Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/321

Hence, in the expression for $T$, the first two terms may be written

$AC^{\prime }-2\mu _{0}CC^{\prime }\log b$ and $A^{\prime }C-2\mu _{0}CC^{\prime }\log b$.

Integrating the two latter terms in the ordinary way, and adding the results, remembering that $C+C^{\prime }=0$, we obtain the value of the kinetic energy $T$. Writing this ${\frac {1}{2}}LC^{2}$, where $L$ is the coefficient of self-induction of the system of two conductors, we find as the value of $L$ for unit of length of the system

	${\frac {L}{l}}=2\mu _{0}\log {\frac {b^{2}}{a_{1}a_{1}^{\prime }}}$	${}+{\frac {1}{2}}{\frac {a_{1}^{2}-3a_{2}^{2}}{a_{1}^{2}-a_{2}^{2}}}+{\frac {4a_{2}^{4}}{(a_{1}^{2}-a_{2}^{2})^{2}}}\log {\frac {a_{1}}{a_{2}}}$
		${}+{\frac {1}{2}}\mu ^{\prime }{\frac {a_{1}^{\prime 2}-3a_{2}^{\prime 2}}{a_{1}^{\prime 2}-a_{2}^{\prime ^{2}}}}+{\frac {4a_{2}^{\prime 4}}{(a_{1}^{\prime 2}-a_{2}^{\prime 2})^{2}}}\log {\frac {a_{1}^{\prime }}{a_{2}^{\prime }}}$.	(22)

If the conductors are solid wires, $a_{2}$ and $a_{2}^{\prime }$ are zero, and

${\frac {L}{l}}=2\mu _{o}\log {\frac {b^{2}}{a_{1}a_{1}^{\prime }}}+{\frac {1}{2}}(\mu +\mu ^{\prime })$.

(23)

It is only in the case of iron wires that we need take account of the magnetic induction in calculating their self-induction. In other cases we may make $\mu _{0}$, $\mu$, and $\mu ^{\prime }$ all equal to unity. The smaller the radii of the wires, and the greater the distance between them, the greater is the self-induction.

To find the Repulsion, $X$, between the Two Portions of Wire.

686.] By Art. 580 we obtain for the force tending to increase $b$,

$X$	${}={\frac {1}{2}}{\frac {dL}{db}}C^{2}$,
	${}=2\mu _{0}{\frac {l}{b}}C^{2}$,

(24)

which agrees with Ampère's formula, when $\mu _{0}=1$, as in air.

687.] If the length of the wires is great compared with the distance between them, we may use the coefficient of self-induction to determine the tension of the wires arising from the action of the current.

If $Z$ is this tension,

		$Z$	${}={\frac {1}{2}}{\frac {dL}{dl}}C^{2}$,
			${}=C^{2}\left\{\mu _{0}\log {\frac {c^{2}}{a_{1}a_{1}^{\prime }}}+{\frac {\mu }{2}}\right\}$.		(25)

In one of Ampère's experiments the parallel conductors consist of two troughs of mercury connected with each other by a floating bridge of wire. When a current is made to enter at the extremity of one of the troughs, to flow along it till it reaches one extremity