Classical Electrodynamics · Problem 8.18

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · waveguide-mode-orthogonality-normalization

Problem

(a) From the use of Green's theorem in two dimensions show that the TM and TE modes in a waveguide defined by the boundary-value problems (8.34) and (8.36) are orthogonal in the sense that

\int_A E_{z\lambda} E_{z\mu}\, da = 0 \quad \text{for } \lambda \neq \mu

for TM modes, and a corresponding relation for

H_z

for TE modes. (b) Prove that the relations (8.131)--(8.134) form a consistent set of normalization conditions for the fields, including the circumstances when

\lambda

is a TM mode and

\mu

is a TE mode.

Browse the problem library