emsol12 - Electromagnetic Theory I Problem Set 12 Due: 21...

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Unformatted text preview: Electromagnetic Theory I Problem Set 12 Due: 21 November 2011 45. J, Problem 6.1 Solution : a) (2 Dimensional Space) ( r , t ) = Z d 3 x ( x ) ( y ) ( t- | r- r | /c ) 4 | r- r | = Z dz ( t- p x 2 + y 2 + ( z- z ) 2 /c ) 4 p x 2 + y 2 + ( z- z ) 2 (1) If 2 = x 2 + y 2 > c 2 t 2 the argument of the delta function never vanishes so = 0 in that case. When < ct , the delta function contributes at | z- z | = p c 2 t 2- 2 , and the derivative of its argument w.r.t. z is ( z- z ) /c p 2 + ( z- z ) 2 Then = 2 c ( ct- ) 4 p c 2 t 2- 2 (2) where the factor of two because both z- z = p c 2 t 2- 2 contribute. b)(1 Dimensional Space) ( r , t ) = Z d 3 x ( x ) ( t- | r- r | /c ) 4 | r- r | = Z dy dz ( t- p x 2 + ( y ) 2 + ( z ) 2 /c ) 4 p x 2 + ( y ) 2 + ( z ) 2 = Z 2 d Z rdr ( t- x 2 + r 2 /c ) 4 x 2 + r 2 = Z rdr ( t- x 2 + r 2 /c ) 2 x 2 + r 2 (3) If | x | > ct the argument of the delta function never vanishes so = 0 in that case. When | x | < ct , the delta function contributes at r = c 2 t 2- x 2 . and the derivative of its argument w.r.t. r is r/c x 2 + r 2 Then = c ( ct- | x | ) 2 (4) 46. J, Problem 6.3. In part d), omit relating the discussion to Sections 5.18B and Problems 5.35 and 5.36. Solution : 1 a) We write the Fourier transform of A as A ( r , t ) = Z d 3 ke i a ( k , t ) (5) Then ( 2- t ) A = 0 implies that a ( k , t ) = a ( k , 0) e- k 2 t/ = Z d 3 x (2 ) 3 A ( r , 0) e- k 2 t/ e- i Comparing with the desired form we read off G ( r- r , t ) = Z d 3 k (2 ) 3 e i...
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This note was uploaded on 01/08/2012 for the course PHY 6346 taught by Professor Staff during the Spring '08 term at University of Florida.

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emsol12 - Electromagnetic Theory I Problem Set 12 Due: 21...

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