Classical Electrodynamics · Problem 14.12

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · radiation-simple-harmonic-motion-relativistic

Problem

As in Problem 14.4a a charge

e

moves in simple harmonic motion along the

z

axis,

z(t') = a\cos(\omega_0 t')

. (a) Show that the instantaneous power radiated per unit solid angle is

\frac{dP(t')}{d\Omega} = \frac{e^2 c\beta^4}{4\pi a^2}\frac{\sin^2\theta\cos^2(\omega_0 t')}{(1 + \beta\cos\theta\sin\omega_0 t')^5}

where

\beta = a\omega_0/c

. (b) By performing a time averaging, show that the average power per unit solid angle is

\frac{dP}{d\Omega} = \frac{e^2 c\beta^4}{32\pi a^2}\left[\frac{4 + \beta^2\cos^2\theta}{(1 - \beta^2\cos^2\theta)^{7/2}}\right]\sin^2\theta