Classical Electrodynamics · Problem 16.5

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · radiation-damping-classical-models-charged-particles

Problem

The particle of Problem 16.4 is initially at rest in a spatially uniform, but time-varying electric field

E(t) = E_0 \Theta(t)

. (a) Show that its speed in the direction of the field is given by the integral

v(t) = -\frac{eE_0 a}{\pi c}\int_{-\infty}^{\infty} d\xi\, \frac{e^{-iT\xi}}{\xi^2 M(\xi)}

where

T = ct/2a

. (b) From the analytic properties of

M(\xi)

established in part c of Problem 16.4, show that

v(t) = 0

for

t \leq 0

(no preacceleration).