find the potential and electric field

**szpengchao** · January 27th 2009, 12:55 AM

Calculate the electrostatic potential a distance z from the center of a charged disk, charge
density p Coulombs per square meter, radius a.

**Rincewind** · January 27th 2009, 03:58 AM

I assume you mean a perpendicular distance. Lets say the disk lies in the $xy$ -plane (centered at the origin) and we want the electrostatic potential along the $z$ -axis. The the distance from the disk area element to point $(0,0,Z)$ is

$\delta = \sqrt{r^2+Z^2}$

The the electrostatic potential is given by

$\phi_E = \frac{\rho}{4\pi\epsilon_0}\int_0^{2\pi}\int_0^a\f rac{r\,dr\,d\theta}{\sqrt{r^2+Z^2}}$

This is a straightforward integral and so the potential is given by

$\phi_E = \frac{\rho}{2\epsilon_0}\left(\sqrt{a^2+Z^2}-|Z|\right)$

If you need to find the potential off the $z$ -axis then the integral is a little tricker. It is given by

$\phi_E = \frac{\rho}{4\pi\epsilon_0}\int_0^{2\pi}\int_0^a\f rac{r\,dr\,d\theta}{\sqrt{(r-R)^2+Z^2}},$
where $R$ is the off-axis distance. If you need this in closed form it can be done but from the wording of the first post it sounds like you only need the on-axis solution

Hope that helps.

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