Net Charge within Electrically Neutral Body of Matter
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Definition
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.
$\blacksquare$
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: $1.3.1$ The atomic charge density