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

We obtain M by considering the following conditions: –

1st. M must fulfil the differential equation

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

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:—

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

(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${\displaystyle }$, 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,

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

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.

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

Here	$a=$ mean radius of the coil.
„	$r=$ diagonal of the rectangular section $={\sqrt {b^{2}+c^{2}}}$
„	$\theta =$ angle between $r$ and the plane of the circle.
„	n$=$ number of windings.

The logarithms are Napierian, 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.