# Electrodynamics

## Table of Contents

## 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

- "Imagine" a charge s.t. we get a homogenous PDE (equiv. Laplace's equation)
- Solve homogenous PDE / Laplacian using separation of variables, i.e. assume a solution of the form
- Then find a special solution to the non-homogenous PDE
- 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

**Gauss's law****Guass's law for magnetism****Maxwell-Faraday equation****AmpĂ¨re's circuital law**

where:

- is the
*displacement field*, which accounts for the effect of free and bound charge within materials while its sources are the free charges only - is the
*magnetizing field* - is the
*electric field* - is the
*magnetic field* - is the
*free charge density*