Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/336

 

699.] It is sometimes convenient to express the series for $M$ in terms of linear quantities as follows:—

Let $a$ be the radius of the smaller circuit, $b$ the distance of its plane from the origin, and $c={\sqrt {a^{2}+b^{2}}}$.

Let $A$, $B$, and $C$ be the corresponding quantities for the larger circuit.

The series for $M$ may then be written,

	$M={}$	$1.2.\pi ^{2}{\frac {A^{2}}{C^{3}}}a^{2}\cos \theta$
		${}+2.3.\pi ^{2}{\frac {A^{2}B}{C^{5}}}a^{2}b(\cos ^{2}\theta -{\frac {1}{2}}\sin ^{2}\theta )$
		${}+3.4.\pi ^{2}{\frac {A^{2}(B^{2}-{\frac {1}{4}}A^{2})}{C^{7}}}a^{2}(b^{2}-{\frac {1}{4}}a^{2})(\cos ^{3}\theta -{\frac {3}{2}}\sin ^{2}\theta \cos \theta )$
		${}+\mathrm {\&c.}$

If we make $\theta =0$, the two circles become parallel and on the same axis. To determine the attraction between them we may differentiate $M$ with respect to $b$. We thus find

700.] In calculating the effect of a coil of rectangular section we have to integrate the expressions already found with respect to $A$, the radius of the coil, and $B$, the distance of its plane from the origin, and to extend the integration over the breadth and depth of the coil.

In some cases direct integration is the most convenient, but there are others in which the following method of approximation leads to more useful results.

Let $P$ be any function of $x$ and $y$, and let it be required to find the value of ${\overline {P}}$ where

In this expression ${\overline {P}}$ is the mean value of $P$ within the limits of integration.

Let $P_{0}$ be the value of $P$ when $x=0$ and $y=0$, then, expanding $P$ by Taylor's Theorem,

Integrating this expression between the limits, and dividing the result by $xy$, we obtain as the value of ${\overline {P}}$,