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emhw22 - Electromagnetic Theory II Problem Set 8 Due 21...

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Electromagnetic Theory II Problem Set 8 Due: 21 March 2011 31. J, Problem 8.2. 32. J, Problem 8.5. 33. In this problem we investigate some of the normal modes for electromagnetic fields with harmonic time dependence ( i.e. e iωt ) in a spherical cavity of radius a in a perfect conductor. Use polar coordinates with origin at the center of the cavity. When k ω/c negationslash = 0, the source-free Maxwell equations can be reduced in two equivalent ways to: I : B = 1 ikc ∇ × E ∇ · E = 0 ( −∇ 2 k 2 ) E = 0 or II : E = c ik ∇ × B ∇ · B = 0 ( −∇ 2 k 2 ) B = 0 a) For ω negationslash = 0 the fields must be strictly zero in the bulk of the conductor. Why? Use Maxwell’s equations to prove that the fields in the cavity must then satisfy the bound- ary conditions E t = B n = 0 at r = a. b) Consider the following forms for TE ( ˆ r · E = 0) and TM ( ˆ r · B = 0) modes: TE : E TE = f ( r ) ˆ r × C e iωt ; TM : B TM = g ( r ) ˆ r × D e iωt , where C and D are constant vectors. It is clearly convenient to use form (I), (II) of Maxwell’s equations for the TE, TM cases respectively.
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