# Find static electric potential problem by solving Laplace equation

Feb 3rd, 2012
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The potential on a sphere is such that V(Q) = +V (0 ≤ Q ≤ pi/2) -V (pi/2 < Q ≤pi/2) Find the potential inside and outside the sphere

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1. The potential on a sphere is such that(q) = +V (0 /2)-V (/2 < /2)Find the potential inside and outside the sphereNote: This is a problem worked out in Jackson but with lots of missing steps. Show thedetails that lead to the answer. [ you need a mathematical handbook that hasproperties of Legandre Polynomials and relations]1.PThe solution to Laplace equation in spherical coordinate with azimuthal symmetric isB(r , ) ( Al r l l l1 ) Pl (cos )rl 0Inside the sphere the potential must be finite, the solution is (r , ) Al r l Pl (cos )l 0and for the same reason potential outside the sphere isB(r , ) l l1 Pl (cos )l 0 rIf the boundary condition at r =a is given by ( a, ) V ( )Then we can determine Al and Bl by using the fact that Pl(cos) are orthogonal.Al (2l 1)V ( ) Pl (cos ) sind2a l 0and(2l 1) l 1a V ( ) Pl (cos ) sind20Plug in V (0 2 )V ( ) (a, ) V ( )2We have /2(2l 1)VAl (P(cos)sind0 l Pl (cos ) sind )2a l /2Bl and /2(2l 1) l 1Va ( Pl (cos )d sin Pl (cos ) sin d )20 /2Now evaluate the integrationBl /20110 Pl (cos )

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