Potential of a Charged Spheroid

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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.


Contributed by: S. M. Blinder (August 2010)
Open content licensed under CC BY-NC-SA



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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