Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/316

Hence, if there is no iron core, the diameter of the wire of the interior coil should be inversely as the square root of the radius, but if there is a core of iron having a high capacity for magnetization, the diameter of the wire should be more nearly directly proportional to the square root of the radius of the layer.

An Endless Solenoid.

681.] If a solid be generated by the revolution of a plane area $A$ about an axis in its own plane, not cutting it, it will have the form of a ring. If this ring be coiled with wire, so that the windings of the coil are in planes passing through the axis of the ring, then, if $n$ is the whole number of windings, the current-function of the layer of wire is $\phi ={\frac {1}{2\pi }}n\gamma \theta$, where $\theta$ is the angle of azimuth about the axis of the ring.

If $\Omega$ is the magnetic potential inside the ring and $\Omega ^{\prime }$ that outside, then

$\Omega -\Omega ^{\prime }=4\pi \phi +C=2n\gamma \theta +C$.

Outside the ring $\Omega ^{\prime }$, must satisfy Laplace's equation, and must vanish at an infinite distance. From the nature of the problem it must be a function of $\theta$ only. The only value of $\Omega ^{\prime }$ which fulfils these conditions is zero. Hence

$\Omega ^{\prime }=0$,⁠$\Omega =2n\gamma \theta +C$.

The magnetic force at any point within the ring is perpendicular to the plane passing through the axis, and is equal to $2n\gamma {\frac {1}{r}}$ where $r$ is the distance from the axis. Outside the ring there is no magnetic force.

If the form of a closed curve be given by the coordinates $z$, $r$, and $\theta$ of its tracing point as functions of $s$, its length from a fixed point, the magnetic induction through the closed curve is

$2n\gamma \int _{0}^{s}{\frac {z}{r}}{\frac {dr}{ds}}ds$

taken round the curve, provided the curve is wholly inside the ring. If the curve lies wholly without the ring, but embraces it, the magnetic induction through it is

$2n\gamma \int _{0}^{s^{\prime }}{\frac {z^{\prime }}{r^{\prime }}}{\frac {dr^{\prime }}{ds^{\prime }}}ds^{\prime }=2n\gamma a$,

where the accented coordinates refer not to the closed curve, but to a single winding of the solenoid.

The magnetic induction through any closed curve embracing the