Classical Electrodynamics · Problem 8.10

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · variational-te111-mode-cylindrical-cavity

Problem

Use the variational principle of Problem 8.9 in terms of the electric field

\mathbf{E}

to find an estimate of the eigenvalue for

k^2

for the

\text{TE}_{111}

mode in a right circular cylinder cavity of radius

R

and length

d

with perfectly conducting walls. Use as a trial function

B_z = B_0(\rho/R)(1 - \rho/2R)\cos\phi\sin(\pi z/d)

. [This function satisfies the boundary conditions of

B_z = 0

z = 0

and

z = d

, and

\partial B_z/\partial\rho = 0

\rho = R

.] (a) First show that the variational principle can be reexpressed as

k^2 = \frac{\displaystyle\int_V (\boldsymbol{\nabla} \times \mathbf{E}^*)\cdot(\boldsymbol{\nabla} \times \mathbf{E})\, d^3x}{\displaystyle\int_V \mathbf{E}^* \cdot \mathbf{E}\, d^3x}

(b) Show that the (transverse) components of the (trial) electric field are

E_\rho = E_0(1 - \rho/2R)\sin\phi\sin(\pi z/d); \quad E_\phi = E_0(1 - \rho/R)\cos\phi\sin(\pi z/d)

\mathbf{E}

and show that the approximation for

k^2

k^2 = \frac{18}{5R^2} + \frac{\pi^2}{d^2}

Compare with the exact result. For small enough

d/R

, this mode has a larger eigenvalue than the

\text{TM}_{010}

mode. Why should the present variational estimate be at all reliable? (d) The original variational expression in Problem 8.9 has an equivalent integrand in the numerator,

\mathbf{E}^*\cdot[\boldsymbol{\nabla}(\boldsymbol{\nabla}\cdot\mathbf{E}) - \boldsymbol{\nabla}^2\mathbf{E}]

. Discuss the relative merits of this integrand compared with the square of the curl of

\mathbf{E}

in part a for the present problem.

Related theory

Standing Wave Cavity Modes Qed Cavity

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