Problem.Set.8soln

Problem.Set.8soln - Problem Set 8 An antenna is constructed...

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Problem Set 8. An antenna is constructed from conducting wire wrapped in the shape of a sphere with a current density given by J ( r , t )=[ ̂ I 0 a ( r a ) sin e i ot ] =[ J 0 ( r ) e i ot ] a) Set up the general expression for the vector and electrostatic potentials, A ( r , t ) , and ( r , t ) for this localized radiation source in the Coulomb gauge. b) Find A ( r , t ) in the radiation limit. c) Find the magnetic induction, B ( r , t ) , and the electric field, E ( r , t ) , in the radiation limit. d) Find the averaged Poynting vector, S ( r ) ave , and the power radiated by the antenna. Solution: a). First we note that J ( r , t )= 0 J ( r , t )= J t ( r , t ) J ( r , t )= 0 Thus A ( r , t ) is given by A ( r , t ) = 1 c ∫∫∫ ∫ J t ( r , t ) G ( r r , t t ) d 3 r dt = 1 c ∫∫∫ ∫ J 0 ( r ) e i ot Θ ( t t ) c | r r | (| r r | − c ( t t )) d 3 r dt = exp (− i o t ) c ∫∫∫ J 0 ( r ) exp ( i o | r r |/ c ) | r r | d 3 r =[ A o ( r ) exp (− i o t )] with G 0 ( r r , o )= exp ( i o | r r |/ c ) | r r | = 4 k l = 0 m =− l l j l ( kr < )[ ij
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Problem.Set.8soln - Problem Set 8 An antenna is constructed...

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