Page:On Faraday's Lines of Force.pdf/65

Rh The whole couple about the axis of $$x$$ is therefore

$\frac{3k'(k_3-k_2)}{(2k_3+k')(2k_2+k')}mnI^2a^3$,

tending to turn the sphere round from the axis of $$y$$ towards that of $$z$$. Suppose the sphere to be suspended so that the axis of $$x$$ is vertical, and let $$I$$ be horizontal, then if $$\theta$$ be the angle which the axis of $$y$$ makes with the direction of $$I$$, $$m=cos \theta$$, $$n= -sin \theta$$, and the expression for the moment becomes

$\frac{3}{2} \frac{k'(k_2-k_3)}{(2k_2+k')(2k_3+k')}I^2a^3 sin 2\theta$

tending to increase $$\theta$$. The axis of least resistance therefore sets axially, but with either end indifferently towards the north.

Since in all bodies, except iron, the values of $$k$$ are nearly the same as in a vacuum, the coefﬁcient of this quantity can be but little altered by changing the value of $$k'$$ to $$k$$, the value in space. The expression then becomes

$\frac{1}{6} \frac{k_2-k_3}{k}I^2a^3 sin 2\theta$

independent of the external medium.

VII. Permanent magnetism in a spherical shell.

The case of a homogeneous shell of a diamagnetic or paramagnetic substance presents no difﬁculty. The intensity Within the shell is less than what it would have been if the shell were away, whether the substance of the shell be diamagnetic or paramagnetic. When the resistance of the shell is inﬁnite, and when it vanishes, the intensity within the shell is zero.

In the case of no resistance the entire effect of the shell on any point, internal or external, may be represented by supposing a superﬁcial stratum of