1. The surface of a sphere of radius R has a potential . If there are no charges outside the sphere, obtain an expression for the potential outside.

One has to first express in terms of Legendre polynomials. It can be checked that . The general expression for the potential outside the sphere is . Comparing this with the boundary condition at r=R, ; all other coefficients are zero. Thus,

2. A sphere of radius R has a surface potential given by .Obtain an expression for the potential inside the sphere.

The potential inside has the form . At r=R, the potential can be expressed in terms of spherical harmonics as

Thus only terms are there in the expression for the potential. We have,