The Poisson integral formula gives the potential at any point r inside a sphere if we specify the potential (r S ) at every

The Poisson integral formula

gives the potential at any point r inside a sphere if we specify the potential ¯ϕ (r_S) at every point on the surface of the sphere. Derive this formula by summing the general solution of Laplace’s equation inside the sphere using the derivatives (with respect to r and R) of the identity.

o(r) = Sa ds' (rs) |rs|=R (R - r) 4 R r-r's r < R