Macroscopic Electric Field in Body

Theorem
Let $B$ be a body of matter.

Let $P$ be a point inside $B$ whose position vector is $\mathbf r$.

The macroscopic electric field at $P$ is given by:


 * $\ds \map {\mathbf E} {\mathbf r} = \dfrac 1 {4 \pi \varepsilon_0} \int_{\text {all space} } \dfrac {\paren {\mathbf r - \mathbf r'} \map \rho {\mathbf r'} } {\size {\mathbf r - \mathbf r'}^3} \rd V' + \dfrac 1 {4 \pi \varepsilon_0} \int_{\text {all surfaces} } \dfrac {\paren {\mathbf r - \mathbf r'} \map \sigma {\mathbf r'} } {\size {\mathbf r - \mathbf r'}^3} \rd S'$

where:
 * $\d V'$ is an infinitesimal volume element
 * $\d S'$ is an infinitesimal area element
 * $\mathbf r'$ is the position vector of $\d V'$ or $\d S'$ as appropriate
 * $\map \rho {\mathbf r'}$ is the macroscopic charge density of the macroscopic electric field at $\mathbf r'$
 * $\varepsilon_0$ denotes the vacuum permittivity.

Proof
From Field Generated by Macroscopic Charge Density, the electric field at $P$ generated by the macroscopic charge density within $B$ is given by:


 * $\ds \map {\mathbf E} {\mathbf r} = \dfrac 1 {4 \pi \varepsilon_0} \int_{\text {all space} } \dfrac {\paren {\mathbf r - \mathbf r'} \map \rho {\mathbf r'} } {\size {\mathbf r - \mathbf r'}^3} \rd V'$

From Field Generated by Surface Charge Density, the electric field at $P$ generated by the surface charge density over $B$ is given by:


 * $\ds \map {\mathbf E} {\mathbf r} = \dfrac 1 {4 \pi \varepsilon_0} \int_{\text {all surfaces} } \dfrac {\paren {\mathbf r - \mathbf r'} \map \sigma {\mathbf r'} } {\size {\mathbf r - \mathbf r'}^3} \rd S'$

We have that the Electric Field satisfies Principle of Superposition.

Hence the total macroscopic electric field at $P$ is given by the sum of these:
 * $\ds \map {\mathbf E} {\mathbf r} = \dfrac 1 {4 \pi \varepsilon_0} \int_{\text {all space} } \dfrac {\paren {\mathbf r - \mathbf r'} \map \rho {\mathbf r'} } {\size {\mathbf r - \mathbf r'}^3} \rd V' + \dfrac 1 {4 \pi \varepsilon_0} \int_{\text {all surfaces} } \dfrac {\paren {\mathbf r - \mathbf r'} \map \sigma {\mathbf r'} } {\size {\mathbf r - \mathbf r'}^3} \rd S'$