret We observe at this point that the magnetic induction is related simply to _.,.,, P U Fig. 14.1 [Sect. 14.1 ] Radiation by Moving Charles 467 the electric field by the relation , B = n x E (14,12) where the equation is understood to be in terms of the retarded quantities in square brackets. The remaining derivatives needed in (14 .1 1) are d,P = 0 (14 .13) Then the electric field can be written (n P ( 1 - #2) + e n E(x, t) = e - ) x {(n - (3) x 01(14.14) K3R2 ret e OR re t while the magnetic induction is given by (14 .12). Fields (14.12) and (14 .14) divide themselves naturally into "velocity fields," which are independent of acceleration, and "acceleration fields," which depend linearly on 0 . The velocity fields are essentially static fields falling off as Rf2, whereas the acceleration fields are typical radiation fields, both E and B being transverse to the radius vector and varying as R-1 . For a particle in uniform motion the velocity fields must be the same as those obtained in Section 11 .10 by means of a Lorentz transformation on the static Coulomb field. For example, the transverse electric field E1 at a point a perpendicular distance b from the straight line path of the charge was found to be El(t) = eYb (14.15) (b 2 + Y2U2 t2)% The origin of the time t is chosen so that the charge is closest to the observation point at t = 0. The electric field El(t} given by (14.15) bears little resemblance to the velocity field in (14.14). The reason for this apparent difference is that field (14 .15) is expressed in terms of the present position of the charge rather than its retarded position . To show the equivalence of the two expressions we consider the geometrical configura- tion shown in Fig. 14.2. Here 0 is the observation point, and the points P and P are the present and apparent or retarded positions of the charge at time t. The distance PQ is ~R cos 0 = (n • P)R. Therefore the distance OQ is KR. But from triangles OPQ and PP'Q we find 0)2 (KR)2 = r2 - (PQ)2 = r2 -#2(R sin Then from triangle OMP' we have R sin 0 = b, so that (KR)2 2 + Y200) = b 2 + v2t2 T N 2b2 = ~ (I? (14.16) Y 468 Classical Electrodynamics Fig. 14.2 Present and retarded positions of a charge in uniform motion . The transverse component of the velocity field in (14 .14) i s El(t) = e ~ b (14.17) Y~(KR)3 ret With substitution (14.16) for KR in terms of the charge's present position, we find that (14.17) is equal to (14 .15) . The other components of E and B come out similarly . 14.2 Total Power Radiated by an Accelerated Charge-Larmor's Formula and Its Relativistic Generalization If a charge is accelerated but is observed in a reference frame where its velocity is small compared to that of light, then in that coordinate frame the acceleration field in (14 .14) reduces to Ea e ~n x (n x ~3) = (14 .18) c R ret The instantaneous energy flux is given by the Poynting's vector , S 4~ E X B = 4~ jEaj2n (14 .19) This means that the power radiated per unit solid angle is * 2 IREQ12 = 4~~ I n x (n x ~3)~2 (14.20) M 47 T * In writing angular distributions of radiation we will always exhibit the polarization explicitly by writing the absolute square of a vector which is proportional to the electric field. [Sect. 14.2] Radiation by Moving Charges 469 If 0 is the angle between the acceleration v and n, as shown in Fig. 14.3, then the power radiated can be writte n dP e2 3 v2 sin2 0 ( 14.2 1 ) d Q 47rc This exhibits the characteristic sin 2 0 angular dependence which is a well- known result . We note from (14.18) that the radiation is polarized in the plane containing v and n. The total instantaneous power radiated is obtained by integrating (14.21) over all solid angle . Thus 2 e2v2 P3 = ~3 (14.22) This is the familiar Larmor result for a nonrelativistic, accelerated charge . Larmor's formula (14 .22) can be generalized by arguments abou t covariance under Lorentz transformations to yield a result which is valid for arbitrary velocities of the charge.

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