Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/350

 

exactly at the point thus defined. The position of the centre of the magnet, however, is always uncertain, and this uncertainty introduces a factor of correction of unknown amount depending on $G_{2}$ and of the form $\left(1-{\frac {6}{5}}{\frac {z}{A}}\right)$, where $z$ is the unknown excess of distance of the centre of the magnet from the plane of the coil. This correction depends on the first power of ${\frac {z}{A}}$. Hence Gaugain's coil with eccentrically suspended magnet is subject to far greater uncertainty than the old form.

Helmholtz's Arrangement.

713.] Helmholtz converted Gaugain's galvanometer into a trustworthy instrument by placing a second coil, equal to the first, at an equal distance on the other side of the magnet.

By placing the coils symmetrically on both sides of the magnet we get rid at once of all terms of even order.

Let $A$ be the mean radius of either coil, the distance between their mean planes is made equal to $A$, and the magnet is suspended at the middle point of their common axis. The coefficients are

$G_{1}={\frac {16\pi n}{5{\sqrt {5}}}}{\frac {1}{A}}\left(1-{\frac {1}{60}}{\frac {\xi ^{2}}{A^{2}}}\right)$,

$G_{2}=0$,

$G_{3}=0.0512{\frac {\pi n}{3{\sqrt {5}}A^{5}}}(31\xi ^{2}-36\eta ^{2})$,

$G_{4}=0$,

$G_{5}=-0.73728{\frac {\pi n}{{\sqrt {5}}A^{5}}}$,

where $n$ denotes the number of windings in both coils together.

It appears from these results that if the section of the coils be rectangular, the depth being $\xi$ and the breadth $\eta$, the value of $G_{3}$, as corrected for the finite size of the section, will be small, and will vanish, if $\xi$ is to $\eta$ as 36 to 31.

It is therefore quite unnecessary to attempt to wind the coils upon a conical surface, as has been done by some instrument makers, for the conditions may be satisfied by coils of rectangular section, which can be constructed with far greater accuracy than coils wound upon an obtuse cone.

The arrangement of the coils in Helmholtz's double galvanometer is represented in Fig. 54, Art. 725.