Classical Electrodynamics · Problem 9.23

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · spherical-cavity-q-factor

Problem

The spherical resonant cavity of Problem 9.22 has nonpermeable walls of large, but finite, conductivity. In the approximation that the skin depth

\delta

is small compared to the cavity radius

a

, show that the

Q

of the cavity, defined by equation (8.86), is given by

\begin{aligned} Q &= \frac{a}{\delta} \qquad \text{for all TE modes} \\ Q &= \frac{a}{\delta}\left(1 - \frac{l(l+1)}{x_{lm}^2}\right) \qquad \text{for TM modes} \end{aligned}

where

x_{lm} = (a/c)\omega_{lm}

for TM modes.