Field Generated by Surface Charge Density/Examples
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Examples of Use of Field Generated by Surface Charge Density
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.