The electromagnetic field expressions for TMmn mode in rectangular waveguide are given by; mi nπ E₂(x, y, z) = Amn sin x

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The electromagnetic field expressions for TMmn mode in rectangular waveguide are given by; mi nπ E₂(x, y, z) = Amn sin x sin- -ye-jßz a b mi NIT -jpmn k²a Amn cos x sinye-ißz a b -jpnn mi nπ Amnsin- x cos k²b a bye-jßz mi пл Amn sin -x cos joenn k²b -joemn a bye-ißz nπ Hy(x, y, z)= Amncos- mл -x sin kza a where; Amn is an arbitrary constant for the TMmn mode and ß 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. [5 marks] ii. [5 marks] If the frequency of operation is 8 GHz, find all possible propagating modes, find also the corresponding propagation constant for each propagating mode. iii. Plot the surface current on both the upper and right-side walls corresponding to the TM₁1 mode. [5 marks] iv. Plot the electric field along the cross section (x-y plan) of the WG for the TM₁1 mode. [5 marks] [5 marks] 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 TM₁1 mode along the direction of propagation without affecting the field distribution. Ex(x, y, z) = Ey (x, y, z) = H₂(x, y, z)= Eye-ibz