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HW04'sol - Classical Electrodynamics II PHY 506 Spring 2011...

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1 C:\User\Teaching\505-506\S11\HW04'sol.doc PHY 506 Classical Electrodynamics II Spring 2011 Homework 4 with solutions Problem 4.1 (to be graded of 15 points). Calculate the skin-effect contribution to the attenuation coefficient defined by equation (7.196) of the lecture notes, for the basic ( H 10 ) mode propagating in a waveguide with the rectangular cross-section (Fig. 7.21). Use the results to evaluate for a 10 GHz wave propagating in the standard X-band waveguide WR-90 (with copper walls, a = 23 mm, b = 10 mm, and no dielectric filling) at room temperature. Compare the estimate with that for the standard coaxial cable, at the same frequency – see Sec. 7.9. Solution : As discussed in class, in the H 10 mode the electric field has just one Cartesian component, with complex amplitude a x ZH ka i x E l y sin ) ( , while the magnetic field has two components: a x H x H a x H a k i x H l z l z x cos ) ( , sin ) ( . Of those two components, only H x contributes to the longitudinal ( z ) component of the time-averaged Poynting vector a x H Z a kk H E H E S l z x y y x z 2 2 2 2 sin 2 2 * * , which gives the total power flow along the waveguide: . 4 2 2 3 0 0 l z a b z H Z b a kk S dy dx P (*) In order to find losses per unit length, we have to integrate losses per unit area, given by Eq. (7.206) of the lecture notes, over the cross-section’s perimeter: . 2 1 4 2 cos sin 2 4 ) ( ) 0 ( ) ( ) ( 2 4 ) ( 4 2 2 0 0 2 2 2 2 0 0 2 0 2 0 2 2 0 2 0 0 loss b a a k H b dx a x a x a k H dy b H dy H dx x H x H dl x H dz d z l a z l b z b z a z x C     P According to the waveguide’s dispersion relation (see Eqs. (7.126) and (7.128) of the lecture notes), a k k k k H t t z  10 2 2 2 2 , , the expression in the last square brackets is just ( ka/ ) 2 , so that, using Eq. (*), for the attenuation constant we finally get
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