Classical Electrodynamics · Problem 14.15

Radiating Systems, Multipole Fields and Radiation — John David Jackson

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

Problem

A particle of charge

e

moves in a circular path of radius

R

in the

x

y

plane with a constant angular velocity

\omega_0

. (a) Show that the exact expression for the angular distribution of power radiated into the

m

th multiple of

\omega_0

\frac{dP_m}{d\Omega} = \frac{e^2\omega_0^4 R^2}{2\pi c^3}\,m^2\left\{\left[\frac{dJ_m(m\beta\sin\theta)}{d(m\beta\sin\theta)}\right]^2 + \frac{\cot^2\theta}{\beta^2}\,J_m^2(m\beta\sin\theta)\right\}

where

\beta = \omega_0 R/c

, and

J_m(x)

is the Bessel function of order

m

. (b) Assume nonrelativistic motion and obtain an approximate result for

dP_m/d\Omega

. Show that the results of Problem 14.4b are obtained in this limit. (c) Assume extreme relativistic motion and obtain the results found in the text for a relativistic particle in instantaneously circular motion. [Watson (pp. 79, 249) may be of assistance to you.]

Related theory

Circular Motion Dynamics

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