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EEE303_hw1 - EEE-303 HOMEWORK 1(Due 8th of October class...

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EEE-303 HOMEWORK # 1 (Due: 8 th of October, class hour) 1) Check whether the following fields satisfy Maxwell’s equations. Assume that the fields exist in charge-free regions. (Note: r and R are in cylindrical and spherical coordinates, respectively.) (a) ( ) ˆ 40sin 10 z A t ω = + x a (b) ( ) 10 ˆ cos 2 B t r a r φ ω = (c) ( ) 2 cos ˆ ˆ 3 cot sin r C r a a r φ φ t φ ω = + (d) ( ) 1 ˆ sin sin 5 D t R a R θ θ ω = 2) In a charge-free region for which 0 σ = , 0 r ε ε ε = , and 0 μ μ = , ( ) 9 ˆ 5cos 10 4 A/m z H t y a = , find: (a) d J and D , (b) r ε . 3) Suppose that the following H-field exists in a source-free vacuum region: ( ) ( ) ( ) ( ) 0 0 0 0 ˆ ˆ sin cos cos sin x z H E x t z a E x t z a β α α ω β α ω β μ ω μ ω = − (a) Use Ampere’s law to find the E associated with the H-field. (b) By substituting the E found in part (a) into Maxwell’s curl- E equation, show that these E- and H-fields are valid only when 2 2 0 0 2 α β μ ε ω + = (c) Prove that these E- and H-fields also satisfy Maxwell’s divergence equations. 4) In a region where 1 r r μ ε = = and 0 σ = , the retarded potentials are given by V and ( ) V x z ct =
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