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 $B$.
Symbol
The usual symbol used to denote surface charge density is $\sigma$ (Greek letter sigma).
Dimension
The dimension of surface charge density is $\mathsf {I T L}^{-2}$: electric charge per unit area.
Units
The SI units of surface charge density are $\mathrm C \, \mathrm m^{-2}$ (coulombs per square metre).
Also see
- Results about surface charge density can be found here.
Sources
- 1990: I.S. Grant and W.R. Phillips: Electromagnetism (2nd ed.) ... (previous) ... (next): Chapter $1$: Force and energy in electrostatics: $1.3$ Electric Fields in Matter: $1.3.3$ The macroscopic electric field: $(1.9)$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): density: 2.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): density: 2.