Electrostatic Equilibrium

An electrical conductor has the ability to be in electrostatic equilibrium. Electrostatic equilibrium is defined as when all the charged particles inside are no longer moving. It changes it’s own charge density to achieve this.

Properties §

Some important properties of a conductor in electrostatic equilibrium are:

• $\vec{E}$, the Electric Field inside the conductor is always 0.
• All excess charge is on the surface of the conductor
• The electric field on the surface is always perpendicular to the surface.
• The surface is Equipotential.
• The surface charge density, $\sigma$, and the electrical field strength, is greatest at sharp points (where surface area is minimised.)

Electrostatic Shielding §

A conductor that is in electrostatic equilibrium can ‘shield’ anything inside it from electric fields, i.e. $E$ inside conductor is zero.

Let’s start with a metal bar through a uniform electric field. Obviously, due to the effects of the field, the free-moving electrons will move, causing the metal to become polarised:

However, the polarisation causes an electric field inside the metal as well!

Eventually, the electric fields ${\vec{E}}_{ext}$ and ${\vec{E}}_{int}$ balance each other out, creating a net field of zero inside the metal.

As such, even a conducting object that is hollow can demonstrate the same properties, and can ‘shield’ anything inside:

Charge Accumulation on Surface §

Using Gauss’ Law, we can show that the charges on a conductor accumulate on the surface of a conductor in electrostatic equilibrium:

Take a conducting metal sphere in electrostatic equilibrium, and take a smaller sphere to act as the Gaussian Surface.

We’ve shown above that the net electric field of a conductor in electrostatic equilibrium is 0, i.e. $\sum {\vec{E}}_{inside}=0$. Let’s use Gauss’ Law:

$$\Phi =\oint \vec{E}\cdot \text{d}\vec{A}=\frac{q_{enclosed}}{{ϵ}_0}$$

And because the gaussian sphere is inside the conductor, the field is 0!

$$\Phi =0=\frac{q}{{ϵ}_0}\to q_{enclosed}=0$$

So the charged inside the gaussian sphere is 0, meaning that all the charge must reside on the surface of the sphere.