4. Consider guides waved moving along the z axis in a resonant cavity, which is constructed by closing off
the two ends of a z-aligned rectangular wave guide (of cross-sectional dimensions a \times b in the xy plane) at
z = 0 and at z = d to make a perfectly conducting empty box.
(a) Start with the TE and TM solutions for guided waves in an infinitely long z-aligned wave guide, then
impose the boundary conditions \(\vec{E}_{\parallel} = 0\) and \(\vec{B}_{\perp} = 0\) at z = 0 and z = d to obtain a new restriction on the
possible values of \(k_z\) (which had been unrestricted when the z extent was infinity).
(b) From your answer in part (a), show that the resonant frequencies in the cavity (for both TE and TM
modes!) are given by
\(\omega_{mnl} = \pi c \sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2 + \left(\frac{l}{d}\right)^2}\)
for integers m, n and l.
(c) For mode integers m, n and l (for the x, y and z axes, respectively), find the Cartesian components of
the associated electric field \(\vec{E}(x, y, z, t)\) and magnetic field \(\vec{B}(x, y, z, t)\) within the hollow box. (Six answers!)
