Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/313

In these expressions $r$ is the distance of the point ($x$, $y$, $z$) from the centre of one of the circular ends of the solenoid, and the zonal harmonics, $Q_{1}$, $Q_{2}$, &c., are those corresponding to the angle $\theta$ which $r$ makes with the axis of the cylinder.

The first of these expressions is discontinuous when $\theta ={\frac {\pi }{2}}$, but we must remember that within the solenoid we must add to the magnetic force deduced from this expression a longitudinal force $4\pi n\gamma$.

677.] Let us now consider a solenoid so long that in the part of space which we consider, the terms depending on the distance from the ends may be neglected.

The magnetic induction through any closed curve drawn within the solenoid is $4\pi n\gamma A^{\prime }$, where $A^{\prime }$ is the area of the projection of the curve on a plane normal to the axis of the solenoid.

If the closed curve is outside the solenoid, then, if it encloses the solenoid, the magnetic induction through it is $4\pi n\gamma A$, where $A$ is the area of the section of the solenoid. If the closed curve does not surround the solenoid, the magnetic induction through it is zero.

If a wire be wound $n^{\prime }$ times round the solenoid, the coefficient of induction between it and the solenoid is

By supposing these windings to coincide with $n$ windings of the solenoid, we find that the coefficient of self-induction of unit of length of the solenoid, taken at a sufficient distance from its extremities, is

Near the ends of a solenoid we must take into account the terms depending on the imaginary distribution of magnetism on the plane ends of the solenoid. The effect of these terms is to make the coefficient of induction between the solenoid and a circuit which surrounds it less than the value $4\pi nA$, which it has when the circuit surrounds a very long solenoid at a great distance from either end.

Let us take the case of two circular and coaxal solenoids of the same length $l$. Let the radius of the outer solenoid be $c_{1}$, and let it be wound with wire so as to have $n_{1}$ windings in unit of length. Let the radius of the inner solenoid be $c_{2}$, and let the number of windings in unit of length be $n_{2}$, then the coefficient of induction between the solenoids, neglecting the effect of the ends, is

		$M$	${}=Gg$,	(18)
where		$G$	${}=4\pi n$,	(19)
and		$g$	${}=\pi c_{2}^{2}ln_{2}$.	(20)