Electrostatic equilibrium

A conductor is said to have reached electrostatic equilibrium when its charge distribution becomes stable.

For example, all conductors in the picture above have reached electrostatic equilibrium.

Properties

Consider a solid conductor $(S)$ at electrostatic equilibrium.

By combination of Newton's 2nd law and Coulomb's law, for any macroscopic charge $A$:

$$\vec a = \frac{Q_A\vec E}{m}$$

By definition of electrostatic equilibrium,

$$\begin{gathered} \forall A \in (S), \vec a=0 \\ \therefore \vec E =0 \text{ for every point inside (S) (1)} \end{gathered}$$

As a result, if conductor $(S)$ is charged, any charge must be destributed on the surface of $(S)$, $S(S)$, which creates an electric field on $S(S)$.

This field must satisfy $\vec E \bot S(S)$ at all points. (3) Otherwise, the macroscopic charges would move along the surface.

Summary

To summarise, the three properties for a conductor at electrostatic equilibrium are:

1. $\vec E_{internal}=0$
2. charges may exist only on the surface of the conductor
3. on the surface, $\vec E =0 \lor \vec E \bot \text {surface}$