Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/338

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306

CIRCULAR CURRENTS.

[701.

is moderate as compared with the radii of the smaller, the series already given do not converge rapidly. In every case, however, we may find the value of $M$ for two parallel circles by elliptic integrals.

For let $b$ be the length of the line joining the centres of the circles, and let this line be perpendicular to the planes of the two circles, and let $A$ and $a$ be the radii of the circles, then

M=\iint {\frac {\cos \epsilon }{r}}ds\,ds^{\prime }

,

the integration being extended round both curves. In this case,

r^{2}=A^{2}+a^{2}+b^{2}-2Aa\cos(\phi -\phi ^{\prime })

,

\epsilon =\phi -\phi ^{\prime }

, ⁠

ds=a\,d\phi

, ⁠

ds^{\prime }=A\,d\phi ^{\prime }

,

	$M$	${}=\int _{0}^{2\pi }\int _{0}^{2\pi }{\frac {Aa\cos(\phi -\phi ^{\prime })\,d\phi \,d\phi ^{\prime }}{\sqrt {A^{2}+a^{2}+b^{2}2Aa\cos(\phi -\phi ^{\prime })}}}$ }}
		${}=2\pi {\sqrt {Aa}}\left\{\left(c-{\frac {2}{c}}\right)F+{\frac {2}{c}}E\right\}$ ,

where

c={\frac {\sqrt {Aa|}}{\sqrt {(A+a)^{2}+b^{2}}}}

,

and $F$ and $E$ are complete elliptic integrals to modulus $c$ . From this we get, by differentiating with respect to $b$ and remembering that $c$ is a function of $b$ ,

{\frac {dM}{db}}={\frac {4\pi bc^{\frac {1}{2}}}{\sqrt {Aa(1-c^{2})^{2}}}}\{E(1+c^{2})-F(1-c^{2})\}

.

If $r_{1}$ and $r_{2}$ denote the greatest and least values of $r$ ,

r_{1}^{2}=(A+a)^{2}+b^{2}

, ⁠

r_{2}^{2}=(A-a)^{2}+b^{2}

,

and if an angle $\gamma$ be taken such that $\cos \gamma ={\frac {r_{2}}{r_{1}}}$ ,

{\frac {dM}{db}}=\pi {\frac {b\sin \gamma }{\sqrt {Aa}}}\{2F_{\gamma }-(1+\sec ^{2}\gamma )E_{\gamma }\}

,

where $F_{\gamma }$ and $E_{\gamma }$ denote the complete elliptic integrals of the first and second kind whose modulus is $\sin \gamma$ . If $A=a$ , $\cot \gamma ={\frac {b}{2a}}$ , and

{\frac {dM}{db}}=2\pi \cos \gamma \{2F_{\gamma }-(1+\sec ^{2}\gamma )E_{\gamma }\}

.

The quantity ${\frac {dM}{db}}$ represents the attraction between two parallel circular currents, the current in each being unity.