The self and mutual-inductances of linear conductors

Abstract

Formulae for the self and. mutual inductances of straight wires and rectangles are to be found in various books and papers, but their demonstrations are usually omitted and often the approximate formulae are given as though they were exact. I have thought that a discussion of these formulae, with the derivation of a number of new expressions, would be of interest, and that illustrations of the formulae, with some examples, would be of service in making such numerical calculations as are often made in scientific and technical work. I have derived the formulae in the simplest possible manner, using the law of Biot and Savart in the differential form instead of Neumann's equation, as it gives a better physical view of the various problems considered. This law has not, of course, been experi- mentally verified for unclosed circuits; but the self-inductance of an unclosed circuit means simply its self-inductance as a part of a closed circuit, the total inductance of which can not be determined until the entire circuit is specified. In this sense the use of the law of Biot and Savart to obtain the self-inductance of an unclosed cir- cuit is perfectly legitimate. I have also shown how, by the use of certain arithmetical mean distances in addition to geometrical mean distances, the accuracy of some of the formulae can be increased. In the following demonstrations the magnetic field is assumed to be instantaneous ; in other words, the dimensions of the circuit are assumed to be small enough and the frequency of the current slow so that it is not necessary to take account of the finite velocity of propagation of the field. This may be done even when the field is integrated to infinity, as the distant magnetic field is canceled when two or more open circuits are combined into a closed circuit.