Macroscopic Electric Field in Body

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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:

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

$\blacksquare$


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