Consider a sphere of radius a with dielectric constant epsilon. A point charge q lies a distance d from the center of the sphere. Assume d > a, calculate the potential for all points inside the sphere, and outside the sphere. The problem is to be solved with an expansion of Legendre Polynomials. Note the azimuthal symmetry of the problem. This is a boundary condition problem. Coefficients are solved by equating the tangent electric field inside and outside the sphere at its boundary, and equating the normals of the of D also at the boundaries. The main problem is finding a way to express the potential outside the sphere. The point charge of course introduces a singularity that makes it difficult (you can express the potential as a Legendre expansion of the sphere plus the point charge potential, but this creates a mess for applying the boundary conditions. Any ideas?