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

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

$\rho =3Icos\theta$

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 $p_{1}=-Ircos\theta$

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 $p_{1}=(I+A)rcos\theta$.

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