Hence, in the expression for , the first two terms may be written
and .
Integrating the two latter terms in the ordinary way, and adding the results, remembering that , we obtain the value of the kinetic energy . Writing this , where is the coefficient of self-induction of the system of two conductors, we find as the value of for unit of length of the system
.
(22)
If the conductors are solid wires, and are zero, and
.
(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 , , and 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, , between the Two Portions of Wire.
686.] By Art. 580 we obtain for the force tending to increase ,
,
,
(24)
which agrees with Ampère's formula, when , 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 is this tension,
,
.
(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