Classical Electrodynamics · Problem 9.24

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · conducting-sphere-normal-modes

Problem

Discuss the normal modes of oscillation of a perfectly conducting solid sphere of radius

a

in free space. (This problem was solved by J. J. Thomson in the 1880s.) (a) Determine the characteristic equations for the eigenfrequencies for TE and TM modes of oscillation. Show that the roots for

\omega

always have a negative imaginary part, assuming a time dependence of

e^{-i\omega t}

. (b) Calculate the eigenfrequencies for the

l = 1

and

l = 2

TE and TM modes. Tabulate the wavelength (defined in terms of the real part of the frequency) in units of the radius

a

and the decay time (defined as the time taken for the energy to fall to

e^{-1}

of its initial value) in units of the transit time

(a/c)

for each of the modes.

Related theory

Standing Waves And Normal Modes

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