§19.34 Mutual Inductance of Coaxial Circles

Notes:: For (19.34.1) see Becker and Sauter (1964, p. 194). For (19.34.7) see Carlson (1977b, Ex. 9.3-2 and p. 313); alternatively, substitute Carlson (1977b, (9.2-3) and (9.2-2)) in (19.34.6) and use Carlson (1977b, Table 9.3-2).
Keywords:: coaxial circles, elliptic integrals, inductance, mutual inductance of coaxial circles, symmetric elliptic integrals
Referenced by:: §19.29(ii)
Permalink:: http://dlmf.nist.gov/19.34

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 $\frac{\lightspeed^{2}M}{2\pi}=ab\int _{0}^{{2\pi}}(h^{2}+a^{2}+b^{2}-2ab\mathop{\cos\/}\nolimits\theta)^{{-1/2}}\mathop{\cos\/}\nolimits\theta d\theta=2ab\int _{{-1}}^{1}\frac{tdt}{\sqrt{(1+t)(1-t)(a_{3}-2abt)}}=2abI(\mathbf{e}_{5}),$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $I(\mathbf{m})$ : integral, $M$ : mutual inductance and $h$ : distance
Referenced by:: §19.34
Permalink:: http://dlmf.nist.gov/19.34.E1
Encodings:: TeX, pMML, png

where $\lightspeed$ is the speed of light, and in (19.29.11),

19.34.2

$a_{3}=h^{2}+a^{2}+b^{2},$

$a_{5}=0,$

$b_{5}=1.$

Symbols:: $h$ : distance
Permalink:: http://dlmf.nist.gov/19.34.E2
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

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

19.34.3 $2abI(\mathbf{e}_{5})=a_{3}I(\boldsymbol{{0}})-I(\mathbf{e}_{3})=a_{3}I(\boldsymbol{{0}})-r_{{+}}^{2}r_{{-}}^{2}I(-\mathbf{e}_{3})=2ab(I(\boldsymbol{{0}})-r_{{-}}^{2}I(\mathbf{e}_{1}-\mathbf{e}_{3})),$

Symbols:: $I(\mathbf{m})$ : integral and $r_{\pm}$
Permalink:: http://dlmf.nist.gov/19.34.E3
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where $a_{1}+b_{1}t=1+t$ and

19.34.4 $r_{{\pm}}^{2}=a_{3}\pm 2ab=h^{2}+(a\pm b)^{2}$

Symbols:: $h$ : distance and $r_{\pm}$
Permalink:: http://dlmf.nist.gov/19.34.E4
Encodings:: TeX, pMML, png

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 $\alpha=1$ , $a_{\beta}+b_{\beta}t=1-t$ , $\delta=3$ , and $a_{\gamma}+b_{\gamma}t=1$ yields

19.34.5 $\frac{3\lightspeed^{2}}{8\pi ab}M=3\!\mathop{R_{F}\/}\nolimits\!\left(0,r_{{+}}^{2},r_{{-}}^{2}\right)-2r_{{-}}^{2}\mathop{R_{D}\/}\nolimits\!\left(0,r_{{+}}^{2},r_{{-}}^{2}\right),$

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables, $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $M$ : mutual inductance and $r_{\pm}$
Permalink:: http://dlmf.nist.gov/19.34.E5
Encodings:: TeX, pMML, png

or, by (19.21.3),

19.34.6 $\frac{\lightspeed^{2}}{2\pi}M=(r_{{+}}^{2}+r_{{-}}^{2})\mathop{R_{F}\/}\nolimits\!\left(0,r_{{+}}^{2},r_{{-}}^{2}\right)-4\!\mathop{R_{G}\/}\nolimits\!\left(0,r_{{+}}^{2},r_{{-}}^{2}\right).$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind, $M$ : mutual inductance and $r_{\pm}$
Referenced by:: §19.34
Permalink:: http://dlmf.nist.gov/19.34.E6
Encodings:: TeX, pMML, png

A simpler form of the result is

19.34.7 $M=(2/\lightspeed^{2})(\pi a^{2})(\pi b^{2})\mathop{R_{{-\frac{3}{2}}}\/}\nolimits\!\left(\tfrac{3}{2},\tfrac{3}{2};r_{{+}}^{2},r_{{-}}^{2}\right).$

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function, $M$ : mutual inductance and $r_{\pm}$
Referenced by:: §19.34
Permalink:: http://dlmf.nist.gov/19.34.E7
Encodings:: TeX, pMML, png

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