Overview

Up to this point, we have examined electric fields and magnetic fields in an infinite space. The usual situation that one encounters in practice is a finite space where the fields transition from one region to another, each with different electrical material characteristics, or regions that are enclosed with specific boundaries. We first describe the transitioning of the fields from one region to another and then examine techniques to solve the boundary value problems. We will focus our attention on examining boundary value problems for electric fields although the techniques can also be used for magnetic fields.

In Chapter 2, we learned that a static electric field would be created from a charge distribution. In addition, it was possible to determine this static electric field from a scalar potential. We also showed there that the potential V could be obtained directly in terms of the charge distribution ? _v via one partial differential equation: Poisson's equation. This equation reduces to Laplace's equation if the charge density in the region of interest is equal to zero. The general procedure for solving these equations for the cases where the potential V depends on only one or two spatial coordinates is given here. In this chapter, we will introduce analytical and numerical techniques that will allow us to examine such complicated problems. We also introduce modern numerical techniques for solving such problems (the method of moments, the finite element method, and the finite difference method