The above expressions for the capacity of an ellipsoid of three unequal axes are in general elliptic integrals, but they can be evaluated for the reduced cases when the ellipsoid is one of revolution, and hence in the limit either takes the form of a long rod or of a circular disk.
Thus if the ellipsoid is one of revolution, and ds is an element of arc which sweeps out the element of surface dS, we have
| dS = 2πyds = 2πydx / ( | dx | ) = 2πydx / ( | py | ) = | 2πb² | dx. |
| ds | b | p |
Hence, since σ = Qp / 4πab², σdS = Qdx / 2a.
Accordingly the distribution of electricity is such that equal parallel slices of the ellipsoid of revolution taken normal to the axis of revolution carry equal charges on their curved surface.
The capacity C of the ellipsoid of revolution is therefore given by the expression
| 1 | = | 1 | ∫ | dx |
| C | 2a | √(x² + y²) |
(7).
If the ellipsoid is one of revolution round the major axis a (prolate) and of eccentricity e, then the above formula reduces to
| 1 | = | 1 | logε ( | 1 + e | ) |
| C1 | 2ae | 1 − e |