Solution of two mutually-reducible problems in electrostatics

Abstract

There are two well-known problems in electrostatics whose solutions reduce to each other. One of them is that of a grounded conductor containing a cavity with given boundary $\bar S$. A charge distribution is specified on another surface S inside the cavity, or within the volume enclosed by S. It is required to find the charge density induced on $\bar S$. The other problem is that of finding “equivalent” sets of charges (producing identical external fields). Here again there are surfaces $\bar S$ and S and the same original distributed charge as in the first problem, but the system is now in empty space and the problem is to find the charge distribution on $\bar S$ that produces the same external field as the given distribution on S. Mutual reducibility means that it is sufficient to consider one of the two problems, say, the second. The problem examined in this paper is that of confocal ellipsoids S and $\bar S$ and charge distributions described in terms polynomial functions of Cartesian coordinates. The method of multipole moments which leads directly to the solution (i.e., without the need to evaluate the field) is described. Analytical solutions are given for simple surface and volume charge distributions. Special and limiting cases are examined, including degenerate surfaces S and $\bar S$ in the form of confocal elliptic cylinders.