The electroquasistatic laws were discussed in Chap. 4. The electric field intensity E is irrotational and represented by the negative gradient of the electric potential.
Gauss' law is then satisfied if the electric potential
is related to the charge density
by Poisson's equation
In charge-free regions of space,
The last part of Chap. 4 was devoted to an "opportunistic" approach to finding boundary value solutions. An exception was the numerical scheme described in Sec. 4.8 that led to the solution of a boundary value problem using the source-superposition approach. In this chapter, a more direct attack is made on solving boundary value problems without necessarily resorting to numerical methods. It is one that will be used extensively not only as effects of polarization and conduction are added to the EQS laws, but in dealing with MQS systems as well.
Once again, there is an analogy useful for those familiar with the description of linear circuit dynamics in terms of ordinary differential equations. With time as the independent variable, the response to a drive that is turned on when t = 0 can be determined in two ways. The first represents the response as a superposition of impulse responses. The resulting convolution integral represents the response for all time, before and after t = 0 and even when t = 0. This is the analogue of the point of view taken in the first part of Chap. 4.
The second approach represents the history of the dynamics prior to when t = 0 in terms of initial conditions. With the understanding that interest is confined to times subsequent to t = 0, the response is then divided into "particular" and "homogeneous" parts. The particular solution to the differential equation representing the circuit is not unique, but insures that at each instant in the temporal range of interest, the differential equation is satisfied. This particular solution need not satisfy the initial conditions. In this chapter, the "drive" is the charge density, and the particular potential response guarantees that Poisson's equation, (2), is satisfied everywhere in the spatial region of interest.
In the circuit analogue, the homogeneous solution is used to satisfy the initial conditions. In the field problem, the homogeneous solution is used to satisfy boundary conditions. In a circuit, the homogeneous solution can be thought of as the response to drives that occurred prior to when t = 0 (outside the temporal range of interest). In the determination of the potential distribution, the homogeneous response is one predicted by Laplace's equation, (2), with
The development of these ideas in Secs. 5.1-5.3 is self-contained and does not depend on a familiarity with circuit theory. However, for those familiar with the solution of ordinary differential equations, it is satisfying to see that the approaches used here for dealing with partial differential equations are a natural extension of those used for ordinary differential equations.
Although it can often be found more simply by other methods, a particular solution always follows from the superposition integral. The main thrust of this chapter is therefore toward a determination of homogeneous solutions, of finding solutions to Laplace's equation. Many practical configurations have boundaries that are described by setting one of the coordinate variables in a three-dimensional coordinate system equal to a constant. For example, a box having rectangular cross-sections has walls described by setting one Cartesian coordinate equal to a constant to describe the boundary. Similarly, the boundaries of a circular cylinder are naturally described in cylindrical coordinates. So it is that there is great interest in having solutions to Laplace's equation that naturally "fit" these configurations. With many examples interwoven into the discussion, much of this chapter is devoted to cataloging these solutions. The results are used in this chapter for describing EQS fields in free space. However, as effects of polarization and conduction are added to the EQS purview, and as MQS systems with magnetization and conduction are considered, the homogeneous solutions to Laplace's equation established in this chapter will be a continual resource.
A review of Chap. 4 will identify many solutions to Laplace's equation. As long as the field source is outside the region of interest, the resulting potential obeys Laplace's equation. What is different about the solutions established in this chapter? A hint comes from the numerical procedure used in Sec. 4.8 to satisfy arbitrary boundary conditions. There, a superposition of N solutions to Laplace's equation was used to satisfy conditions at N points on the boundaries. Unfortunately, to determine the amplitudes of these N solutions, N equations had to be solved for N unknowns.
The solutions to Laplace's equation found in this chapter can also be used as the terms in an infinite series that is made to satisfy arbitrary boundary conditions. But what is different about the terms in this series is their orthogonality. This property of the solutions makes it possible to explicitly determine the individual amplitudes in the series. The notion of the orthogonality of functions may already be familiar through an exposure to Fourier analysis. In any case, the fundamental ideas involved are introduced in Sec. 5.5.
