Conductor in electrostatic equilibrium

The electric field vanishes inside a conductor at electrostatic equilibrium, so each connected conductor is equipotential and the exterior field is normal to its surface.

A Gaussian pillbox givesSince $E_{\mathrm{in}}=0$, the surface charge density is $\sigma ={ϵ}_0{E}_{\mathrm{out}}\cdot n$.

Surface charge density is electric charge per unit area. At a conductor in electrostatic equilibrium it equals ${ϵ}_0{\mathbf{E}}_{\mathrm{out}}\cdot \mathbf{n}$ for the normal pointing out of the conductor.

Two widely separated conducting spheres of radii $R_1,R_2$ connected by a thin wire have equal potentials, so a total charge $Q$ divides as

Two connected concentric spherical shells form one equipotential conductor. In the absence of charge in the inner cavity, all net charge lies on the exterior surface of the outer shell.

A neutral conducting sphere of radius $R$ in the field $E\hat{z}$ has exterior potential and induced surface chargeThe surface integral of $\sigma$ is zero.