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.