Field Generated by Surface Charge Density
Theorem
Let $P$ be a point inside $B$ whose position vector is $\mathbf r$.
The electric field at $P$ generated by the surface charge density over $B$ is given by:
- $\ds \map {\mathbf E} {\mathbf r} = \dfrac 1 {4 \pi \varepsilon_0} \int_{\text {all surfaces} } \dfrac {\paren {\mathbf r - \mathbf r'} \map \sigma {\mathbf r'} } {\size {\mathbf r - \mathbf r'}^3} \rd S'$
where:
- $\d S'$ is an infinitesimal area element
- $\mathbf r'$ is the position vector of $\d S'$
- $\map \sigma {\mathbf r'}$ is the surface charge density at $\mathbf r'$
- $\varepsilon_0$ denotes the vacuum permittivity.
Proof
From Electric Field Strength from Assemblage of Point Charges, the electric field strength caused by an assemblage of point charges $q_1, q_2, \ldots, q_n$ is given by:
- $\ds \map {\mathbf E} {\mathbf r} = \dfrac 1 {4 \pi \epsilon_0} \sum_i \dfrac {\paren {\mathbf r - \mathbf r_i} q_i} {\size {\mathbf r - \mathbf r_i}^3}$
where $\mathbf r_1, \mathbf r_2, \ldots, \mathbf r_n$ are the position vectors of $q_1, q_2, \ldots, q_n$ respectively.
We apply the same principle to the surface charge density and convert the summation into a definite integral, as follows:
Consider an area element $\d S'$.
The electric field strength caused by $\d S'$ is:
- $\map {\mathbf E} {\mathbf r'} = \dfrac 1 {4 \pi \epsilon_0} \dfrac {\paren {\mathbf r - \mathbf r'} \rd q} {\size {\mathbf r - \mathbf r'}^3}$
where $\d q$ is the electric charge on $\d S'$.
By definition of surface charge density:
- $\map \sigma {\mathbf r'} = \dfrac {\d q} {\d S'}$
and so:
- $\map {\mathbf E} {\mathbf r'} = \dfrac 1 {4 \pi \epsilon_0} \dfrac {\paren {\mathbf r - \mathbf r'} \map \sigma {\mathbf r'} } {\size {\mathbf r - \mathbf r'}^3} \rd S'$
Integrating over all surfaces of $B$:
- $\ds \map {\mathbf E} {\mathbf r} = \dfrac 1 {4 \pi \varepsilon_0} \int_{\text {all surfaces} } \dfrac {\paren {\mathbf r - \mathbf r'} \map \rho {\mathbf r'} } {\size {\mathbf r - \mathbf r'}^3} \rd S'$
Hence the result.
$\blacksquare$
Examples
Arbitrary Rectangular Area
Consider a rectangular surface $S$ embedded in the $x$-$y$ plane in a Cartesian $3$-space.
Let the corners of $S$ be at $x = \pm a$ and $y = \pm b$.
The electric field at $P$ generated by the surface charge density over $S$ is given by:
- $\ds \map {\mathbf E} {\mathbf r} = \dfrac 1 {4 \pi \varepsilon_0} \int_{x \mathop = -a}^a \int_{y \mathop = -b}^b \dfrac {\paren {\mathbf r - \mathbf r'} \map \sigma {\mathbf r'} } {\size {\mathbf r - \mathbf r'}^3} \rd y' \rd x'$
where:
- $\d y' \rd x'$ is an infinitesimal area element of $S$
- $\mathbf r'$ is the position vector of $\d y' \rd x'$
- $\map \sigma {\mathbf r'}$ is the surface charge density at $\mathbf r'$
- $\varepsilon_0$ denotes the vacuum permittivity.
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