§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		$$\begin{array}{ll}\frac{c^{2}M}{2\pi }& =ab{\int }_0^{2\pi }{(h^2+a^2+b^2-2ab\cos \theta )}^{-1/2}\cos \theta d\theta \\ & =2ab{\int }_{-1}^1\frac{tdt}{\sqrt{(1+t)(1-t)(a_3-2abt)}}=2abI({\mathbf{e}}_5),\end{array}$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: 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 See also: Annotations for §19.34 and Ch.19

where $c$ 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 See also: Annotations for §19.34 and Ch.19

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

19.34.3		$$\begin{array}{ll}2abI({\mathbf{e}}_5)& =a_{3}I(\mathbf{0})-I({\mathbf{e}}_3)=a_{3}I(\mathbf{0})-r_{+}^2{r}_{-}^{2}I(-{\mathbf{e}}_3)\\ & =2ab(I(\mathbf{0})-r_{-}^{2}I({\mathbf{e}}_1-{\mathbf{e}}_3)),\end{array}$$
ⓘ Symbols: $I(\mathbf{m})$: integral and $r_{\pm }$ Permalink: http://dlmf.nist.gov/19.34.E3 Encodings: TeX, pMML, png See also: Annotations for §19.34 and Ch.19

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 See also: Annotations for §19.34 and Ch.19

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		$$\frac{3c^2}{8\pi ab}M=3R_F(0,r_{+}^2,r_{-}^2)-2r_{-}^2{R}_D(0,r_{+}^2,r_{-}^2),$$
ⓘ Symbols: $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $M$: mutual inductance and $r_{\pm }$ Permalink: http://dlmf.nist.gov/19.34.E5 Encodings: TeX, pMML, png See also: Annotations for §19.34 and Ch.19

or, by (19.21.3),

19.34.6		$$\frac{c^2}{2\pi }M=(r_{+}^2+r_{-}^2)R_F(0,r_{+}^2,r_{-}^2)-4R_G(0,r_{+}^2,r_{-}^2).$$
ⓘ Symbols: $R_F(x,y,z)$: symmetric elliptic integral of first kind, $R_G(x,y,z)$: symmetric elliptic integral of second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $M$: mutual inductance and $r_{\pm }$ Referenced by: §19.34 Permalink: http://dlmf.nist.gov/19.34.E6 Encodings: TeX, pMML, png See also: Annotations for §19.34 and Ch.19

A simpler form of the result is

19.34.7		$$M=(2/c^2)(\pi a^2)(\pi b^2)R_{-\frac{3}{2}}(\frac{3}{2},\frac{3}{2};r_{+}^2,r_{-}^2).$$
ⓘ Symbols: $R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(\mathbf{b};\mathbf{z})$: multivariate hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $M$: mutual inductance and $r_{\pm }$ Referenced by: §19.34 Permalink: http://dlmf.nist.gov/19.34.E7 Encodings: TeX, pMML, png See also: Annotations for §19.34 and Ch.19

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