hw9 - mode TM mn 1 Phys 3327 Fall 2010 HW 9 Due Wednesday...

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Phys 3327 Fall 2010 HW 9 Due Wednesday 11/10/10 in lecture Homework 9 Wave Guides (2) Ex. 9.1: Dielectrics in Capacitors (Conceptual) Make careful sketches of the D and E fields for a charged capacitor with a wedge of dielectric with permittivity inserted, see diagram below. Make sure you explain your diagrams, either with other supplementary diagrams and/or a short written explanation. Ignore edge effects. - + (a) Ex. 9.2: TM Modes in a rectangular waveguide Consider a rectangular waveguide with walls made of an ideal conductor. The rectangular cross section has sides of length a and b < a . The goal of this problem is to find the transverse magnetic (TM) modes of this waveguide, identify the cut-off frequency for each mode, and the amount of energy transmitted. In this exercise, we choose coordinate axes as shown in the figure below. a) What are the boundary conditions that E z ( x, y ) has to satisfy? b) The z dependence of the fields is described using the phase factor e i ( kz - ωt ) . What is the dispersion relation k ( ω ) for this waveguide? What is the cut-off frequency for the
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Unformatted text preview: mode TM mn ? 1 Phys 3327 Fall 2010 HW 9 Due Wednesday 11/10/10 in lecture c) Find the transverse components of the electric and magnetic ±elds in the waveguide. d) What is the magnitude and direction of the Poynting vector for the mode TM mn , averaged over time and over the cross section of the wave guide? Ex. 9.3: Waveguide with circular cross section Heald & Marion ex. 7-13. Suggested problems for further study: • Loss A rectangular hollow-conductor waveguide of dimensions a × b has walls made of good, but not perfectly conducting metal. Estimate the loss per unit length for the funda-mental ( n = 1) mode in this waveguide. Your answer should contain the dependence on the skin depth δ of the wall conductor, the frequency ω of the EM wave, a and b (if applicable), but you need not specify the precise numerical prefactor. • Resonant Cavity : Heald & marion, ex. 7-14. 2...
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