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Unformatted text preview: 2 JACOB LEWIS BOURJAILY We should note that this more-or-less agrees with the expressions for the potential worked out in the text for similar circumstances. In particular, Jacksons equation (3.36) describes a similar problem, where 7 = - V/ 2 on the inner sphere and the outer sphere is removed completely. Making this redefinition of the potential on the inner two hemispheres and taking the limit where ab , (keeping only terms O ( a i b j ) where j i ), we obtain Jacksons equation (3.36) as desired. Problem 3.2 Let us consider the potential induced by a uniform, spherical charge distribution of radius R , with density Q/ 4 R 2 except for a spherical cap centered on = 0 defined by the cone = , where there is no charge. a) We are to show that the potential inside the spherical surface can be expressed as ( , ) = Q 8 X =0 1 2 + 1 R +1 [ P +1 (cos )- P - 1 (cos )] P (cos ) . Quite generally, we can express the potential as a Poisson integral over the charge dis- tribution. Specifically, ( x ) = 1 4 Z ( x ) d 3 x | x- x | ....
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