The electrostatic potential created by a static charge distribution at a charge-free region is given by the following Laplace equation:
Here, is the potential within the region.
The solution of this problem for particular charge distributions concerns the subject of partial differential equations.
However, the dimensions of the problem can be reduced if the charge distribution exhibit a spatial symmetry.
For example, in a system with spherical symmetry, the solution of the problem becomes easier if the partial derivatives in Laplace's equation are expressed in terms of spherical coordinates:
(5.4)
Let the two concentric conductive spherical shells of radii and be held at constant potentials and (Figure 5.10).
Because of spherical symmetry, the potential distribution in the region between the two spheres (
) will be a function of distance only.
Accordingly, the derivatives with respect to the variables (
) in Equation 5.4 become zero, and the partial derivative in the remaining term becomes the full derivative:
Figure 5.10:
The region between two spherical shells of different potential.
The boundary conditions of this differential equation become:
First, we transform this equation into a linear system of equations:
with these values (
), the system of equations to be solved
and the boundary conditions are:
0
Now, we have a set of equations.
The analytical solution to this spherically symmetric problem would be:
We had a set of equations.
We transformed this boundary value problem into a first-order system of equations.