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Unformatted text preview: Physics 498/MMA Handout 10 Fall 2007 Mathematical Methods in Physics I http://w3.physics.uiuc.edu/ m-stone5 Prof. M. Stone 2117 ESB University of Illinois 1) Dielectric Sphere : Consider a solid dielectric sphere of radius a and permittivity . The sphere is placed in a electric field which is takes the constant value E = E z a long distance from the sphere. Recall that Maxwells equations require that D and E k be continuous across the surface of the sphere. a) Use the expansions in = X l A l r l P l (cos ) out = X l ( B l r l + C l r- l- 1 ) P l (cos ) and find all non-zero coefficents A l , B l , C l . b) Show that the E field inside the sphere is uniform and of magnitude 3 +2 E . c) Show that the electric field is unchanged if the dielectric is replaced by the polarization- induced surface charge density induced = 3- + 2 E cos . (Some systems of units may require extra 4 s in this last expression. In SI units D E = E + P , and the polarization induced charge density is induced =- P ) 2) Hollow Sphere : The potential on a spherical surface of radius a is ( , ). We want to express the potential inside the sphere as an integral over the surface in a manner analagous to the Poisson kernel in two dimensions. a) By using the generating function for Legendre polynomials, show that 1- r 2 (1 + r 2- 2 r cos ) 3 / 2 = X l =0 (2 l + 1) r l P l (cos ) , r < 1 b) Starting from the expansion...
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- Fall '07