Capacitance 3: Parallel Plane Capacitor

From HvWiki

Fringing field near parallel plane capacitor

Here, we consider two parallel planes separated by an air dielectric ( $\epsilon_r=1\,$ . This problem lacks the elegant (and convenient) symmetry of the coaxial line or dielectric sphere, so a closed form solution is difficult to find. (Although this problem can be solved exactly using conformal transformation methods []). Our approach will be to apply a computer method to generate the potential distribution, the fields and finally, the capacitance per unit length.

Figure 2 shows the geometry of the problem. The top plate is at a fixed positive potential (say, 1V) and the lower at -1V.

Geometry of the parallel plane capacitor.

How do we find the resulting distribution of charge on the plates? The answer lies in using the generalised integral equation solution for the Laplace Equation (except, here, in two dimensions) that was introduced earlier

$V(\textbf{r}) = \frac{1}{2\pi\epsilon_0} \oint_C \rho_s\ln\left (\vert\textbf{r}- \textbf{r}^\prime\vert\right ) dl^\prime$ .

Notice, however, because we are solving a two-dimensional problem, instead of the $1/\vert\textbf{r}-\textbf{r}^\prime\vert$ Green's function previously encountered, we use a logarithmic relationship (which, in fact, represents the spatial impulse response for a line source).