An infinitely long rectangular waveguide has a coaxial line terminating in the short side of the guide with the thin central conductor forming a semicircular loop of radius R whose center is a height h above the floor of the guide, as shown in the accompanying cross-sectional view. The half-loop is in the plane z = 0 and its radius R is sufficiently small that the current can be taken as having a constant value I0 everywhere on the loop.

(a) Prove that to the extent that the current is constant around the half-loop, the TM modes are not excited. Give a physical explanation of this lack of excitation.

(b) Determine the amplitude for the lowest ?? mode in the guide and show that its value is independent of the height h.

(c) Show that the power radiated in either direction in the lowest ?? mode is

where Z is the wave impedance of the TE₁₀ mode. Here assume R

-2R