A waveguide is constructed from a rectangular tube of perfectly conduct- ing material. Its hollow core occupies the region 0 < x < a, 0 < y b. The z axis runs along the tube. The tube is filled with a dielectric with permittivity and permeability . You will investigate the properties of transverse electric, or "TE" modes, in which the longitudi- nal component Ez of the electric field is identically zero. You are therefore seeking solutions to Maxwell's equations of the fom E(x, y, z, t) B(x, y, z, t) = = (Ex(x, y), Ey(x, y), 0) exp{ikzz - iwt}, (Br(x, y), By(x, y), B(x, y)) exp{ikz - iwt}. a) What boundary conditions are obeyed by the E and B fields at the surface of the perfect conductor? b) It is possible to introduce a scalar function (x, y) in terms of which one (and only one) of the following pairs of equations holds Br (i) (ii) Which pair, (i) or (ii), holds as consequence of Maxwell's equations and the condition E = 0? ET = - ???? dx 1 By Ey = = c) Use your result for part (b) to derive the equation obeyed by the scalar function (x, y) in the interior of the waveguide, and also the boundary condition that obeys at surface (r = 0, a or y = 0, b) of the perfect conductor. d) Waves with an angular frequency w below some cutoff frequency @min are unable to propagate in the waveguide. Solve your equation from part (c) so as to find the mode with the lowest cutoff frequency and express wmin in terms of a, b, u and . e) Use your result from part (d) to write down the relation between w and k, for frequencies above min Show that the phase velocity w/k is greater that the speed of light for all such frequencies. Explain why can we not make use of this fact for faster-than-light communication.