The electromagnetic field expressions for TMmn mode in rectangular waveguide are given by; mi n E(x, y, z) = Amn sin x sin- b a Ex(x, y, z) = Ey (x, y, z)= H(x, y, z) = -jpmn ka -jpnn mi Amn cos x sin- a mi Amnsin- x cos a mi kb joenn kb -joemn ka Amn sin -ye-jz NIT a -x cos bye-jz n bye-jz bye-1Bz m -X xsin ye a n Hy(x, y, z)= Amncos- where; Amn is an arbitrary constant for the TMmn mode and is the corresponding propagation constant. The waveguide has dimensions: a-19.05 mm and (b/a) = 0.65. The waveguide is filled with dielectric material of er=4. i. Find the cutoff frequency of the first TMmn mode. ii. If the frequency of operation is 8 GHz, find all possible propagating modes, find also the corresponding propagation constant for each propagating mode. bye-jz iii. Plot the surface current on both the upper and right-side walls corresponding to the TM1 mode. iv. Plot the electric field along the cross section (x-y plan) of the WG for the TM1 mode. v. Find the appropriate placement for a narrow slot on the metallic walls which might be needed to measure the electric field inside the waveguide for the TM1 mode along the direction of propagation without affecting the field distribution. [5 marks] [5 marks] [5 marks] [5 marks] [5 marks]