Electrostatic boundary conditions at a conductor

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.