Conductors in electrostatic equilibrium

Conductors and mobile charge

A conductor contains charges that can move freely through the material. In metals, these mobile charges are electrons. When an electric field exists inside a conductor, these charges experience forces and begin to move.

Electrostatic equilibrium means charges have stopped moving macroscopically. Once equilibrium is reached, several powerful properties follow.

Electric field inside a conductor

In electrostatic equilibrium, the electric field inside the conducting material is zero:

ec{E}=0

If it were not zero, mobile charges would feel a force

ec{F}=qec{E}

and would continue moving. That would contradict electrostatic equilibrium.

This statement applies inside the conducting material, not necessarily inside an empty cavity unless additional conditions are considered.

Charge resides on the surface

Any excess charge placed on a conductor moves to the outer surface in electrostatic equilibrium. Gauss's law explains why. Imagine a Gaussian surface just inside the conducting material. Since ec{E}=0 everywhere on that surface, the electric flux is zero:

oint ec{E}cdot dec{A}=0

Gauss's law then gives

$Q_{enc}=0$

So there can be no net excess charge inside the conducting material. Excess charge must reside on surfaces.

Field at the surface

Just outside a conductor, the electric field is perpendicular to the surface. If there were a tangential component, surface charges would move along the surface. Therefore electrostatic equilibrium requires

$E_{parallel}=0$

The normal component just outside a conductor is related to surface charge density:

E_{perp}=rac{sigma}{epsilon_0}

for a surface in vacuum. This result comes from applying Gauss's law to a small pillbox crossing the surface.

Conductors are equipotentials

Since ec{E}=0 inside the conductor, there is no potential difference between points inside it. Also, because the tangential field at the surface is zero, the surface is an equipotential. Thus a conductor in electrostatic equilibrium has constant potential throughout:

$V=constant$

This does not mean the conductor has zero potential. It means every point on and inside the conductor has the same potential.

Charge concentration at sharp points

Surface charge density is greater where a conductor has smaller radius of curvature, such as sharp points. Stronger surface charge density means stronger electric field nearby. This is why lightning rods are pointed: they enhance local electric fields and can promote controlled discharge.

Electrostatic shielding

A conductor can shield its interior from external electric fields. In electrostatic equilibrium, charges rearrange on the surface so that the field inside the conducting material vanishes. A closed conducting shell can protect its interior region from external static fields. This is the principle of a Faraday cage.

If there is no charge inside a cavity within a conductor, the electric field inside the cavity is zero. If a charge is placed inside the cavity, induced charges appear on cavity walls and outer surfaces.

Grounding

Grounding connects a conductor to Earth, allowing charge to flow to or from a large reservoir. A grounded conductor is often taken to have potential

$V=0$

Grounding can neutralize excess charge or hold an object at a fixed reference potential.

The big idea

Conductors in electrostatic equilibrium have zero electric field inside the conducting material, excess charge on surfaces, perpendicular surface fields, and constant potential. These properties explain shielding, grounding, charge concentration, and the behavior of capacitors and electrical devices.