Classical Electrodynamics · Problem 14.14

Radiating Systems, Multipole Fields and Radiation — John David Jackson

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

Problem

(a) Show that for the simple harmonic motion of a charge discussed in Problem 14.12 the average power radiated per unit solid angle in the

m

th harmonic is

\frac{dP_m}{d\Omega} = \frac{e^2 c\beta^2}{2\pi a^2}\,m^2\tan^2\theta\, J_m^2(m\beta\cos\theta)

(b) Show that in the nonrelativistic limit the total power radiated is all in the fundamental and has the value

P \simeq \frac{2}{3}\frac{e^2}{c^3}\omega_0^4\overline{a^2}

where

\overline{a^2}

is the mean square amplitude of oscillation.