Homework6 - Homework 6 Answers 95.657 Electromagnetic...

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Homework 6 Answers, 95.657 Electromagnetic Theory I Dr. Christopher S. Baird, UMass Lowell Jackson 3.4 The surface of a hollow conducting sphere of inner radius a is divided into an even number of equal segments by a set of planes; their common line of intersection is the z axis and they are distributed uniformly in the angle . (The segments are like the skin on wedges of an apple, or the earth's surface between successive meridians of longitude.) The segments are kept at fixed potentials ± V , alternately. (a) Set up a series representation for the potential inside the sphere for the general case of 2 n segments, and carry the calculation of the coefficients in the series far enough to determine exactly which coefficients are different from zero. For the nonvanishing terms, exhibit the coefficients as an integral over cos θ . SOLUTION: There is no charge present, so we seek to solve Laplace's equation. In spherical coordinates this becomes: 2 = 0 1 r 2 r 2 r  1 r 2 sin ∂ sin ∂ ∂ 1 r 2 sin 2 2 ∂ 2 = 0 Using the method of separation of variables, and when the full azimuthal range needs a valid solution, the general solution is expressed in terms of the spherical harmonics Y lm :  r , , = l = 0 m =− l l A l , m r l B l ,m r l 1 Y lm  ,  In this problem, we require a valid solution at the origin, so that we must have B l = 0 to keep those terms from blowing up. The solution now becomes:  r , , = l = 0 m =− l l A l ,m r l Y lm  ,  The boundary condition on the surface of the sphere is described mathematically as:  r = a = V  where V = { V if 2 i n  2 i 1  n V if 2 i 1  n  2 i 2  n } where i is any of 0,1,...( n -1) We apply this boundary condition: V = l = 0 m =− l l A l , m a l Y lm  , 
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Multiply both sides by Y l ' m ' *  ,  and integrate over the surface of the sphere: 0 2 0 V  Y l ' m ' *  ,  sin d d = l = 0 m =− l l A l, m a l 0 2 0 Y l ' m ' *  ,  Y lm  ,  sin d d Use the orthogonality of the spherical harmonics to pick one term from the double series: 0 2 0 V  Y l ' m ' *  ,  sin d d = l = 0 m =− l l A l, m a l l ' l m ' m 0 2 0 V  Y lm *  ,  sin d d = A l ,m a l A l , m = a l 0 2 0 V  Y l m *  ,  sin d d Expand the definition of the spherical harmonics: A l , m = a l 2 l 1 4 l m ! l m ! 0 2 V  e i m d 0 P l m cos  sin d Break the integral over the azimuthal angle into a sum over 2 n integral pieces and plug in the explicit value of the potential on the boundary: A l , m = a l 2 l 1 4 l m !
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