(Solution) - a From the use of Green s theorem in two dimensions -(2025 Original AI-Free Solution)

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(a) From the use of Green's theorem in two dimensions show that the TM and ?? modes in a waveguide defined by the boundary-value problems (8.34) and (8.36) are orthogonal in the sense that ?A Ez?Ez?da = 0 for ? ? ? for TM modes, and a corresponding relation for Hz for ?? modes.
(b) Prove that the relations (8.131)-(8.134) form a consistent set of normalization conditions for the fields, including the circumstances when ? is a TM mode and ? is ? ?? mode.