Classical Electrodynamics · Problem 8.11

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · variational-breadbox-cavity-tm-mode

Problem

Apply the variational method of Problem 8.9 to estimate the resonant frequency of the lowest TM mode in a ``breadbox'' cavity with perfectly conducting walls, of length

d

in the

z

direction, radius

R

for the curved quarter-circle ``front'' of the breadbox, and the ``bottom'' and ``back'' of the box defined by the plane segments

(y = 0,\, 0 < x < R)

and

(x = 0,\, 0 < y < R)

, respectively. Use the trial function,

E_z = E_0(\rho/R)^\nu(1 - \rho/R)\sin 2\phi

for the only component of electric field present. This function gives vanishing tangential component of

\mathbf{E}

on the boundary surfaces; the index

\nu

is a variational parameter. Show that

k^2 R^2 = \frac{(\nu+2)(2\nu+3)(\nu^2+\nu+4)}{\nu(2\nu+1)}

Minimize with respect to

\nu

to find the best estimate of

kR

from the given trial function. Compare with the exact answer,

kR = 5.13562

, the first root of

J_2(x)

Related theory

Standing Wave Cavity Modes Qed Cavity

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