Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/348

 

The values of the coefficients for a magnet of length $2l$ magnetized in any other way are smaller than when it is magnetized uniformly.

710.] When the apparatus is used as a tangent galvanometer, the coil is fixed with its plane vertical and parallel to the direction of the earth's magnetic force. The equation of equilibrium of the magnet is in this case

$mg_{1}H\cos \theta =m\gamma \sin \theta \{G_{1}g_{1}+G_{2}g_{2}Q_{1}^{\prime }(\theta )+\mathrm {\&c.} \}$,

where $mg_{1}$ is the magnetic moment of the magnet, $H$ the horizontal component of the terrestrial magnetic force, and $\gamma$ the strength of the current in the coil. When the length of the magnet is small compared with the radius of the coil the terms after the first in $G$ and $\gamma$ may be neglected, and we find

$\gamma ={\frac {H}{G_{1}}}\cot \theta$.

The angle usually measured is the deflexion, $\delta$, of the magnet which is the complement of $\theta$, so that $\cot \theta =\tan \delta$.

The current is thus proportional to the tangent of the deviation, and the instrument is therefore called a Tangent Galvanometer.

Another method is to make the whole apparatus moveable about a vertical axis, and to turn it till the magnet is in equilibrium with its axis parallel to the plane of the coil. If the angle between the plane of the coil and the magnetic meridian is $\delta$, the equation of equilibrium is

$mg_{1}H\sin \delta =M\gamma {\frac {1}{g}}\{G_{1}g_{1}-{\frac {3}{2}}G_{3}g_{3}+\mathrm {\&c.} \}$,

Since the current is measured by the sine of the deviation, the instrument when used in this way is called a Sine Galvanometer.

The method of sines can be applied only when the current is so steady that we can regard it as constant during the time of adjusting the instrument and bringing the magnet to equilibrium.

711.] We have next to consider the arrangement of the coils of a standard galvanometer.

The simplest form is that in which there is a single coil, and the magnet is suspended at its centre.

Let $A$ be the mean radius of the coil, $\xi$ its depth, $\eta$ its breadth, and $n$ the number of windings, the values of the coefficients are