Answer Happy

In this problem we will consider a rectangular metallic waveguide. The z-direction points down the propagation direction in the waveguide, and the waveguide extends from 0 to a in the x-direction and from 0 to b in the y-direction. The inside of the waveguide is filled with air, and as usual, you may assume that the metallic walls of the waveguide are perfect conductors. We are interested in operating this waveguide at a frequency of 3 GHz. (a) You are told that we want our rectangular waveguide to operate in a single mode. Specifically, we want the waveguide to operate in the TE10 mode, and we want the operating frequency (3 GHz) to be 1.3 times the cutoff frequency associated with the TE10 mode. Additionally, we want the operating frequency to be 0.8 times the cutoff frequency of the next higher mode. Find a and b to meet these two constraints. (b) Consider the first two modes with cutoff frequencies above that of the TE10 mode. Find the attenuation (in dB) that these modes will experience after propagating 1 m down the waveguide. (c) Calculate the time-average power carried in our waveguide by a TE10 mode with a peak electric field of Eo. (HINT: Calculate the time-average power density of the TE10 mode and integrate over the waveguide cross-section.) (d) Large electric fields in dielectrics can result in "electrical breakdown". During electrical breakdown in air, electric fields ionize molecules in the air and turns the air into a conducting plasma (that is, the air turns from a dielectric into a conductor). Electrical breakdown occurs in air at a field strength of around 2 x 106 V/m (at 3 GHz). Given this information, what is the maximum time-average power that our waveguide can carry?