The Effective Resistance and Inductance of a Helical Coil

This paper deals with a determination of the effective resistance and inductance of a helical coil of great length, composed of thin wire, wound on a cylinder whose radius is large in comparison with that of the wire. The pitch of the winding is not small, so that the problem cannot be treated by the method of Cohen. The method employed depends upon the use of a type of "helical coordinates" defining the position of any point, and of the general theorem relating to orthogonal systems of coordinates. A solution is obtained for the internal and external forces, corresponding to a given impressed electro-motive force, in the form of a Fourier series of which only the initial terms require calculation. The value of the effective current across any section is obtained, and thence the inductance and resistance. The results are of simple character, and are expressed in general in terms of the ber and bei functions of Lord Kelvin, whose results for a straight wire appear as a limiting case. Certain particular cases are worked out in detail and formulæ obtained in terms of elementary functions which are suitable for a very high or a low frequency. Their necessary limitations are also examined numerically. For a high frequency, it is found that the change of self-inductance due to twisting of the wire tends to vanish, and that the change of resistance tends towards a value independent of the frequency.