Page:A Dynamical Theory of the Electromagnetic Field.pdf/50

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We obtain M by considering the following conditions: –

1st. M must fulfil the differential equation

$\frac{d^{2}M}{da^{2}}+\frac{d^{2}M}{db^{2}}+\frac{1}{a}\frac{dM}{da}=0$|undefined

This equation being true for any magnetic field symmetrical with respect to the common axis of the circles, cannot of itself lead to the determination of M as a function of $$a, a_{1}$$, and $$b$$. We therefore make use of other conditions.

2ndly. The value of M must remain the same when $$a$$ and $$a_{1}$$ are exchanged.

3rdly. The first two terms of M must be the same as those given above.

M may thus be expanded in the following series:—

$\begin{array}{r} M=4\pi a\log\frac{8a}{r}\left\{ 1+\frac{1}{2}\frac{a-a_{1}}{a}+\frac{1}{16}\frac{3b^{2}+\left(a_{1}-a\right)^{2}}{a^{2}}-\frac{1}{32}\frac{\left(3b^{2}+\left(a-a_{1}\right)^{2}\right)\left(a-a_{1}\right)}{a^{3}}+etc.\right\} \\ \\-4\pi a\left\{ 2+\frac{1}{2}\frac{a-a_{1}}{a}+\frac{1}{16}\frac{b^{2}-3\left(a-a_{1}^{2}\right)}{a^{2}}-\frac{1}{48}\frac{\left(6b^{2}-\left(a-a_{1}\right)^{2}\right)\left(a-a_{1}\right)}{a^{3}}+etc.\right\} \end{array}$|undefined

(113) We may apply this result to find the coefficient of self-induction (L) of a circular coil of wire whose section is small compared with the radius of the circle.

Let the section of the coil be a rectangle, the breadth in the plane of the circle being c$$$$, and the depth perpendicular to the plane of the circle being $$b$$.

Let the mean radius of the coil be $$a$$, and the number of windings $$n$$; then we find, by integrating,

$L=\frac{n^{2}}{b^{2}c^{2}}\iiiint M(xy\ x'y')dx\ dy\ dx'dy'$|undefined

where $$M(xy\ x'y')$$ means the value of M for the two windings whose coordinates are $$xy$$ and $$x'y'$$ respectively; and the integration is performed first with respect to $$x$$ and $$y$$ over the rectangular section, and then with respect to $$x'$$ and $$y'$$ over the same space.

$\begin{array}{l} L=4\pi n^{2}a\left\{ \log_{e}\frac{8a}{r}+\frac{1}{12}-\frac{4}{3}\left(\theta-\frac{\pi}{4}\right)\cot2\theta-\frac{\pi}{3}\cos2\theta-\frac{1}{6}\cot^{2}\theta\log\cos\theta-\frac{1}{6}\tan^{2}\theta\log\sin\theta\right\} \\ \\\qquad+\frac{\pi n^{2}r^{2}}{24a}\left\{ \log\frac{8a}{r}\left(2\sin^{2}\theta+1\right)+3.45+27.475\cos^{2}\theta-3.2\left(\frac{\pi}{2}-\theta\right)\frac{\sin^{3}\theta}{\cos\theta}+\frac{1}{5}\frac{\cos^{4}\theta}{\sin^{2}\theta}\log\cos\theta\right.\\ \\\qquad\qquad\left.+\frac{13}{3}\frac{\sin^{4}\theta}{\cos\theta}\log\sin\theta\right\} +etc.\end{array}$|undefined

The logarithms are, and the angles are in circular measure.

In the experiments made by the Committee of the British Association for determining a standard of Electrical Resistance, a double coil was used, consisting of two nearly equal coils of rectangular section, placed parallel to each other, with a small interval between them.