Page:On Faraday's Lines of Force.pdf/63

The expression for the potential, the middle of the line joining the poles Being the origin, is

$p=M\left({\frac {1}{\sqrt {c^{2}+r^{2}-2cr\ cos\theta }}}-{\frac {1}{\sqrt {c^{2}+r^{2}+2cr\ cos\theta }}}\right)$

From this we find as the value of $I^{2}$,

$I^{2}={\frac {4M^{2}}{c^{4}}}\left(1-3{\frac {r^{2}}{c^{2}}}+9{\frac {r^{2}}{c^{2}}}cos^{2}\theta \right)$;

$\therefore \ I{\frac {dI}{d\theta }}=-18{\frac {M^{2}}{c^{4}}}r^{2}sin\ 2\theta$

and the moment to turn a pair of spheres {radius $a$, distance $2b$} in the direction in which $\theta$ is increased is

$-36{\frac {k-k'}{2k+k'}}{\frac {M^{2}a^{3}b^{2}}{c^{6}}}sin\ 2\theta$

This force, which tends to turn the line of centres equatoreally for diamagnetic and axially for magnetic spheres, varies directly as the square of the strength of the magnet, the cube of the radius of the spheres and the square of the distance of their centres, and inversely as the sixth power of the distance of the poles of the magnet, considered as points. As long as these poles are near each other this action of the poles will be much stronger than the mutual action of the spheres, so that as a general rule we may say that elongated bodies set axially or equatoreally between the poles of a magnet according as they are mag— netic or diamagnetic. If, instead of being placed between two poles very near to each ether, they had been placed in a uniform field such as that of terrestrial magnetism or that produced by a spherical electro-magnet {sec Ex. VIII.), an elongated body would set axially whether magnetic or diamagnetic.

In all these cases the phenomena depend on $k-k'$, so that the sphere conducts itself magnetically or diamagnetically according as it is more or less magnetic, or less or more diamagnetic than the medium in which it is placed.

VI. On the Magnetic Phenomena of a Sphere cut from a substance whose coefficient of assistance is different in different directions.

Let the axes of magnetic resistance be parallel throughout the sphere, and let them be taken for the axes of $x,y,z$. Let $k_{1},k_{2},k_{3}$ be the coefficients of resistance in these three directions, and let $k'$ be that of the external medium,