Chapter 3

This chapter we deal with the potential V.

Using the equations

Which is Poisson's equation. For r =0 we get Laplace's equation.

We saw the triangle diagram that shows you how to go between the quantities

r , V, E. This chapter develops the tools for finding V given r .

How do we get solutions for a potential that satisfies Poisson's/Laplace's equations ?

1. Find which conditions restrict the solution to a single result. What needs to be specified in order to get a unique result?
2. In a problem where this information is provided how do we generate a solution?

Starting with the divergence theorem

we can consider B to be f Ñ y to arrive at Green's theorem.

Derivatives wrt to n are along the normal to the surface.

Green's theorem can be used to show that a general solution for the potential can be found which has the form shown below and that satisfies Poisson's equation.

We evaluate the potential at a point r in the volume. The integrals are performed over the volume and the surface enclosing the volume using r'.

The potential in a volume therefore depends on the charge density and the value of the potential and its derivative on the enclosing surface. The derivative of V wrt the surface normal is the normal electric field on the surface. The potential therefore depends on the value of V and the value of the normal electric field on the surface. Providing both of these conditions over specifies the problem. We find two general conditions that provide a unique function for the potential inside a closed volume.

1. Dirichlet boundary conditions: V specified on the enclosing surface.
2. Neumann boundary conditions: E normal specified on the enclosing surface.
3. Cauchy boundary conditions: E normal on some parts, V given on other parts.

In the chapter we consider only a subset of the most general boundary conditions; condition 1 and condition 2 for conductors.

Image charges:

Because you only need to specify the boundary values to uniquely obtain a solution, there must be only one solution independent of how the boundary conditions are set. If you can imagine any external charge distribution that produces the boundary values provided in a problem then the solution to that problem must match the solution of the original problem inside the enclosing surface. This sometimes allows one to solve an otherwise difficult problem.

This is a technique that you store in your tool box.

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The real important and new part of this chapter is the application of the separation of variables technique to find potentials when boundary values are given.

For Laplace's equation the differential equation determines the solution in the interior based on the BC, boundary conditions. Because you have complete freedom to set the potential on the surface you need a way to represent any arbitrary function on the surface. The separation of variables provides a complete set of functions for the surface and gives the solution in the interior for each function. Using the superposition you can combine these functions to match you BC. The same combination provides the potential in the interior.

We will work through these methods in class.

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Finally we come to an important way of evaluating fields for localized charge distributions. You can expand the potential about a point. This expansion generates mathematical entites of increasing complexity. The more complex objects however have associated fields that fall off quickly.

 monopole scalar spin 0 1/r => falloff dipole vector spin 1 1/r2 => falloff quadrapole tensor spin 2 1/r3 => falloff

This type of expansion is a common way to handle problems in physics.

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