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19 Elliptic IntegralsApplications

§19.34 Mutual Inductance of Coaxial Circles

The mutual inductance M of two coaxial circles of radius a and b with centers at a distance h apart is given in cgs units by

19.34.1 c2M2π=ab02π(h2+a2+b2-2abcosθ)-1/2cosθdθ=2ab-11tdt(1+t)(1-t)(a3-2abt)=2abI(e5),

where c is the speed of light, and in (19.29.11),

19.34.2 a3 =h2+a2+b2,
a5 =0,
b5 =1.

The method of §19.29(ii) uses (19.29.18), (19.29.16), and (19.29.15) to produce

19.34.3 2abI(e5)=a3I(0)-I(e3)=a3I(0)-r+2r-2I(-e3)=2ab(I(0)-r-2I(e1-e3)),

where a1+b1t=1+t and

19.34.4 r±2=a3±2ab=h2+(a±b)2

is the square of the maximum (upper signs) or minimum (lower signs) distance between the circles. Application of (19.29.4) and (19.29.7) with α=1, aβ+bβt=1-t, δ=3, and aγ+bγt=1 yields

19.34.5 3c28πabM=3RF(0,r+2,r-2)-2r-2RD(0,r+2,r-2),

or, by (19.21.3),

19.34.6 c22πM=(r+2+r-2)RF(0,r+2,r-2)-4RG(0,r+2,r-2).

A simpler form of the result is

19.34.7 M=(2/c2)(πa2)(πb2)R-32(32,32;r+2,r-2).

References for other inductance problems solvable in terms of elliptic integrals are given in Grover (1946, pp. 8 and 283).