5.11Summary

There are two themes in this chapter. First is the division of a solution to a partial differential equation into a particular part, designed to balance the "drive" in the differential equation, and a homogeneous part, used to make the total solution satisfy the boundary conditions. This chapter solves Poisson's equation; the "drive" is due to the volumetric charge density and the boundary conditions are stated in terms of prescribed potentials. In the following chapters, the approach used here will be applied to boundary value problems representing many different physical situations. Differential equations and boundary conditions will be different, but because they will be linear, the same approach can be used.

Second is the theme of product solutions to Laplace's equation which by virtue of their orthogonality can be superimposed to satisfy arbitrary boundary conditions. The thrust of this statement can be appreciated by the end of Sec. 5.5. In the configuration considered in that section, the potential is zero on all but one of the natural Cartesian boundaries of an enclosed region. It is shown that the product solutions can be superimposed to satisfy an arbitrary potential condition on the "last" boundary. By making the "last" boundary any one of the boundaries and, if need be, superimposing as many series solutions as there are boundaries, it is then possible to meet arbitrary conditions on all of the boundaries. The section on polar coordinates gives the opportunity to extend these ideas to systems where the coordinates are not interchangeable, while the section on three-dimensional Cartesian solutions indicates a typical generalization to three dimensions.

In the chapters that follow, there will be a frequent need for solving Laplace's equation. To this end, three classes of solutions will often be exploited: the Cartesian solutions of Table 5.4.1, the polar coordinate ones of Table 5.7.1, and the three spherical coordinate solutions of Sec. 5.9. In Chap. 10, where magnetic diffusion phenomena are introduced and in Chap. 13, where electromagnetic waves are described, the application of these ideas to the diffusion and the Helmholtz equations is illustrated.