Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/378

 

point of equilibrium is then $-\phi$, and the magnet will swing to a negative elongation $\theta$, such that

		$-\rho (\phi +\theta _{1})=(\theta _{0}+\phi )$,
or		$-\rho \theta _{1}=\theta _{0}+(\rho +1)\phi$.\

Similarly, if the current is now made positive while the magnet swings to $\theta _{2}$,

		$\rho \theta _{2}$	${}=-\theta _{1}+(\rho +1)\phi$,
or		$\rho ^{2}\theta _{2}$	${}=\theta _{0}+(\rho +1)^{2}\phi$;

and if the current is reversed $n$ times in succession, we find

whence we may find $\phi$ in the form

If $n$ is a number so great that $\rho ^{-n}$ may be neglected, the expression becomes

The application of this method to exact measurement requires an accurate knowledge of $\rho$, the ratio of one vibration of the magnet to the next under the influence of the resistances which it experiences. The uncertainties arising from the difficulty of avoiding irregularities in the value of $\rho$ generally outweigh the advantages of the large angular elongation. It is only where we wish to establish the existence of a very small current by causing it to produce a visible movement of the needle that this method is really valuable.

On the Measurement of Transient Currents.

748.] When a current lasts only during a very small fraction of the time of vibration of the galvanometer-magnet, the whole quantity of electricity transmitted by the current may be measured by the angular velocity communicated to the magnet during the passage of the current, and this may be determined from the elongation of the first vibration of the magnet.

If we neglect the resistance which damps the vibrations of the magnet, the investigation becomes very simple.

Let $\gamma$ be the intensity of the current at any instant, and $Q$ the quantity of electricity which it transmits, then