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 w, and its variation in space can be approximated as I(y) = I₀ sin [(w/c) (h - y)]. The thickness of the probe can be neglected. The frequency is such that only the TE₁₀ mode can propagate in the guide.

(a) Calculate the amplitudes for excitation of both ?? and TM modes for all (m, n) and show how the amplitudes depend on m and n for m, n >> 1 for a fixed frequency w.

(b) For the propagating mode show that the power radiated in the positive z direction is

with an equal amount in the opposite direction. Here ? is the wave number for the TE₁₀ 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 I₀? 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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