Page:On Faraday's Lines of Force.pdf/58

212 This is the value of the potential outside the sphere. At the surface we have

$R=a \ and \ \frac{b_1}{r_1}=\frac{a}{r'_1}, \ \frac{b_2}{r_2}=\frac{a}{r'_2}, \ \&c.$,

so that at the surface

$p=\frac{E}{a}+\frac{e_1}{b_1}+\frac{e_2}{b_2}+\&c.$,

and this must also be the value of p for any point within the sphere.

For the application of the principle of electrical images the reader is referred to Prof. Thomson’s papers in the Cambridge and Dublin Mathematical Journal.

The only case which we shall consider is that in which $$\frac{e_1}{b_1^2}=I$$ and $$b_1$$ is inﬁnitely distant along the axis of $$x$$, and $$E=0$$.

The value $$p$$ outside the sphere becomes then

$p=Ix\left(-\frac{a^3}{r^3}\right)$,

and inside $$p=o$$.

II. On the effect of a paramagnetic or diamagnetic sphere in a uniform ﬁeld of magnetic force.

The expression for the potential of a small magnet placed at the origin of co-ordinates in the direction of the axis of $$x$$ is

$l\frac{d}{dx}\left(\frac{m}{r}\right)=-lm\frac{x}{r^3}$

The effect of the sphere in disturbing the lines of force may be supposed

as a ﬁrst hypothesis to be similar to that of a small magnet at the origin, whose strength is to be determined. (We shall ﬁnd this to be accurately true.)