Conductors

1. $\mathbf E=0$ everywhere inside a conductor, since that would result in charges experiencing a net force (not electrostatic)
2. $(1) \implies \rho = 0$ everywhere inside a conductor. All charges reside on the surface
3. $(1) \implies$ the entire conductor is equipotential
4. $(3) \implies \mathbf E$ is perpendicular to a conductor’s surface.

At the surface of a conductor, $E=\sigma/\epsilon_0$ because $\textbf E=0$ inside conductor so all electric flux must be outside.

Basic theorems

1. Earnshaw’s theorem: maxima and minima of the potential function occur at boundaries. This can be proven using the Laplace equation.

2. Green’s reciprocity theorem: (same coordinate system)

   $$ \iiint \rho_1 V_2 d \Omega_1 = \iiint \rho_2 V_1 d \Omega_2 $$

Uniqueness theorem(s)

Specifying the total charges or potential on conductors completely surrounding a volume and the charge density inside the volume uniquely determines the electric field throughout the volume (a solution is guaranteed). Of course, this implies the boundary condition that each conductor is equipotential. Infinity can be considered a zero-potential conductor.

Specifying potential and total charge on each conductor is actually two different uniqueness theorems according to Ricardo (1st and 2nd), but they can be combined.

Also note that the potential function can only be found if at least one potential of a surface is specified (for example in a conductor cavity, the potential of the conductor depends on external charges which are unknown when only considering the internal system)

• Proof of 1st uniqueness theorem
• Proof of 2nd uniqueness theorem
• Example: a spherical conductor has a cavity with a charge $Q$ inside. What is the electric field outside the conductor?
• Example: prove that an empty volume completely surrounded by a conductor with arbitrary total charge has no electric field.

Note that uniqueness theorems only define the electric field in the volume which it is defined. It will not work outside of that volume.

Image charges

These are fake charges set up to produce an equivalent electric field. This is proven using the uniqueness theorem. If the system of charges produce the same potential at the boundaries and has the same charge density everywhere, the electric field is equivalent.