Classical Electrodynamics · Problem 16.4

Radiating Systems, Multipole Fields and Radiation — John David Jackson

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

Problem

A classical model of an electron is a spherical shell of charge of radius

a

and total charge

e

. (a) Using (16.30) for the ``mass''

M(\omega)

and the angular average of

e^{i\omega R/c}/R

, show that

M(\omega) = m + \frac{2}{3}\frac{e^2}{ac^2}\cdot\left(\frac{e^{i\xi} - 1 - i\xi}{i\xi}\right)

where

\xi = 2\omega a/c

, and

m = m_0 + 2e^2/3ac^2

is the physical mass of the electron. (b) Expand in powers of the frequency (

\xi

) and show that, to lowest nontrivial order,

M(\omega)

has a zero in the upper half-plane at

\omega\tau = i

, where

\tau = 2e^2/3mc^3

. What is the physical significance of such a zero? (c) For the exact result of part a, show that the zeros of

\omega M(\omega)

, if any, are defined by the two simultaneous equations, proportional to the real and imaginary parts of

i\omega M(\omega)

e^{-y}\cos x - 1 + y(1 - a/c\tau) = 0

e^{-y}\sin x - x(1 - a/c\tau) = 0

where

x = \text{Re}\,\xi

and

y = \text{Im}\,\xi

. Find the condition on the radius

a

such that

\omega M(\omega)

has no zeros in the upper half-

\omega

-plane. Express the condition also in terms of the mechanical mass

m_0

for fixed physical mass. What about zeros and/or singularities in the lower half-plane?