Page:Philosophical magazine 21 series 4.djvu/194

Hence the lines of force in a part of space where $\mu$ is uniform, and where there are no electric currents, must be such as would result from the theory of "imaginary matter" acting at a distance. The assumptions of that theory are unlike those of ours, but the results are identical.

Let us first take the case of a single magnetic pole, that is, one end of a long magnet, so long that its other end is too far off to have a perceptible influence on the part of the field we are considering. The conditions then are, that equation (18) must be fulfilled at the magnetic pole, and (19) everywhere else. The only solution under these conditions is

$\phi =-{\frac {m}{\mu }}{\frac {1}{r}},$

(20)

where $r$ is the distance from the pole, and $m$ the strength of the pole.

The repulsion at any point on a unit pole of the same kind is

${\frac {d\phi }{dr}}={\frac {m}{\mu }}{\frac {1}{r^{2}}}.$

(21)

In the standard medium $\mu =1$; so that the repulsion is simply ${\tfrac {m}{r^{2}}}$ in that medium, as has been shown by Coulomb.

In a medium having a greater value of $\mu$ (such as oxygen, solutions of salts of iron, &c.) the attraction, on our theory, ought to be less than in air, and in diamagnetic media (such as water, melted bismuth, &c.) the attraction between the same magnetic poles ought to be greater than in air.

The experiments necessary to demonstrate the difference of attraction of two magnets according to the magnetic or diamagnetic character of the medium in which they are placed, would require great precision, on account of the limited range of magnetic capacity in the fluid media known to us, and the small amount of the difference sought for as compared with the whole attraction.

Let us next take the case of an electric current whose quantity is C, flowing through a cylindrical conductor whose radius is R, and whose length is infinite as compared with the size of the field of force considered.

Let the axis of the cylinder be that of $z$, and the direction of the current positive, then within the conductor the quantity of current per unit of area is

$r={\frac {C}{\pi R^{2}}}={\frac {1}{4\pi }}\left({\frac {d\beta }{dx}}-{\frac {d\alpha }{dy}}\right);$

(22)