mutual inductance of coaxial circles

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1: 19.34 Mutual Inductance of Coaxial Circles

§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 $$\frac{c^{2}M}{2\pi }=ab{\int }_0^{2\pi }{(h^2+a^2+b^2-2ab\cos \theta )}^{-1/2}\cos \theta d\theta =2ab{\int }_{-1}^1\frac{tdt}{\sqrt{(1+t)(1-t)(a_3-2abt)}}=2abI({𝐞}_5),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $I(𝐦)$: integral, $M$: mutual inductance and $h$: distance

Referenced by:

§19.34

Permalink:

http://dlmf.nist.gov/19.34.E1

Encodings:

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

2: Bibliography G

… ►

Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:

ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

… ►

Special functions of matrix argument. I. Algebraic induction, zonal polynomials, and hypergeometric functions. Trans. Amer. Math. Soc. 301 (2), pp. 781–811.

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External Links:

ISSN 0002-9947, FullText, MathReview (Wayne J. Holman, III), ZentralBlatt

Referenced by:

§35.8(i), §35.8(ii), §35.8(iv), Ch.35.

Encodings:

BibTeX

… ►

Inductance Calculations. Van Nostrand, New York.

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Notes:

Reprinted by Dover, New York, 1962 and 2004

External Links:

ZentralBlatt

Referenced by:

§19.34.

Encodings:

BibTeX

…

3: Tom M. Apostol

… ►Tom Apostol and his wife Jane were inducted into the MAA’s Icosahedron Society in 2010. …

4: 10.51 Recurrence Relations and Derivatives

…

5: 31.15 Stieltjes Polynomials

… ►are mutually orthogonal over the set $Q$: …

6: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Note that eigenfunctions for distinct (necessarily real) eigenvalues of a self-adjoint operator are mutually orthogonal. … ►By Weyl’s alternative $n_1$ equals either 1 (the limit point case) or 2 (the limit circle case), and similarly for $n_2$. … A boundary value for the end point $a$ is a linear form $ℬ$ on $𝒟({ℒ}^{\ast })$ of the form … ►The above results, especially the discussions of deficiency indices and limit point and limit circle boundary conditions, lay the basis for further applications. … ►The materials developed here follow from the extensions of the Sturm–Liouville theory of second order ODEs as developed by Weyl, to include the limit point and limit circle singular cases. …

7: 10.6 Recurrence Relations and Derivatives

… ►

10.6.10 $$p_{\nu }{s}_{\nu }-q_{\nu }{r}_{\nu }=4/({\pi }^{2}ab).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\nu$: complex parameter, $p_{\nu }$: cross-product, $q_{\nu }$: cross-product, $r_{\nu }$: cross-product and $s_{\nu }$: cross-product

A&S Ref:

9.1.34

Permalink:

http://dlmf.nist.gov/10.6.E10

Encodings:

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See also:

Annotations for §10.6(iii), §10.6 and Ch.10

8: Frank W. J. Olver

… ►Department of Commerce Gold Medal, the highest honorary award granted by the Department, and was inducted into the NIST Portrait Gallery of Distinguished Scientists, Engineers, and Administrators. …

9: 23.2 Definitions and Periodic Properties

… ►

23.2.14 $${\eta }_3{\omega }_2-{\eta }_2{\omega }_3={\eta }_2{\omega }_1-{\eta }_1{\omega }_2={\eta }_1{\omega }_3-{\eta }_3{\omega }_1=\frac{1}{2}\pi \mathrm{i}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

A&S Ref:

18.3.37

Referenced by:

§23.12, §23.8(ii)

Permalink:

http://dlmf.nist.gov/23.2.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §23.2(iii), §23.2 and Ch.23

…

10: 10.29 Recurrence Relations and Derivatives

…