Macroscopic Electric Field in Body
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Theorem
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:
- $\ds \map {\mathbf E} {\mathbf r} = \dfrac 1 {4 \pi \varepsilon_0} \int_{\text {all space} } \dfrac {\paren {\mathbf r - \mathbf r'} \map \rho {\mathbf r'} } {\size {\mathbf r - \mathbf r'}^3} \rd V' + \dfrac 1 {4 \pi \varepsilon_0} \int_{\text {all surfaces} } \dfrac {\paren {\mathbf r - \mathbf r'} \map \sigma {\mathbf r'} } {\size {\mathbf r - \mathbf r'}^3} \rd S'$
$\blacksquare$
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: $(1.10)$