The self-inductance of a coil of any length and any number of layers of wire

The self-inductance of a coil or short solenoid wound with any number of layers of wire is given by the formula of Weinstein, or Stefan's modification of Weinstein's formula, when one applies the proper corrections for the thickness of the insulation and the shape of the section of the wire. But for a long solenoid this formula is not very accurate. I have elsewhere shown how to obtain the selfinductance of a long solenoid wound with a single layer of round covered wire or with bare wire wound at any given pitch. I propose now to show how one may obtain accurately the self-inductance of a solenoid of any length having a uniform winding of any number of layers; this will include the case of short coils as well as those where the length is too great to be calculated by the formulae of Weinstein or Stefan. Mr. Cohen gives elsewhere in this Bulletin an approximate formula for the self-inductance of relatively long coils of more than one layer. His formula is convenient in calculation when the number of layers is not large, and is accurate enough for most practical cases, notwithstanding it assumes the current to be distributed in current sheets, taking no account of the shape of the cross section of the wire or the thickness of the insulation. I shall now show how to calculate accurately the self-inductance of a coil of any length and any number of layers, wound with insulated round wire, taking account of the shape of the section as well as the thickness of the insulation of the wire. L,et Fig. i be the section of such a winding of mean radius a, length /, and depth of winding /, and having m layers. If n is the number of turns per centimeter,