Page:On Faraday's Lines of Force.pdf/60

214 Suppose this uniform ﬁeld to be that due to terrestrial magnetism, then, if $$k$$ is less than $$k'$$ as in paramagnetic bodies, the marked end of the equivalent magnet will be turned to the north. If k is greater than k’ as in diamagnetic bodies, the unmarked end of the equivalent magnet would be turned to the north.

III. Magnetic ﬁeld of variable Intensity.

Now suppose the intensity in the undisturbed magnetic ﬁeld to vary in magnitude and direction from one point to another, and that its components in $$x, y, z$$ are represented by $$\alpha, \beta, \gamma$$ then, if as a ﬁrst approximation we regard the intensity within the sphere as sensibly equal to that at the centre, the change of potential outside the sphere arising from the presence of the sphere, disturbing the lines of force, Will be the same as that due to three small magnets at the centre, with their axes parallel to $$x, y, and z,$$ and their moments equal to

$\frac{k-k'}{2k+k'}a^3\alpha, \ \frac{k-k'}{2k+k'}a^3\beta, \ \frac{k-k'}{2k+k'}a^3\gamma. $

The actual distribution of potential within and without the sphere may be conceived as the result of a distribution of imaginary magnetic matter on the surface of the sphere; but since the external effect of this superﬁcial magnetism is exactly the same as that of the three small magnets at the centre, the mechanical effect of external attractions will be the same as if the three magnets really existed.

Now let three small magnets whose lengths are $$l_1, l_2, l_3,$$ and strengths $$m_1, m_2, m_3,$$ exist at the point $$x, y, z$$ with their axes parallel to the axes of $$x, y, z$$; then resolving the forces on the three magnets in the direction of $$X$$, we have

$-X=m_1\begin{Bmatrix}\ a+\frac{da}{dx}\frac{l_1}{2} \\ -a+\frac{da}{dx}\frac{l_1}{2} \end{Bmatrix} + m_2\begin{Bmatrix}\ a+\frac{da}{dy}\frac{l_2}{2} \\ -a+\frac{da}{dy}\frac{l_2}{2}  \end{Bmatrix} + m_3\begin{Bmatrix}\ a+\frac{da}{dz}\frac{l_3}{2} \\ -a+\frac{da}{dy}\frac{l_3}{2}  \end{Bmatrix}$

$=m_1 l_1\frac{da}{dx} + m_2 l_2\frac{da}{dy} + m_3 l_3\frac{da}{dz}$.