Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/380

 

The equation of motion is

		$\theta$	${}=Ce^{-\omega _{1}\tan \beta }\sin \omega _{1}t$,			(8)
		${\frac {d\theta }{dt}}$	${}=C\omega _{1}\sec \beta e^{-\omega _{1}t\tan \beta }\cos(\omega _{1}t+\beta )$.			(9)

When $t=0$, $\theta =0$, and ${\frac {d\theta }{dt}}=C\omega _{1}=\nu$.

When $\omega _{1}t+\beta ={\frac {\pi }{2}}$,

			$\theta$	${}=Ce^{-\left({\frac {\pi }{2}}-\beta \right)\tan \beta }\cos \beta =\theta _{1}$.			(10)
Hence			$\theta _{1}$	${}={\frac {\nu }{\omega _{1}}}e^{-\left({\frac {\pi }{2}}-\beta \right)\tan \beta }\cos \beta$.			(11)

Now			${\frac {MH}{A}}$	${}=\omega ^{2}=\omega _{1}^{2}\sec ^{2}\beta$,			(12)
			$\tan \beta$	${}={\frac {\lambda }{\pi }}$, $\omega _{1}={\frac {\pi }{T_{1}}}$,			(13)

Hence			$\theta _{1}$	$={\frac {QG}{H}}{\frac {\sqrt {\pi ^{2}+\lambda ^{2}}}{T_{1}}}e^{-{\frac {\lambda }{\pi }}\tan ^{-1}{\frac {\pi }{\lambda }}}$,			(15)
and			$Q$	${}={\frac {H}{G}}{\frac {T_{1}}{\sqrt {\pi ^{2}+\lambda ^{2}}}}e^{{\frac {\lambda }{\pi }}\tan ^{-1}{\frac {\pi }{\lambda }}}\theta _{1}$,			(16)

which gives the first elongation in terms of the quantity of electricity in the transient current, and conversely, where $T_{1}$ is the observed time of a single vibration as affected by the actual resistance of damping. When $\lambda$ is small we may use the approximate formula

Method of Recoil.

750.] The method given above supposes the magnet to be at rest in its position of equilibrium when the transient current is passed through the coil. If we wish to repeat the experiment we must wait till the magnet is again at rest. In certain cases, however, in which we are able to produce transient currents of equal intensity, and to do so at any desired instant, the following method, described by Weber^[1], is the most convenient for making a continued series of observations.

1. Resultate des Magnetischen Vereins, 1838, p. 98.