Net Charge within Electrically Neutral Body of Matter

Definition
Let $B$ be a body of matter.

Let $B$ be neutral.

Let $\map {\rho_{\text {atomic} } } {\mathbf r}$ denote the atomic charge density at a point within $B$ whose position vector is $\mathbf r$.

Then:
 * $\ds \int_B \map {\rho_{\text {atomic} } } {\mathbf r} \rd V = 0$

where $\d V$ denotes an infinitesimal volume element containing the point whose position vector is $\mathbf r$.

Proof
By definition, $\ds \int_B \map {\rho_{\text {atomic} } } {\mathbf r} \rd V$ denotes the total electric charge on $B$.

Hence the result.