364.
ON THE SELF-INDUCTION OF ELECTRIC CUERENTS IN A THIN ANCHOR-RING.
[Proceedings of the Royal Society, A, Vol. LXXXVI. pp. 562 — 571, 1912.]
IN their useful compendium of " Formulae and Tables for the Calculation of Mutual and Self-Inductance*," Rosa and Cohen remark upon a small discrepancy in the formulas given by myself f and by M. WienJ for the self-induction of a coil of circular cross-section over which the current is uniformly distributed. With omission of n, representative of the number of windings, my formula was
7 2 /. 8a 1\1
(1)
where p is the radius of the section and a that of the circular axis. The first two terms were given long befpre by Kirchhoff§. In place of the fourth term within the bracket, viz., + TJ^ pa/a3, Wien found — 'OOSSpYa2. 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
* Bulletin of the Bureau of Standards, Washington, 1908, Vol. in. No. 1. t Roij. Soc. Proc. 1881, Vol. xxxn. p. 104; Scientific Payers, Vol. n. p. 15. t Ann. d. Physik, 1894, Vol. LIII. p. 934 ; it would appear that Wien did not know of my earlier calculation.
§ Pogg. Ann. 1864, Vol. cxxi. p. 551. || Electricity and Magnetism, § 705.us dispersion, i.e. to be due to resonances lying beyond the visible range. In the simplest form of this theory, as given by Maxwell f and Sellmeier, the resonating bodies take their motion from those parts of the sether with which they are directly connected, but they do not influence one another. In such a case the boundary conditions involve merely the continuity of the displacement and its first derivative, and no complication ensues. When there is no refraction, there is also no reflexion. By introducing a mutual reaction between the resonators, and probably in other ways, it would be possible to modify the situation in such a manner that the boundary conditions would involve higher derivatives, as in the case of the stiff string, and thus to allow reflexion in spite of equality of wave- velocities for a given ray.