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

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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.


Field-from-Surface-Charge.png


Sources