On the self-induction of electric currents in a thin anchor-ring

In their useful compendium of "Formulæ and Tables for the Calculation of Mutual and Self-Inductance," Rosa And Cohen remark upon a small discrepancy in the formulæ given by myself and by M. Wien for the self-induction of a coil of circular cross-section over which the current isuniformly distributed . With omission ofn , representative of the number of windings, my formula was L = 4πa [ log 8a /ρ - 7/4 +ρ 2 /8a 2 (log 8a /ρ + 1/3) ], (1) whereρ is the radius of the section anda that of the circular axis. The first two terms were given long before by Kirchhoff. In place of the fourth term within the bracket, viz., +1/24ρ 2 /a 2 , Wien found -·0083ρ 2 /a 2 . In either case a correction would be necessary in practice to take account of the space occupied by the insulation. Without, so far as I see, giving a reason, Rosa and Cohen express a preference for Wien's number. The difference is of no great importance, but I have thought it worth while to repeat the calculation and I obtain the same result as in 1881. A confirmation after 30 years, and without reference to notes, is perhaps almost as good as if it were independent. I propose to exhibit the main steps of the calculation and to make extension to some related problems. The starting point is the expression given by Maxwell for the mutual induction M between two neighbouring co-axial circuits. For the present purpose this requires transformation, so as to express the inductance in terms of the situation of the elementary circuits relatively to the circular axis. In the figure, O is the centre of the circular axis, A the centre of a section B through the axis of symmetry, and the position of any point P of the section is given by polar co-ordinates relatively to A, viz.