Consider a parallel plate waveguide with plate separation d as shown in Fig. The electric and magnetic fields for the TEM mode are given by $${E_x} = \,{E_0}\,\,{e^{ - j\,k\,z + j\,\omega \,t}},\,{H_y} = {{{E_0}} \over \eta }\,{e^{ - j\,k\,z + j\,\omega \,t}}$$

Where $$k = \,\,\eta \,\omega \,\, \in $$
(a) Determine the surface charge densities $${\rho _s}$$ on the plates at x = 0 and x = d
(b) Determine the surface current densities $$\mathop {{J_s}}\limits^ \to $$ on the same plates.
(c) Prove that $${\rho _s}$$ and $$\mathop {{J_s}}\limits^ \to $$ satisfy the current continuity condition.

A rectangular hollow metal waveguide has dimensions a = 2.29 cm and b = 1.02 cm. Microwave power at 10 GHz is transmitted through the waveguide in the $$T{E_{10}}$$ mode.
(a) Calculate the cut-off wavelength and the guide wavelength for this mode.
(b) What are the other (TE or TM) modes that can propagate through the waveguide?
(c) If a = b = 2.29 cm, what are the modes which can propagate through the waveguide?

A 100 m section of an air-filled rectangular wave-guide operating in the $$T{E_{10}}$$ mode has a cross-sectional dimension of 1.071 cm $$ \times $$ 0.5 cm. Two pulses carriers of 21 GHz and 28 GHz are simultaneously launched at one end of the wave-guide section. What is the time delay difference between the two pulses at the other end of the waveguide?

A rectangular waveguide with inner dimensions 6 cm $$ \times $$ 3 cm has been designed for a single mode operation. Find the possible frequency range of operation such that the lowest frequency is 5% above the cut off and the highest frequency is 5% below the cut off of the next higher mode.