Solution of an electrostatic problem specifying its boundary condition is the only possible solution, irrespective of the method by which this solution is obtained. To prove this theorem, as shown in figure 2.23, we consider a volume Vr and a closed surface Sr encloses this volume. Sr is such that it may also be a surface at infinity. Inside the closed surface Sr , there are charged conducting bodies with surfaces S1 , S2 , S3,....and these charged bodies are at specified potentials.
If possible, let there be two solutions of Poisson’s equation in Vr. Each of these solutions V1 and V2 satisfies Poisson equation as well as the boundary conditions on S1 , S2 ,....Sn and Sr. Since V1 and V2 are assumed to be different, let Vd = V1 -V2 be a different potential function such that,
Fig 2.23: Uniqueness Theorem