Classical Electrodynamics · Problem 16.6

Radiating Systems, Multipole Fields and Radiation — John David Jackson

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

Problem

A particle of bare mass

m_0

and charge

e

has a charge density,

\rho(\mathbf{x}) = e\,e^{-r/a}/4\pi a^2 r

. (a) Show that the charge form factor is

f(\mathbf{k}) = (1 + k^2 a^2)^{-1}

. (b) Show that the mass, (16.33), is

m = m_0 + \frac{mc\tau}{2a}

M(\omega)

, (16.34), in the complex

\omega

plane, are given by

\omega\tau = -i(c\tau/a)\left[1 \pm (1 - 2a/c\tau)^{-1/2}\right]

(d) Find the trajectories of the roots in the complex

\omega

plane for

m_0 > 0

and

m_0 < 0

. Find the limiting form for the roots when

a/c\tau \ll 1

and

a/c\tau \gg 1

. Discuss.