Definition:Surface Charge Density

Definition
Let $B$ be a body made out of an electrically conducting substance.

Let $B$ be under the influence of an electric field $\mathbf E$ under which a surface charge is induced on $B$.

Let $\delta S$ be an area element which is smaller than the scale used for a macroscopic electric field, but still large enough to contain many atoms on the surface of $B$.

Let $P$ be a point in the vicinity of $\delta S$ whose position vector is $\mathbf r$.

Let $\delta V$ be a volume element just thick enough to enclose the whole of the surface charge $\map \sigma {\mathbf r} \delta S$ associated with $\delta S$.

The surface charge density is the charge density of the macroscopic electric field on the surface $P$, defined as:


 * $\ds \map \sigma {\mathbf r} = \dfrac 1 {\delta S} \int_{\delta V} \map {\rho_{\text {atomic} } } {\mathbf r'} \rd \tau'$

where:
 * $\d \tau'$ is an infinitesimal volume element
 * $\mathbf r'$ is the position vector of $\d \tau'$
 * $\map {\rho_{\mathrm {atomic} } } {\mathbf r'}$ is the atomic charge density caused by the electric charges within the atoms that make up $B$.