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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 aa 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.
 Fall '07
 Stone
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