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1-3-2 Radiation
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As an elementary application of Green functions let us recall the computation of the electromagnetic fields generated by a moving point charge. Let xl'(r) be its space-time trajectory andjll(y, t) the associated current given by (1-51). Causality requires to use the retarded Green function to solve for the potential. Moreover, the remarks at the end of the preceding section lead at once to the expression of the potential in the Lorentz gauge:
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(1-186) xO(r)] b{[y - x(rW}xl'(r)
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The dots refer to derivatives with respect to proper time. Let x + == x( r +), the unique y-dependent retarded point on the trajectory, such that (1-187) The following identity holds:
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(1-188)
From Eq. (1-187) and the fact that for a physical trajectory x+ belongs to the forward light cone, it follows that x+ (y - x+) > 0. We thus obtain the LienardWiechert retarded potential:
A~eb) = 4n x+ . (y -
(1-189)
the relativistic generalization of the Coulomb potential to which it reduces in the frame where xtt = (1,0,0,0). Explicitly, ifr = y - x+ and v+ = (dx/dt)(r+):
Aret(y) = 4n(r - r' v+)
(1-190)
Dropping the
+ index, the corresponding fields read
CLASSICAL THEORY
~ (r - rv)(1 - V )
4n rxE B=-r
+ r X [(r - rv) (r - r V)3
dv/dt] (1-191)
and involve, of course, the retarded point. Both E and B contain terms in l/r2 which only contribute at short distance and l/r radiated parts. The energy flux across a sphere is given in terms of the Poynting vector as dlff = dS (E dt s
B) =
fs dO. (r x E)2 > 0
Brad = e
(1-192)
At small velocities the relevant l/r contributions to (1-192) are r x dv/dt 2 r (1-193)
which are the standard dipole fields. In the same limit the radiated power is given by the Larmor formula: 2 2 2 dlff e Jdo. ( dV)2 e (dV)2 2 2 e (dV)2 (1-194) at = (4n 7 r x dt = 8n dt d cos () Slll () = 3 4n dt
These results apply to many interesting situations of which we are only going to consider a few. As an example let a charged particle oscillate nonrelativistically under the influence of a weak electric field of an incident plane wave of low frequency: dv e e . - = -E = -sEoe-lmt+1'k 'X dt m m (1-195)
Here s is the polarization of the wave, Eo its amplitude, and k its wave vector. According to Larmor's formula the power radiated in the solid angle dO. (see
Direction of observation
Figure 1-3 Radiation of a charged particle osciIla ting in a plane wave of wave vector k and polarization 8.
QUANTUM FIELD THEORY
Fig. 1-3 for the definition of angles) is given by
2 e ( dV)2 dO. (4n r x dt -;I
2 e (dV)2 . 2 (4n)2 dt sm dO.
(1-196)
If the particle displacement during a period is only a small fraction of the wavelength of the incident wave, the mean value of (dvjdt)2 is
l(dv)2) \ dt
= 1 Re dv dv* = ~ IE
and thus
dP) ( dO.
1( e ' 2 \4nm )21 Eo 1 sm 2e
(1-197)
The average incident energy flux per unit time and unit area perpendicular to the propagation is given by the average of the incident Poynting vector as i IEo 12. We define the differential scattering cross section da jdo. as the ratio of radiated power per unit solid angle to incident power flux across a unit area:
da - = ( -e
sin 2
r; sin
(1-198)
We have introduced the notation re for the classical electromagnetic radius (restoring the velocity of light e):
r ----2
ha 4nme - me
(1-199)
with a value of 2.82 x 10 - 13 cm for an electron, and a stands for the fine structure constant a = e 2j4nhe ~ 1j137. The expression (1-198) is referred to as the Thomson scattering cross section; it is given here for an incident polarized wave. In terms of the angles {3, 0/, and t/! defined on Fig. 1-3, we have sin 2e = 1 - sin 2{3 cos 2(0/
- t/!)
t/!:
(1-200)
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