Electrostatic boundary value problem

In this section we consider the solution for field and potential in a region where the electrostatic conditions are known only at the boundaries. Finding solution to such problems requires solving Laplace’s or Poisson’s equation satisfying the specified boundary condition. These types of problems are usually referred to as boundary value problem. Boundary value problems for potential functions can be classified as:

• Dirichlet problem where the potential is specified everywhere in the boundary

• Neumann problem in which the normal derivatives of the potential function are specified everywhere in the boundary

• Mixed boundary value problem where the potential is specified over some boundaries and the normal derivative of the potential is specified over the remaining ones.

Usually the boundary value problems are solved using the method of separation of variables, which is illustrated, for different types of coordinate systems.

Boundary Value Problems In Cartesian Coordinates:

Laplace's equation in cartesian coordinate can be written as,

                                     ...............(2.116)