Page:On Faraday's Lines of Force.pdf/64

and a the radius of the sphere. Let $I$ be the undisturbed magnetic intensity of the field into which the sphere is introduced, and let its direction-cosines be $l,m,n$.

Let us now take the case of a homogeneous sphere whose coefficient is $k$, placed in a uniform magnetic field Whose intensity is $lI$ in the direction of $x$. The resultant potential outside the sphere would be

$p'=lI\left(1+{\frac {k_{1}-k'}{2k_{1}+k'}}{\frac {a^{3}}{r^{3}}}\right)\ x$,

and for internal points

$p_{1}=lI{\frac {k_{1}-k'}{2k_{1}+k'}}\ x.$

So that in the interior of the sphere the magnetization is entirely in the direction of $x$. It is therefore quite independent of the coefficients of resistance in the directions of $x$ and $y$, which may be changed from $k_{1}$ into $k_{2}$ and $k_{3}$ without disturbing this distribution of magnetism. We may therefore treat the sphere as homogeneous for each of the three components of $I$, but we must use a different. coefficient for each. We find for external points

$p'=I\left\{lx+my+nz+\left({\frac {k_{1}-k'}{2k_{1}+k'}}lx+{\frac {k_{2}-k'}{2k_{2}+k'}}my+{\frac {k_{3}-k'}{2k_{3}+k'}}nz+\right){\frac {a^{3}}{r^{3}}}\right\}$,

and for internal points

$p_{1}=I\left({\frac {3k_{1}}{2k_{1}+k'}}lx+{\frac {3k_{2}}{2k_{2}+k'}}my{\frac {3k_{3}}{2k_{3}+k'}}nz\right)$.

The external effect is the same as that which would have been produced if the small magnet whose moments are

${\frac {k_{1}-k'}{2k_{1}+k'}}lIa^{3},\ {\frac {k_{2}-k'}{2k_{2}+k'}}mIa^{3},\ {\frac {k_{3}-k'}{2k_{3}+k'}}nIa^{3},\$

had been placed at the origin with their directions coinciding with the axes of $x,y,z$. The effect of the original force $I$ in turning the sphere about the axis of $x$ may be found by taking the moments of the components of that force on these equivalent magnets. The moment of the force in the direction of $y$ acting on the third magnet is

${\frac {k_{3}-k'}{2k_{3}+k'}}mnI^{2}a^{3},$

and that of the force in $z$ on the second magnet is

$-{\frac {k_{2}-k'}{2k_{2}+k'}}mnI^{2}a^{3}.$