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Math Help - find the potential and electric field

  1. #1
    Senior Member
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    find the potential and electric field

    Calculate the electrostatic potential a distance z from the center of a charged disk, charge
    density p Coulombs per square meter, radius a.
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  2. #2
    Junior Member
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    Jan 2009
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    Australia
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    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.
    Last edited by Rincewind; January 27th 2009 at 06:28 AM.
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