Note on the Self-inductance of Small Circular Section.

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1 84 K.TERAZAWA: [Ser, 2, Vol. 5, Note on the Self-inductance of Small Circular Section. of a Ring RY K. TRRAZAWA. [READ MAY 1, 1909.] The formultu for the self-inductance of a circular ring of sallm circular section were given by Kirchholf(1), Maxwell, Max Wien(2), and Rayleigh and Niven(3). Among them, the formula given by Wien has been considered as the most accurate one(4) and is of the form:- where R is the radius of the central circle of the ring, and p that of the cross-section. Rayleigh and Niven's formula agrees with Wien's, except as to one term in place of ; i.e., Wien's formula is deduced on the supposition that the current is distributed uniformly over the section, and is given by the integra tion of where df1, df2 are the areal elements of the cross-section, and the mutual inductance of two coaxial circles passing through M is df1 and f3. The form for M taken by Wien is the Maxwell's series formula d (1) Kirehhof, Pogg. Anna, 121 (1864), p. 551, (2) M, Wien, Wied. Ann,, 53 (1894), p. 928, (3) Rayleigh's Collected Papers U, p. 15. (4) See Bulletin of the Burean of Standards V, p. 37.

2 May, 1909] SELF-INDUCTAXCE OF A RING. where A and A+x are the radii of the two coaxial circles passing through df1 and df2, y the distance between their planes, and r that between df1 and df2. wien takes the polar coordinates, the centre of the section being the pole 85 M is thus transformed into up to terms of the second order in Consequently The integral may be resolved into two terms, one of which does not contain the factor log r, and the other contains it; by integration wien obtains for the former part and for the latter. I have repeated the calculation in the same way as wien, and got the same result for the former, but different one for the latter which is o f the form

3 86. TERAGAWA : [Ser. 2, Vol. 5. so tlia the expression for the self-inductance is of the form which is the same as that of Rayleigh and Niven. I can not find the reason why the same method of calculation gives the different result. It would not be needless to sketch the calculation for the latter part. Since or denoting by T the latter part of integration, where S stands for We know that when a<1, Therefore

4 May, 19 0 SELF-INDUCTANCE OF A,R 87 Thus since Therefore the integration with respect to seasily attained, and have tried another method of calculation, by Prof. Nagaoka's suggestion, employing the toroidal coordinates instead of the polar. This calculation is very complicated compared with that of wien's but it may serve as a check., Using C. Neumaun's notation, the toroidal coordinates ere 2a is the distance between the both poles, the axis of the ring being the z-axis. Eliminating them, we have an equation of the ring containing a parameter we have to put

5 88 K. TERAZAWA: [Ser. 2, Vol,, 5. Thus I take the expression for M the q-series of Prof. Nagaoka (1), instead of the Maxwell's formula, the moduli being given by and If the section of the ring is small, q1 are also small, therefore we may take neglecting thefourth and higher powers of k'. Thus we may put where and Therefore if we put (1) H. Nagaoka, J. Coil, Sci. (Tbkyo), 16,

6 May, 1909] S ELF=INDUCTANCE OF A RIKG. 89 and then Expanding Q and N into power series of and if we stop at the second degree of approximation the result of integration is At first we shall expand Q. Since where Pn is he zonal hermonic of the nth order, and we obtain four expansions of the above quantities; multiplying these we get an expression for Q, rip to the 2nd order, in the form Next to expand N we must consider two cases, in and Since

7 90 K. TERAZAWA : [Ser. 2, Vol. 55 a nd for other K'2 we may take In these summations with respect to n, it is sufficient to stop at n=3, for the terms of order n>3 vanish by the integration with respect to w2. Therefore where ƒö=ƒö1-ƒö2 We cal obtain similar expression for N (ƒé1<ƒé2) by interchanging ƒé1 and ƒé2. Now we go on to the integration of It is very complicated to write down, each step of integration, so that it will be necessary to give only a few important steps of the operations. Omitting the factor for ƒé1>ƒé2,

8 May, 1909] SELF-INDUCTANCE OF ARING. 91 Therefore up to the 2nd degree in ƒé1 and ƒé2, for ƒé1>ƒé2. Similarly for ƒé1<ƒé2 Next integrating with respect to ƒé2, we have

9 2 K. TERAZAWA: [Ser. 2, Vol. 5. so that Since ƒé is small-it is always less than 1-we can expand this by the binomial and logarithmic series and by then But so that we arrive at the result

10 May, 1909] SELF-INDUCTANCE OF A RING. 93 where a and ƒé are related to ƒï and B by or The advantage of using toroidal coordinates lies in the fact that we can build up a system of pings by altering ƒé, and thus get an insight how the self inductance varies with the dimension of the cross-section. The following numbers will illustrate with ƒé. Finally, we shall compare our expression with that of wien. Since

11 94 K. TERAZAWA: [Ser. 2, Vol.5 Substituting theme in the above formula, we get which is that given by Rayleigh and Niven. The difference in the coefficient of does not however much affect the numerial result, as will be seep from the following examples: by our formula, by Wien's,,. by our formula, by wien's,,.

314 H. NAGAOKA: [VOL. W H. NAGAOKA.

314 H. NAGAOKA: [VOL. W H. NAGAOKA. 314 H. NAGAOKA: [VOL. W BY H. NAGAOKA. 1. The practical importance of having an accurate standard of inductance has lately been felt in electrical measurements relating to alternate currents and electric