Classical Electrodynamics · Problem 14.10

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · radiation-sudden-deceleration

Problem

A particle of charge

e

moves at constant velocity

\boldsymbol{\beta}c

for

t < 0

. During the short time interval,

0 < t < \Delta t

, its velocity remains in the same direction, but its speed decreases linearly in time to zero. For

t > \Delta t

, the particle remains at rest. (a) Show that the radiant energy emitted per unit solid angle is

\frac{dE}{d\Omega} = \frac{e^2 \beta^2}{16\pi c\, \Delta t} \frac{(2 - \beta\cos\theta)[1 + (1 - \beta\cos\theta)^2]\sin^2\theta}{(1 - \beta\cos\theta)^4}

where

\theta

is the polar angle relative to the direction of the initial velocity. (b) In the limit of

\gamma \gg 1

, show that the angular distribution can be expressed as

\frac{dE}{d\xi} \approx \frac{e^2 \beta^2 \gamma^4}{c\, \Delta t} \frac{\xi}{(1 + \xi)^4}

where

\xi = (\gamma\theta)^2

. Show that

\langle\theta^2\rangle^{1/2} \approx \sqrt{2}/\gamma

and that the expression for the total energy radiated is in agreement with the result from (14.43) in the same limit.