Definition:Macroscopic Charge Density

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

Let $\delta V$ be a volume element which is smaller than the scale used for a macroscopic electric field, but still large enough to contain many atoms.

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

Then the charge density of the macroscopic electric field at $P$ is defined as:


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

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 $B$.