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

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

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

If the conductors are solid wires, $$a_2$$ and $$a_2^\prime$$ are zero, and

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.

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

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