Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/129

To find the direction of the magnetic axis within the magnet subtract (10) from (9) and add (15),

$l=\lambda +{\tfrac {1}{2}}(\zeta -\zeta ')-{\tfrac {1}{2}}(\nu -\nu ')+{\tfrac {1}{2}}\tau '(\zeta -\zeta '+2\lambda -\pi ).$

(17)

By repeating the experiments with the bar on its two edges, so that the axis of x is vertically upwards and downwards, we can find the value of m. If the axis of collimation is capable of adjustment it ought to be made to coincide with the magnetic axis as nearly as possible, so that the error arising from the magnet not being exactly inverted may be as small as possible ^[1].

On the Measurement of Magnetic Forces.

453.] The most important measurements of magnetic force are those which determine M, the magnetic moment of a magnet, and H, the intensity of the horizontal component of terrestrial magnetism. This is generally done by combining the results of two experiments, one of which determines the ratio and the other the product of these two quantities.

The intensity of the magnetic force due to an infinitely small magnet whose magnetic moment is M, at a point distant r from the centre of the magnet in the positive direction of the axis of the magnet, is

$R=2{\frac {M}{r^{3}}}$

(1)

and is in the direction of r. If the magnet is of finite size but spherical, and magnetized uniformly in the direction of its axis, this value of the force will still be exact. If the magnet is a solenoidal bar magnet of length 2L,

$R=2{\frac {M}{r^{3}}}\left(1+2{\frac {L^{2}}{r^{2}}}+3{\frac {L^{4}}{r^{4}}}+etc.\right).$

(2)

If the magnet be of any kind, provided its dimensions are all small compared with r,

$R=2{\frac {M}{r^{3}}}\left(1+A_{1}{\frac {1}{r}}+A_{2}{\frac {1}{r^{2}}}\right)+etc.,$

(3)

where A₁,A₂,&c. are coefficients depending on the distribution of the magnetization of the bar.

Let H be the intensity of the horizontal part of terrestrial magnetism at any place. H is directed towards magnetic north. Let r be measured towards magnetic west, then the magnetic force at the extremity of r will be H towards the north and R towards

1. See a Paper on Imperfect Inversion. by W. Swan. Trans. R. S. Edin., vol. xxi (1855), p. 349.