Uniqueness Theory Regarding Electric Potential

Statement. There can be only one solution of Laplace’s equation that satisfies the prescribed boundary conditions.

Proof. Consider a system consisting of large number of conductors C1, C2, … Ck having fixed potentials V1, V2, … Vk, and the potential V1, V2,…Vk and the potential Vk of each conductor approaches zero at infinite distance. Let in Cartesian coordinates, V1 (x, y, z) and V2 (x, y, z) be two solution of Laplace’s equation ∂2V/∂x2 + ∂2V/∂y2 + ∂2V/∂z2 = 0 satisfying the same boundary conditions. Laplace’s equation is linear. Hence, the linear combination of these two solutions α1 V₁ + α₂ V2 should also be another solution, where α₁ and α₂ are constants. In particular, the difference between our two solutions V1 – V2 must also be a possible solution. Let V₁ – V₂ = W. Thus,
        W (x, y, z) = V1 (x, y, z) – V2 (x, y, z)

Of course, W does not satisfy the boundary conditions.

Thus we find that W = 0 at the boundary of all the conductors, as V1 and V2 are equal everywhere on them. This is contrary of boundary condition. Thus the solution W does not satisfy the given boundary conditions. Hence it is not the solution of the given problem. In fact it represents a solution for an entirely different boundary value problem where all the conductors are held at zero potential and the potential at infinity is also zero. Thus the uniqueness theorem is proved.