# Electrodynamics

## Equations / Theorems

### Guass' Law for electrical field of closed surface

for any closed surface and is the surface charge-density.

### Poisson's Equation

where we've used Gauss' Law and the fact that .

#### Boundary conditions

We have the standard boundary conditions in Electrostatiscs:

Note that since we basically have boundary conditions for and in the "normal" ODE sense.

#### Properties

##### Uniqueness

For any two different solutions of the Poisson's equation, the solutions only differ by a constant.

#### Laplace's equation

which is a special case of the Poisson's Equation, where we have space absent of free charges .

#### Examples

The charge density within the follow region is , and thus we have the Laplace's equation with the boundary conditions where , i.e. the potential at the boundary of the inner (non-charge) region is equal to the potential of the inner boundary of the conductor (since they're obviously the same boundary).

#### Method of Images

1. "Imagine" a charge s.t. we get a homogenous PDE (equiv. Laplace's equation)
2. Solve homogenous PDE / Laplacian using [BROKEN LINK: No match for fuzzy expression: *Laplace's%20equation%20in%20cylindrical%20coordinates], i.e. assume a solution of the form
3. Then find a special solution to the non-homogenous PDE
4. The linear combination of these two then form the general solution to the problem

There is one issue that one my bring up; is this solution unique? As it turns out, it is!

### Maxwell's equations

1. Gauss's law

2. Guass's law for magnetism