Please answer in detail with all steps and calculations.

2. Electromagnetic waves in a rectangular waveguide Electromagnetic waves travel inside a long rectangular pipe of width at and height I) made from perfectly conducting walls, as shown. We consider modes with angular frequency w where the electric eld is polarized in the a: y plane: E267, t) = 0, Ey(?, t) : E0 sin(k$:1:) cos(kyy) cos(kzz wt) {a} Maxwell's equation requires that: as a as a: + 3y + as vials): =0 Use the Expressions for Er and E3 above, derive Em. [b] The boundary conditions at the surface of the conductors require that Eu = G at r = El and z = a, and E, = [l at y = [l and p = b. Show that the possible values for k: and k1, that satisfy the boundary conditions are k: = "f: and k1, = %, where wan = [1, 1,2,.... For parts (ce} below, consider only the modes corresponding to m = 1, n = I]. {c} Write down the wave equation for electromagnetic wavesj where the speed of light is o. By substituting the solution for E above into the wave equation1 or otherwise, show that the dispersion relation for these modes of the waveguide is: [5 pts} a: = arms} = {:1ka + Wafe [d] If the frequency a: equals Ewcfo, calculate the phase velocity of the mode along 3. {e} If the frequency as equals Ewcfo, calculate the group velocity of the mode along .3