Question

2. A hollow pipe of length L and square cross-section of side a is ter- minated by square faces of side a that are orthogonal to its length dimension which is along the z axis. The walls of the so-formed cavity are perfectly conducting, and the medium filling the cavity is lossless and with a real dielectric permittivity, e'. (a) Calculate the spatial distribution of the longitudinal component of the electric field for the various TM modes of the cavity. (b) What is the resonant frequency of each such mode? (c) Derive an expression for the H field for the lowest-frequency TM mode inside the cavity. (d) Show that the time averaged electromagnetic energy inside the cavity in the lowest-frequency TM mode has the value u = 1 e' a² 140 where yo is the maximum amplitude of the electric field inside the cavity. Hints: (a) Time-averaged electromagnetic energy density - (1/4)(d'|E|-+ 110|H|?). (b) The transverse Er field is related to the combined longitudinal- transverse gradient of the longitudinal component of the electric field as 1 2 Ēr = 72 az TE. Scanned with where y is the so-called transverse propagation constant. CamScanner

Answer 1

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