Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/323

This is a minimum when $a_{1}=a_{2}$, and then

	$L$	${}=2l(\log 4+{\frac {1}{2}})$,	⁠
		${}=2l(1.8863)$,
		${}=3.7726l$.		(27)

This is the smallest value of the self-induction of a round wire doubled on itself, the whole length of the wire being $2l$.

Since the two parts of the wire must be insulated from each other, the self-induction can never actually reach this limiting value. By using broad flat strips of metal instead of round wires the self-induction may be diminished indefinitely.

On the Electromotive Force required to produce a Current of Varying Intensity along a Cylindrical Conductor.

689.] When the current in a wire is of varying intensity, the electromotive force arising from the induction of the current on itself is different in different parts of the section of the wire, being in general a function of the distance from the axis of the wire as well as of the time. If we suppose the cylindrical conductor to consist of a bundle of wires all forming part of the same circuit, so that the current is compelled to be of uniform strength in every part of the section of the bundle, the method of calculation which we have hitherto used would be strictly applicable. If, however, we consider the cylindrical conductor as a solid mass in which electric currents are free to flow in obedience to electromotive force, the intensity of the current will not be the same at different distances from the axis of the cylinder, and the electromotive forces themselves will depend on the distribution of the current in the different cylindric strata of the wire.

The vector-potential $H$, the density of the current $w$, and the electromotive force at any point, must be considered as functions of the time and of the distance from the axis of the wire.

The total current, $C$, through the section of the wire, and the total electromotive force, $E$, acting round the circuit, are to be regarded as the variables, the relation between which we have to find.

Let us assume as the value of $H$,

where $S$, $T_{0}$, $T_{1}$, &c. are functions of the time. Then, from the equation