Page:On Faraday's Lines of Force.pdf/61

Substituting the values of the moments of the imaginary magnets

$-X={\frac {k-k'}{2k+k'}}a^{3}\left(\alpha {\frac {d\alpha }{dx}}+\beta {\frac {d\beta }{dx}}+\gamma {\frac {d\gamma }{dx}}\right)={\frac {k-k'}{2k+k'}}{\frac {a^{3}}{2}}{\frac {d}{dx}}(\alpha ^{2}+\beta ^{2}+\gamma ^{2})$,

The force impelling the sphere in the direction of $x$ is therefore dependent on the variation of the square of the intensity or (\alpha^2+\beta^2+\gamma^2), as we move along the direction of $x$, and the same is true for $y$ and $z$, so that the law is, that the force acting on diamagnetic spheres is from places of greater to places of less intensity of magnetic force, and that in similar distributions of magnetic force it varies as the mass of the sphere and the square of the intensity.

It is easy by means of Laplace’s Coefficients to extend the approximation to the value of the potential as far as we please, and to calculate the attraction. For instance, if a north or south magnetic pole whose strength is $M$, be placed at a distance $b$ from a diamagnetic sphere, radius $a$, the repulsion will be

$R=M^{2}(k-k'){\frac {a^{3}}{b^{3}}}\left({\frac {2.1}{2k+k'}}+{\frac {3.2}{3k+2k'}}{\frac {a^{2}}{b^{2}}}+{\frac {4.3}{4k+3k'}}{\frac {a^{4}}{b^{4}}}+\&c.\right).$

When ${\frac {a}{b}}$ is small, the first term gives a sufficient approximation. The repulsion is then as the square of the strength of the pole, and the mass of the sphere directly and the fifth power of the distance inversely, considering the pole as a point.

IV. Two Spheres in uniform field.

Let two spheres of radius $a$ be connected together so that their centres kept at a distance $b$, and let them be suspended in a uniform magnetic field, then, although each sphere by itself would have been in equilibrium at any part of the field, the disturbance of the field will produce forces tending to make the balls set in a particular direction.

Let the centre of one of the spheres be taken as origin, then the undisturbed potential is

$p=Ircos\theta ,$

and the potential due to the sphere is

$p'=I{\frac {k-k'}{2k+k'}}{\frac {a^{3}}{r^{2}}}cos\theta$.