Problem.Set.2.soln.2009

Problem.Set.2.soln.2009 - Problem Set 2 2.1 The Coulomb...

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Problem Set 2 : 2 . 1 The Coulomb potential for a point charge located at r 0 is given by ( r , r 0 ) = q | r r 0 | a) Find 2 ( r , r 0 ) and write down the partial differential equation satisfied by ( r , r 0 ) . b) Using steps similar to those found on page 12 of Chapter 1 for the wave equation, find the fourier transform, ( k , r 0 ), of the Coulomb potential. c) Find ( k , r 0 ) directly from ( r , r 0 ) = 1 ( 2 ) 3 ∫∫∫ ( k , r 0 ) exp ( i k r ) d 3 k d) Evaluate the following integral 0 sin ( ku ) du Solution: a) 2 ( r , r 0 ) = ∇ 2 q | r r 0 | = − 4 q ( r r 0 ) so ( r , r 0 ) satisfies 2 ( r , r 0 ) = − 4 q ( r r 0 ) b) One can take the Fourier transform of this equation: 2 1 ( 2 ) 3 ∫∫∫ ( k , r 0 ) exp ( i k r ) d 3 k = − 4 q ( 2 ) 3 ∫∫∫ exp ( i k ( r r 0 )) d 3 k ∫∫∫ [ ( k , r 0 )(− k k ) + 4 q exp (− i k r 0 )] exp i k r ) d 3 k = 0 Since the exp i k r are linearly independent functions for each k , ( k , r 0 )(− k k ) + 4 q exp (− i k r 0 ) = 0 and ( k , r 0 ) = 4 q exp (− i k r 0 ) k k c) The direct method for finding ( k , r 0 ) is as follows.: ( r , r 0 ) = q | r r 0 | = 1 ( 2 ) 3 ∫∫∫ ( k , r 0 ) exp ( i k r ) d 3 k and
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