The use of Bessel functions for calculating the self-inductance of single-layer solenoids

Abstract

This paper is a continuation of the paper on “The Field of a Coil between two Parallel Metal Sheets.” It was pointed out in Section 6 of that paper that the general expressions, which had been developed therein, made it possible to calculate both the self-inductance and the high-frequency resistance of any circular coil enclosed in a closed screening-can. This solution was not then developed, but the work is in progress now. As a preliminary to this said new work it seemed desirable to show first that the Bessel function treatment did yield the correct result for the self-inductance of a single-layer solenoid. This exploration has shown that the old problem can be attacked with great elegance by the Bessel process and that it has definite merits of its own. This is perhaps not surprising, since the form of a solenoid suggests that Bessel functions are more appropriate to it than the integral of elliptic functions. This paper sets out the solution of an old problem by a new means.The first stage shows that the extremely general Bessel formulae do lead to the well-known expression H = 4πIT cos ϕ for the magnetic field at the centre of a solenoid. To obtain this result, use has to be made of formulae for the infinite integral of products of Bessel functions, and in particular to one formula due originally to Heaviside. The Bessel treatment turns out to have special advantages for calculating the strength of the field just inside the winding: this evaluation is necessarily peculiarly cumbersome by Legendre functions because they are in a series which is verging on becoming divergent at this radius. Formulae typified by eqn. (10) in the paper have a certain element of novelty and are valuable for calculating the high-frequency resistance of a coil having only one turn. The inductance of the isolated solenoid is derived in Section 2, and the results are tabulated and compared with values derived by Dr. A. Russell: also it is shown that the results can be expressed as t the sum of two integrals, the one or other of which becomes relatively unimportant when the coil is either long or short. The process of addition is exhibited graphically in Fig. 2; this Figure exposes very clearly the structure of the whole calculation, which otherwise remains obscure right up to the final result. Section 3 is devoted to calculating the inductance of any solenoid placed symmetrically between a pair of infinite sheets of metal, perpendicular to its axis. The tabulated results are useful for estimating the order of magnitude of the effect of a large metal sheet; and also they establish that the inductance of a long, isolated solenoid may be calculated by means of a rapidly convergent series without resort to the much more cumbersome process of using the definite integral.Though most of the results are old, the process of deriving them is believed to be novel. The new method is found to have some advantages over the old, and moreover it will solve many problems which are insoluble by the old. It is hoped that the new solutions will follow shortly; this paper is a necessary preliminary to the exposition of them.