Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/94

In most cases the differences between the coefficients of magnet ization in different directions are very small, so that we may put

$L={\frac {4}{3}}\pi a^{3}{\frac {r_{2}-r_{3}}{(1+{\frac {4}{3}}\pi r)^{2}}}F^{2}\sin 2\theta .$

(13)

This is the force tending to turn a crystalline sphere about the axis of x from y towards z. It always tends to place the axis of greatest magnetic coefficient (or least diamagnetic coefficient) parallel to the line of magnetic force.

The corresponding case in two dimensions is represented in Fig. XVI.

If we suppose the upper side of the figure to be towards the north, the figure represents the lines of force and equipotential surfaces as disturbed by a transversely magnetized cylinder placed with the north side eastwards. The resultant force tends to turn the cylinder from east to north. The large dotted circle represents a section of a cylinder of a crystalline substance which has a larger coefficient of induction along an axis from north-east to south-west than along an axis from north-west to south-east. The dotted lines within the circle represent the lines of induction and the equipotential surfaces, which in this case are not at right angles to each other. The resultant force on the cylinder is evidently to turn it from east to north.

437.] The case of an ellipsoid placed in a field of uniform and parallel magnetic force has been solved in a very ingenious manner by Poisson.

If V is the potential at the point (x, y, z) due to the gravitation of a body of any form of uniform density ρ, then $-{\frac {dV}{dx}}$ is the potential of the magnetism of the same body if uniformly magnetized in the direction of x with the intensity I = ρ.

For the value of $-{\frac {dV}{dx}}\delta x$ at any point is the excess of the value of V, the potential of the body, above V', the value of the potential when the body is moved –δx in the direction of x.

If we supposed the body shifted through the distance –δx, and its density changed from ρ to –ρ (that is to say, made of repulsive instead of attractive matter,) then $-{\frac {dV}{dx}}\delta x$ would be the potential due to the two bodies.

Now consider any elementary portion of the body containing a volume δv. Its quantity is ρ δv, and corresponding to it there is