Potential of a Charged Spheroid

This Demonstration shows the electrostatic potential of a uniformly charged spheroid. We consider both prolate spheroids, with , and oblate spheroids, with . Here , , are the semi-axes, with the axis oriented horizontally. The potential is cylindrically symmetrical and it suffices to show just the plane containing the axis. The potential external to the spheroid is given by , the sum representing a multipole expansion over the charge distribution. For an oblate or prolate spheroid, the monopole contribution is dominant, with only the quadrupole term making a significant additional contribution to the potential. The quadrupole moment of a charged spheroid is given by .
You can select the semi-axes and to display a scaled contour plot of the potential. Multiply by to find the actual potential. The same result pertains to a gravitational potential, with as the scaling factor. You can isolate the quadrupole contribution with the checkbox.


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  • [Snapshot]
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The potential of a spheroid with unit charge , thus density , expressed in cylindrical coordinates , is given by
where is determined by the quadratic equation , taking the positive sign of the square root.
Snapshot 1: quadrupole contribution for an oblate spheroid
Snapshot 2: potential for a prolate spheroid
Snapshot 3: limiting case of a sphere
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