Field Generated by Surface Charge Density/Examples/Arbitrary Rectangular Area

Example of Use of Field Generated by Surface Charge Density
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.