Equal parallel cylindrical conductors in electrical problems

Abstract

In a recent paper Dr. Alexander Russell has called attention to the practical importance of the electrostatic problem of determining the distribution of charge and of potential when parallel cylindrical conductors are electrified. Russell points out that the same analysis will serve for the calculation of the current density and the magnetic flux when currents of high frequency pass along the parallel conductors. The solution he gives for the case of equal and opposite charges (or currents) in the two conductors is exact, but in the general case a more elaborate investigation is necessary. In the present paper the problem is solved by the method of conjugate functions. For the most part, the results are given in forms convenient for numerical calculation. The investigation is confined to the case in which the two cylinders are of the same size; for this case ordinary Jacobian elliptic functions suffice though in general theta-functions are required and the algebra is rather heavy. It is hardly to be supposed that a problem of such a simple character in a favourite field of investigation has not been solved before. It is desirable, however, to have the solution in an accessible form.