Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/349

 

$G_{1}={\frac {2\pi n}{A}}\left\{1+{\frac {1}{12}}{\frac {\xi ^{2}}{A^{2}}}={\frac {1}{}}{\frac {\eta ^{2}}{A^{2}}}\right\}$,

$G_{2}=0$,

$G_{3}=-{\frac {\pi n}{A^{3}}}\left\{1+{\frac {1}{2}}{\frac {\xi ^{2}}{A^{2}}}-{\frac {5}{8}}{\frac {\eta ^{2}}{A^{2}}}\right\}$,

$G_{4}=0$, &c.

The principal correction is that arising from $G_{3}$. The series

The factor of correction will differ most from unity when the magnet is uniformly magnetized and when $\theta =0$. In this case it becomes $1-{\frac {1}{2}}{\frac {l^{2}}{A^{2}}}$. It vanishes when $\tan \theta =2$, or when the deflexion is $\tan ^{-1}{\frac {1}{2}}$, or 26°34′. Some observers, therefore, arrange their experiments so as to make the observed deflexion as near this angle as possible. The best method, however, is to use a magnet so short compared with the radius of the coil that the correction may be altogether neglected.

The suspended magnet is carefully adjusted so that its centre shall coincide as nearly as possible with the centre of the coil. If, however, this adjustment is not perfect, and if the coordinates of the centre of the magnet relative to the centre of the coil are $x$, $y$, $z$, $z$ being measured parallel to the axis of the coil, the factor of correction is

When the radius of the coil is large, and the adjustment of the magnet carefully made, we may assume that this correction is insensible.

Gaugain's Arrangement

712.] In order to get rid of the correction depending on $G_{3}$ Gaugain constructed a galvanometer in which this term was rendered zero by suspending the magnet, not at the centre of the coil, but at a point on the axis at a distance from the centre equal to half the radius of the coil. The form of $G_{3}$ is

and, since in this arrangement $B={\frac {1}{2}}A$, $G_{3}=0$.

This arrangement would be an improvement on the first form if we could be sure that the centre of the suspended magnet is