Definition:Macroscopic Charge Density
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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 in the vicinity of $\delta V$ whose position vector is $\mathbf r$.
The macroscopic charge density is the charge density of the macroscopic electric field at $P$, 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 $B$.
Also see
- Results about macroscopic 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: $(1.8)$