Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/360

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328

ELECTROMAGNETIC INSTRUMENTS.

[724.

724.] Sir William Thomson has constructed a single instrument by means of which the observations required to determine $H$ and $\gamma$ may be made simultaneously by the same observer.

The coil is suspended so as to be in equilibrium with its plane in the magnetic meridian, and is deflected from this position when the current flows through it. A very small magnet is suspended at the centre of the coil, and is deflected by the current in the direction opposite to that of the deflexion of the coil. Let the deflexion of the coil be $\theta$ , and that of the magnet $\phi$ , then the energy of the system is

H\gamma g\sin \theta +m\gamma G\sin(\theta -\phi )-Hm\cos \phi -F\cos \theta

.

Differentiating with respect to $\theta$ and $\phi$ , we obtain the equations of equilibrium of the coil and of the magnet respectively,

H\gamma g\cos \phi +m\gamma G\cos(\theta -\phi )+F\sin \theta =0

,

-m\gamma G\cos(\theta -\phi )+Hm\sin \phi =0

.

From these equations we find, by eliminating $H$ or $\gamma$ , a quadratic equation from which $\gamma$ or $H$ may be found. If $m$ , the magnetic moment of the suspended magnet, is very small, we obtain the following approximate values

H={\frac {\pi }{T}}{\sqrt {\frac {-AG\sin \theta \cos(\theta -\phi )}{g\cos \theta \sin \phi }}}-{\frac {1}{2}}{\frac {mG}{g}}{\frac {\cos(\theta -\phi )}{\cos \theta }}

,

\gamma ={\frac {\pi }{T}}{\sqrt {\frac {-A\sin \theta \sin \phi }{Gg\cos \theta \cos(\theta -\phi )}}}-{\frac {1}{2}}{\frac {m}{g}}{\frac {\sin \phi }{\cos \theta }}

.

In these expressions $G$ and $g$ are the principal electric constants of the coil, $A$ its moment of inertia, $T$ its time of vibration, $m$ the magnetic moment of the magnet, $H$ the intensity of the horizontal magnetic force, $\gamma$ the strength of the current, $\theta$ the deflexion of the coil, and $\phi$ that of the magnet.

Since the deflexion of the coil is in the opposite direction to the deflexion of the magnet, these values of $H$ and $\gamma$ will always be real.

Weber's Electrodynamometer.

725.] In this instrument a small coil is suspended by two wires within a larger coil which is fixed. When a current is made to flow through both coils, the suspended coil tends to place itself parallel to the fixed coil. This tendency is counteracted by the moment of the forces arising from the bifilar suspension, and it is also affected by the action of terrestrial magnetism on the suspended coil.