Classical Electrodynamics · Problem 16.13

Radiating Systems, Multipole Fields and Radiation — John David Jackson

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

Problem

A single particle oscillator under the action of an applied electric field

\mathbf{E}_0 e^{-i\omega t}

has a dipole moment given by

\mathbf{p} = \alpha(\omega)\mathbf{E}_0 e^{-i\omega t}

(a) Show that the total dipole cross section can be written as

\sigma_t(\omega) = \frac{2\pi}{c}\left[-i\omega\alpha(\omega) + \text{c.c.}\right]

(b) Using only the facts that all the normal modes of oscillation must have some damping and that the polarizability

\alpha(\omega)

must approach the free-particle value

(-e^2/m\omega^2)

at high frequencies, show that the cross section satisfies the dipole sum rule,

\int_0^\infty \sigma_t(\omega)\,d\omega = \frac{2\pi^2 e^2}{mc}

(The discussion of Kramers--Kronig dispersion relations in Chapter 7 is clearly relevant.)