Conductors in Electrostatics

A conductor is full of mobile electrons. The moment you place it in any external field, those electrons rearrange until they have nowhere left to go — that is, until the field inside the conductor vanishes. This single fact has profound consequences.

Concept

Properties of a conductor in electrostatic equilibrium:

Field inside is zero. $\vec E_\text{inside} = 0$ . If not, mobile electrons would still be in motion, contradicting "equilibrium".
Potential is constant throughout. Since $\vec E = -\nabla V$ and $\vec E = 0$ inside, $V$ is the same at every interior point and on the surface — the whole conductor is one big equipotential.
All net charge resides on the surface. Gauss's law applied to a tiny interior volume gives zero enclosed charge.
Surface field is perpendicular to the surface. Any tangential component would push surface charges sideways — but they would already have moved.
Surface field magnitude: $E_\text{just outside} = \sigma/\varepsilon_0$ , where $\sigma$ is local surface charge density.
Charge density is higher where curvature is higher (sharp points). This is why lightning rods are pointy.

Electrostatic Shielding

A hollow conductor with no charge inside has $\vec E = 0$ in its cavity, regardless of external fields. The induced charges on the outer surface rearrange to exactly cancel the external field inside. This is Faraday cage shielding — used to protect sensitive electronics, MRI rooms, even cars in lightning storms.

If there is a charge $q$ inside the cavity, the inner surface develops induced charge $-q$ and the outer surface acquires the remaining $+q$ (assuming initially neutral conductor). The outer field looks exactly as if $q$ sat at the centre — completely independent of where $q$ actually is inside the cavity.

Derivation

$E_\text{inside} = 0$ : suppose $\vec E_\text{inside} \neq 0$ . Free electrons (charge $-e$ ) feel force $-e\vec E$ , accelerate, and produce current. But in electrostatic equilibrium there is no current. So $\vec E_\text{inside}$ must equal zero.

Surface field: apply Gauss's law to a tiny pillbox straddling the surface, with the top outside and bottom inside the conductor.

Bottom face: $\vec E_\text{inside} = 0$ , contributes $0$ to flux.
Side: tangential, contributes $0$ for small thickness.
Top: $\vec E_\text{outside}\cdot A$ .

Charge enclosed: $\sigma\,A$ . So $E\cdot A = \sigma A/\varepsilon_0 \Longrightarrow E_\text{just outside} = \frac{\sigma}{\varepsilon_0}.$

Worked Example

A solid metal sphere of radius $5\,cm$ carries a total charge of $+4\,nC$ .

Field inside ( $r < 5\,cm$ ): $0$ .
Field at the surface: charges distribute uniformly by symmetry. Total surface area $4\pi R^2 = 4\pi (0.05)^2 = 0.0314\,m^2$ . Surface density $\sigma = 4\times 10^{-9}/0.0314 = 1.27\times 10^{-7}\,C/m^2$ . Field just outside: $E = \sigma/\varepsilon_0 \approx 1.44\times 10^4\,N/C$ .
Field outside ( $r > R$ ): $kQ/r^2$ , equal to point-charge field.
Potential inside or on surface: constant $V = kQ/R = (9\times 10^9)(4\times 10^{-9})/0.05 = 720\,V$ .

Common Confusions

Field inside a charged conductor is zero, but field just outside is $\sigma/\varepsilon_0$ , NOT $\sigma/(2\varepsilon_0)$ . The factor of 2 of an isolated sheet does not apply because all flux exits one side.
Net charge on a neutral conductor with internal cavity charge $q$ : outer surface gets $+q$ , inner surface $-q$ .
Shielding works even in time-varying fields if the conductor is thick enough (high frequencies penetrate via skin effect).
Sharp points have high charge density — discharge can happen there easily (corona discharge).

Key Takeaways

Inside conductor: $\vec E = 0$ , $V$ constant.
Net charge sits on outer surface.
Surface field $= \sigma/\varepsilon_0$ , perpendicular.
Hollow conductor shields its interior from external fields.
Sharp points concentrate charge.