Conductors in electrostatic equilibrium

A conductor contains free charges, typically electrons, able to move within it. When no current flows, the conductor is said to be in an electrostatic regime: the charges have redistributed themselves until reaching an equilibrium configuration in which nothing moves any more. From this simple equilibrium condition, three fundamental properties follow necessarily.

Principle — Conductors in equilibrium

1. Zero internal field. Inside the conductor $\ev{\vec{E}=0}$.
2. Charges on the surface. All the excess charge accumulates on the outer surface.
3. Perpendicular field. Just outside the surface the field is perpendicular to it and equals $\ev{E_{\text{sup}} = \frac{\sigma}{\varepsilon_0}}$ where $\sigma$ is the local surface charge density.

Key formula

Conductor in equilibrium: $\vec{E}_{\text{interno}} = 0$ $E_{\text{sup}} = \sigma/\varepsilon_0 \qquad (\sigma = \text{surface density})$

Why is $\vec{E}=0$ inside? If a field existed inside the conductor, the free charges would feel a force and start moving, contradicting the equilibrium hypothesis. The only possible stationary configuration is therefore one in which the internal field is zero: the charges arrange themselves so as to cancel, point by point, any residual field inside the metal.

Why do the charges sit on the surface? It suffices to apply Gauss’s theorem to a closed surface drawn just below the conductor’s surface. Since $\vec{E}=0$ everywhere inside, the flux through that surface is zero, and hence the enclosed charge is also zero. It follows that all the excess charge must lie on the outer surface, the only region not covered by this reasoning.

Internal cavities and the Faraday cage. If the conductor has an empty cavity inside it, the same Gaussian argument shows that the field in the cavity is zero and that the cavity walls are uncharged. This is the principle of the Faraday cage: a conducting shell completely shields its interior from any external field. The applications are numerous: electromagnetic shielding of sensitive equipment, high-voltage cabins that protect operators, coaxial screens that wrap signal cables.

Partial shielding

The Faraday cage also works partially: a conducting shell with small holes still significantly reduces the internal field, even without cancelling it entirely.

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Topics: Electric field and potential Concepts: Conductors in electrostatic equilibrium · Electric field · Gauss’s theorem

Related exercises: Shielding and charge in a hollow conductor · Which graph of E · True or false on field and potential