solutionhw9 - Solutions to Homework Set 9 1) Conducting...

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Unformatted text preview: Solutions to Homework Set 9 1) Conducting strip : A ( k ) = Z + - V ( x ) e- ikx dx = V Z a- a e- ikx dx = 2 V sin ka k From this we have V ( x, , y ) = 2 V Z - dk 2 sin( ka ) k e ikx e-| k || y | . Taking the y gradients to get E y , and then getting = ( E y | y =+ - E y | y =- ) gives us ( x ) = 4 V Z - dk 2 sgn( k ) sin( ka ) e ikx e- | k | . The integral is elementary, and gives ( x ) = 4 V 4 1 x + a- i- 1 x- a- i + 1 x + a + i- 1 x- a + i We can take safely take the regulator to zero. We end up with ( x ) = 2 V 1 a + x + 1 a- x = 4 V a a 2- x 2 . x a-a A sketch of the charge distribution on the strips. 2) Qual Problem : a) We have, in general b l = 2 l + 1 2 R l +1 Z 1- 1 d (cos ) V ( ) P l (cos ) . 1 Plugging in the explicit expression for the relevent P l gives b 1 = 3 4 R 2 ( V 1- V 2 ) b 2 = (1) b 3 = 7 16 R 4 ( V 2- V 1 ) b) If V ( r, , ) = 1 4 Q r + d r r 2 + , then Q is the total charge, and d is the dipole moment. Thus our divided sphere has | d | = 3 R 2 ( V 1- V 2 ) . c) For the sphere immersed in the external field we have V ( r, , ) =-| E | r cos + V sphere ( r, , ) , and so V ( r, , ) =-| E | r cos + 1 4 Q r + | d | cos r 2 + ....
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This homework help was uploaded on 01/29/2008 for the course PHYS 598 taught by Professor Stone during the Fall '07 term at University of Illinois at Urbana–Champaign.

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solutionhw9 - Solutions to Homework Set 9 1) Conducting...

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