In spherical polar coordinate system,

...............(2.104)

At points in simple media, where no free charge is present, Poisson’s equation reduces to

                                                              ...................................(2.105)

which is known as Laplace’s equation.

Laplace’s and Poisson’s equation are very useful for solving many practical electrostatic field problems where only the electrostatic conditions (potential and charge) at some boundaries are known and solution of electric field and potential is to be found throughout the volume. We shall consider such applications in the section where we deal with boundary value problems.

Uniqueness Theorem

Solution of Laplace’s and Poisson’s Equation can be obtained in a number of ways. For a given set of boundary conditions, if we can find a solution to Poisson’s equation ( Laplace’s equation is a special case), we first establish the fact that the solution is a unique solution regardless of the method used to obtain the solution. Uniqueness theorem thus can stated as: