19 Elliptic Integrals Applications 19.33 Triaxial Ellipsoids 19.35 Other Applications

§19.34 Mutual Inductance of Coaxial Circles

ⓘ

Keywords:: coaxial circles, elliptic integrals, inductance, mutual inductance of coaxial circles, symmetric elliptic integrals
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).
Referenced by:: §19.29(ii)
Permalink:: http://dlmf.nist.gov/19.34
See also:: Annotations for Ch.19

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		$\frac{{c}^{2}M}{2\pi}=ab\int_{0}^{2\pi}(h^{2}+a^{2}+b^{2}-2ab\cos\theta)^{-1/2% }\cos\theta\,\mathrm{d}\theta=2ab\int_{-1}^{1}\frac{t\,\mathrm{d}t}{\sqrt{(1+t% )(1-t)(a_{3}-2abt)}}=2abI(\mathbf{e}_{5}),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : 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	$\displaystyle a_{3}$	$\displaystyle=h^{2}+a^{2}+b^{2},$
	$\displaystyle a_{5}$	$\displaystyle=0,$
	$\displaystyle b_{5}$	$\displaystyle=1.$
ⓘ 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		$2abI(\mathbf{e}_{5})=a_{3}I(\boldsymbol{{0}})-I(\mathbf{e}_{3})=a_{3}I(% \boldsymbol{{0}})-r_{+}^{2}r_{-}^{2}I(-\mathbf{e}_{3})=2ab(I(\boldsymbol{{0}})% -r_{-}^{2}I(\mathbf{e}_{1}-\mathbf{e}_{3})),$
ⓘ Symbols: $I(\mathbf{m})$ : integral and $r_{\pm}$ Permalink: http://dlmf.nist.gov/19.34.E3 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		$\frac{3{c}^{2}}{8\pi ab}M=3R_{F}\left(0,r_{+}^{2},r_{-}^{2}\right)-2r_{-}^{2}R% _{D}\left(0,r_{+}^{2},r_{-}^{2}\right),$
ⓘ Symbols: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : 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		$\frac{{c}^{2}}{2\pi}M=(r_{+}^{2}+r_{-}^{2})R_{F}\left(0,r_{+}^{2},r_{-}^{2}% \right)-4R_{G}\left(0,r_{+}^{2},r_{-}^{2}\right).$
ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : 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		$M=(2/{c}^{2})(\pi a^{2})(\pi b^{2})R_{-\frac{3}{2}}\left(\tfrac{3}{2},\tfrac{3% }{2};r_{+}^{2},r_{-}^{2}\right).$
ⓘ Symbols: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : 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).

19.33 Triaxial Ellipsoids 19.35 Other Applications

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§19.34 Mutual Inductance of Coaxial Circles

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