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# aklec04 - 4 Time harmonic sources and retarded potentials...

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4 Time harmonic sources and retarded potentials In the last lecture we have seen that for the general case of time- varying sources ) , ( and ) , ( t t r J J r the solution for ) , ( ), , ( ), , ( ), , ( t t t t r H H r B B r D D r E E may be obtained using retarded potentials V ( r , t ) and A ( r , t ): 0 2 2 0 0 2 SOLUTION t V V r r r r r r r 3 0 4 ) , ( ) , ( d c t t V  0 2 2 0 0 2 SOLUTION J A A t r r r r r r J r A 3 0 4 ) , ( ) , ( d c t t Fields may be calculated using V ( r , t ) and A ( r , t ) as: and A B A E  t V

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We will obtain these solutions in the frequency domain and work back to time domain. Consider equation for V . When the source is a time-harmonic function, the response will also be time harmonic: t j t j e V t V e t ) ( ~ Re ) , ( ) ( ~ Re ) , ( r r r r For harmonic time dependence time derivatives change according to: 2 2 2 and t j t When these are applied, wave equation for V becomes the wave equation in frequency domain. ~ ~ ~ 0 2 0 0 2 0 2 2 0 0 2 V V t V V
Note that wave equation reduces to Poisson’s equation for =0 (static case), and we know the impulse response relationship for Poisson’s equation as: 4 1 4 1 ~ ~ 0 0 r πε πε ) ( V ) δ( ) ( ρ r r r r The impulse response is r / 1 , and is symmetric w.r.t. the origin.

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aklec04 - 4 Time harmonic sources and retarded potentials...

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