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