Classical Electrodynamics · Problem 8.7

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · spherical-shell-resonant-cavity

Problem

A resonant cavity consists of the empty space between two perfectly conducting, concentric spherical shells, the smaller having an outer radius

a

and the larger an inner radius

b

. As shown in Section 8.9, the azimuthal magnetic field has a radial dependence given by spherical Bessel functions,

j_l(kr)

and

n_l(kr)

, where

k = \omega/c

. (a) Write down the transcendental equation for the characteristic frequencies of the cavity for arbitrary

l

. (b) For

l = 1

use the explicit forms of the spherical Bessel functions to show that the characteristic frequencies are given by

\frac{\tan kh}{kh} = \frac{\left(k^2 + \dfrac{1}{ab}\right)}{k^2 + ab\left(k^2 - \dfrac{1}{a^2}\right)\!\left(k^2 - \dfrac{1}{b^2}\right)}

where

h = b - a

. (c) For

h/a \ll 1

, verify that the result of part b yields the frequency found in Section 8.9, and find the first order correction in

h/a

. [The result of part b seems to have been derived first by J. J. Thomson and published in his book Recent Researches in Electricity and Magnetism, Oxford Clarendon Press, 1893, pp. 373 ff.]

Related theory

Field Due To A Uniformly Charged Spherical Shell

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