Page:On Faraday's Lines of Force.pdf/66

220 magnetic matter spread over the outer surface, the density being given by the equation

$\rho=3I cos \theta$

Suppose the shall now to be converted into a permanent magnet, so that the distribution of imaginary magnetic matter is invariable, then the external potential due to the shell will be

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

and the internal potential $$p_1=-Ir cos\theta$$

Now let us investigate the effect of ﬁlling up the shell with some substance of which the resistance is $$k$$, the resistance in the external medium being $$k'$$. The thickness of the magnetized shell may be neglected. Let the magnetic moment of the permanent magnetism be $$Ia^3$$, and that of the imaginary superﬁcial distribution due to the medium $$k=Aa^3$$. Then the potentials are

external $p'=(I+A)\frac{a^3}{r^2}cos \theta$, internal $p_1=(I+A) r cos \theta$.

The distribution of real magnetism is the same before and after the introduction of the medium $$k$$, so that

$\frac{1}{k'}I+\frac{2}{k'}I=\frac{1}{k'}(I+A )+\frac{2}{k'}(I+A)$

or $ A=\frac{k-k'}{2k+k'}I$.

The external effect of the magnetized shell is increased or diminished according as $$k$$ is greater or less than $$k'$$ It is therefore increased by ﬁlling up the shell with diamagnetic matter, and diminished by ﬁlling it with paramagnetic matter, such as iron.

VIII. Electro-magnetic spherical shell.

Let us take as an example of the magnetic effects of electric currents, an electro—magnet in the form of a thin spherical shell. Let its radius be $$a$$, and its thickness $$t$$, and let its external effect be that of a magnet whose moment is $$Ia^3$$. Both within and without the shell the magnetic effect may be represented by a potential, but within the substance of the shell, where there