The surface area of an ellipsoid with semiaxes , and volume is given by
For additional geometrical properties of ellipsoids (and ellipses), see Carlson (1964, p. 417).
If a conducting ellipsoid with semiaxes bears an electric charge , then the equipotential surfaces in the exterior region are confocal ellipsoids:
The potential is
and the electric capacity is given by
A conducting elliptic disk is included as the case .
Let a homogeneous magnetic ellipsoid with semiaxes , volume , and susceptibility be placed in a previously uniform magnetic field parallel to the principal axis with semiaxis . The external field and the induced magnetization together produce a uniform field inside the ellipsoid with strength , where is the demagnetizing factor, given in cgs units by
Ellipsoidal distributions of charge or mass are used to model certain atomic nuclei and some elliptical galaxies. Let the density of charge or mass be
where are dimensionless positive constants. The contours of constant density are a family of similar, rather than confocal, ellipsoids. In suitable units the self-energy of the distribution is given by
Subject to mild conditions on this becomes