4.0
Introduction

The reason for taking up electroquasistatic fields first is the relative ease with which such a vector field can be represented. The EQS form of Faraday's law requires that the electric field intensity E be irrotational.

The electric field intensity is related to the charge density by Gauss' law.

Thus, the source of an electroquasistatic field is a scalar, the charge density . In free space, the source of a magnetoquasistatic field is a vector, the current density. Scalar sources, are simpler than vector sources and this is why electroquasistatic fields are taken up first.

Most of this chapter is concerned with finding the distribution of E predicted by these laws, given the distribution of . But before the chapter ends, we will be finding fields in limited regions bounded by conductors. In these more practical situations, the distribution of charge on the boundary surfaces is not known until after the fields have been determined. Thus, this chapter sets the stage for the solving of boundary value problems in Chap. 5.

We start by establishing the electric potential as a scalar function that uniquely represents an irrotational electric field intensity. Byproducts of the derivation are the gradient operator and gradient theorem.

The scalar form of Poisson's equation then results from combining (1) and (2). This equation will be shown to be linear. It follows that the field due to a superposition of charges is the superposition of the fields associated with the individual charge components. The resulting superposition integral specifies how the potential, and hence the electric field intensity, can be determined from the given charge distribution. Thus, by the end of Sec. 4.5, a general approach to finding solutions to (1) and (2) is achieved.

The art of arranging the charge so that, in a restricted region, the resulting fields satisfy boundary conditions, is illustrated in Secs. 4.6 and 4.7. Finally, more general techniques for using the superposition integral to solve boundary value problems are illustrated in Sec. 4.8.

For those having a background in circuit theory, it is helpful to recognize that the approaches used in this and the next chapter are familiar. The solution of (1) and (2) in three dimensions is like the solution of circuit equations, except that for the latter, there is only the one dimension of time. In the field problem, the driving function is the charge density.

One approach to finding a circuit response is based on first finding the response to an impulse. Then the response to an arbitrary drive is determined by superimposing responses to impulses, the superposition of which represents the drive. This response takes the form of a superposition integral, the convolution integral. The impulse response of Poisson's equation that is our starting point is the field of a point charge. Thus, the theme of this chapter is a convolution approach to solving (1) and (2).

In the boundary value approach of the next chapter, concepts familiar from circuit theory are again exploited. There, solutions will be divided into a particular part, caused by the drive, and a homogeneous part, required to satisfy boundary conditions. It will be found that the superposition integral is one way of finding the particular solution.