Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/397

 

The kinetic energy of the system is

$T={\frac {1}{2}}L\gamma ^{2}-Hg\gamma \sin \theta -MG\gamma \sin(\theta -\phi )+MH\cos \phi +{\frac {1}{2}}A{\dot {\phi }}^{2}$.

(1)

The first term, ${\frac {1}{2}}L\gamma ^{2}$, expresses the energy of the current as depending on the coil itself. The second term depends on the mutual action of the current and terrestrial magnetism, the third on that of the current and the magnetism of the suspended magnet, the fourth on that of the magnetism of the suspended magnet and terrestrial magnetism, and the last expresses the kinetic energy of the matter composing the magnet and the suspended apparatus which moves with it.

The potential energy of the suspended apparatus arising from the torsion of the fibre is

$V={\frac {MH}{2}}\tau (\phi ^{2}-2\phi a)$.

(2)

The electromagnetic momentum of the current is

$p={\frac {dT}{d\gamma }}=L\gamma -Hg\sin \theta -MG\gamma \sin(\theta -\phi )$,

(3)

and if $R$ is the resistance of the coil, the equation of the current is

$R\gamma +{\frac {d^{2}T}{d\gamma \,dt}}=0$,

(4)

or, since

$\theta =\omega t$

(5)

$\left(R+L{\frac {d}{dt}}\right)\gamma =Hg\omega \cos \theta +MG(\omega -{\dot {\phi }})\cos(\theta -\phi )$.

(6)

765.] It is the result alike of theory and observation that $\phi$, the azimuth of the magnet, is subject to two kinds of periodic variations. One of these is a free oscillation, whose periodic time depends on the intensity of terrestrial magnetism, and is, in the experiment, several seconds. The other is a forced vibration whose period is half that of the revolving coil, and whose amplitude is, as we shall see, insensible. Hence, in determining $\gamma$, we may treat $\phi$; as sensibly constant.

We thus find

��y = (R cos B + Zo> sin 0) (7)

+)im(*--4)) J (8)

��+ Ce . (9)

The last term of this expression soon dies away when the rotation is continued uniform.