Ellipsoidal harmonics and their application in electrostatics

Ellipsoidal functions and ellipsoidal co-ordinates, naturally adapted to the treatment of potential problems involving ellipsoidal boundaries, are due to a relatively complicated mathematical appearance avoided in the majority of textbooks on electromagnetism, leading to an unjustified ignorance of the whole system. The paper at hand is devoted to showing that the system is worthwhile using and investigating. By first recollecting the key results of the co-ordinate system and the related potential functions, including the indispensable orthogonality result and the reciprocal distance formula, basic potential problems of electrostatics are reviewed. Closed-form expressions are thereupon derived for the induced charge and its centroid in the case of a point charge influencing a grounded ellipsoid. An image source interpretation, generalising the image solution for a sphere, follows easily from these expressions, as well as an approximate expression for the mutual capacitance between two ellipsoids or a parallel disk capacitor. An expression is also given for the total polarisation induced in a dielectric ellipsoid by the influence of a point dipole.