Problem — Conductor in an external field

Problem

A metal conductor is placed in a uniform electric field. What happens to the electric field inside the conductor and at its surface, at electrostatic equilibrium?

Solution

Rearrangement of charges. The conductor contains free electrons. The external field $\vv{E}_{\text{est}}$ sets them in motion: negative charges build up on the face facing the field, and positive charges on the opposite face. These induced charges generate an internal field $\vv{E}_{\text{ind}}$ opposite to the external one.

Equilibrium condition. The motion of electrons continues until the induced field exactly cancels the external one inside the metal. At electrostatic equilibrium there is no more charge motion, so the total field inside the conductor is zero:

$\vv{E}_{\text{int}} = \vv{E}_{\text{est}} + \vv{E}_{\text{ind}} = \vv 0$

Consequences.

• The whole volume of the conductor is at the same potential: it is an equipotential volume (if there were a potential difference, the internal field would not be zero).
• The surface is an equipotential surface, so the field just outside is perpendicular to the surface: any tangential component would set the surface charges in motion, contradicting equilibrium.
• The induced charges settle only on the surface, with maximum density where the curvature is greatest (points, edges).

$\ev{\vv{E}_{\text{int}} = 0,\ \ \vv{E}_{\text{sup}}\perp\text{surface}}$

Links

Topics: Electrostatics Concepts: Conductors in electrostatic equilibrium · Electric field