One of the many mathematical techniques that are useful in the study of electrostatics is called the separation of variables method. This topic covers the application of this method to a situation in which the electrostatic potential in a spherically symmetric system is known. In such a system, the potential (potential and electrostatic potential are interchangeable here) everywhere in space is a function described by the Legendre polynomials. To remind yourself of why this is so, review treatments of Laplace's equation in spherical coordinates. The following discussion takes the form of the potential as the starting point and then demonstrates a way to use this method for determining more about the system.

Consider a grounded sphere of radius a that is made of a conducting material. This conducting, and solid, sphere is then covered by a shell made of an insulating material. The insulation material is a shell with radius b, in which it is required that b > a (treat the shell as though it has no thickness). The electrostatic potential on the insulating shell, Φ_shell, is known to be,

where α is a constant and θ is the angular coordinate of the spherical coordinate system. Figure 1 shows the setup for this topic.

For this system we will find the electrostatic potential everywhere in space and then determine the charge densities on the spherical objects.

Figure 1: Setup for this topic. The central, solid, sphere is grounded to achieve a potential of zero. The enclosing shell potential is also shown. While not shown in this figure, the potential at r = infinity is also zero, this time by definition.