Page:On Faraday's Lines of Force.pdf/68

222 IX. Effect of the core of the electro-magnet.

Now let us suppose a sphere of diamagnetic or paramagnetic matter introduced into the electro-magnetic coil. The result may be obtained as in the last case, and the potentials become

$p'=I_2\frac{n}{6}\frac{3k'}{2k+k'}\frac{a^2}{r^2} cos \theta,\ p_1=-2I_2\frac{n}{6}\frac{3k}{2k+k'}\frac{r}{a} cos \theta$.

The external effect is greater or less than before, according as $$k'$$, is greater or less than $$k$$, that is, according as the interior of the sphere is magnetic or diamagnetic with respect to the external medium, and the internal effect is altered in the opposite direction, being greatest for a diamagnetic medium.

This investigation explains the effect of introducing an iron core into an electro-magnet. If the value of $$k$$ for the core were to vanish altogether, the effect of the electro-magnet would be three times that which it has without the core. As $$k$$ has always a ﬁnite value, the effect of the core is less than this.

In the interior of the electro-magnet we have a uniform ﬁeld of magnetic force, the intensity of which may be increased by surrounding the coil with a shell of iron. If $$k'=0$$, and the shell inﬁnitely thick, the effect on internal points would be tripled.

The effect of the core is greater in the case of a cylindric magnet, and greatest of all when the core is a ring of soft iron.

X. Electro-tonic functions in spherical electro-magnet.

Let us now ﬁnd the electro-tonic functions due to this electro-magnet.

They will be of the form

$\alpha_0=0,\ \ \beta_0=\omega z,\ \ \gamma_0=-\omega y$

where $$\omega$$ is some function of $$r$$. Where there are no electric currents, we must have $$a_2,\ b_2,\ c_2$$ each $$=0$$, and this implies

$\frac{d}{dr}\left(3\omega+r\frac{d\omega}{dr}=0\right)$

the solution of which is

$\omega=C_1+\frac{C_2}{r_3}$.