Classical Electrodynamics · Problem 8.19

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · probe-excitation-rectangular-waveguide

Problem

The figure shows a cross-sectional view of an infinitely long rectangular waveguide with the center conductor of a coaxial line extending vertically a distance

h

into its interior at

z = 0

. The current along the probe oscillates sinusoidally in time with frequency

\omega

, and its variation in space can be approximated as

I(y) = I_0\sin[(\omega/c)(h - y)]

. The thickness of the probe can be neglected. The frequency is such that only the

\text{TE}_{10}

mode can propagate in the guide. (a) Calculate the amplitudes for excitation of both TE and TM modes for all

(m, n)

and show how the amplitudes depend on

m

and

n

for

m, n \gg 1

for a fixed frequency

\omega

. (b) For the propagating mode show that the power radiated in the positive

z

direction is

P = \frac{\mu c^2 I_0^2}{\omega k ab}\sin^2\!\left(\frac{\pi X}{a}\right)\sin^4\!\left(\frac{\omega h}{2c}\right)

with an equal amount in the opposite direction. Here

k

is the wave number for the

\text{TE}_{10}

mode. (c) Discuss the modifications that occur if the guide, instead of running off to infinity in both directions, is terminated with a perfectly conducting surface at

z = L

. What values of

L

will maximize the power flow for a fixed current

I_0

? What is the radiation resistance of the probe (defined as the ratio of power flow to one-half the square of the current at the base of the probe) at maximum?

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