Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/382

 

− and +, then sending a positive current, and so on, we obtain a series consisting of sets of four elongations, in each of which

${\frac {d-b}{a-c}}=e^{-\lambda }$,

(29)

and

$KQ={\frac {(a-b)e^{-2\lambda }+d-c}{1+e^{-\lambda }}}$;

(30)

If $n$ series of elongations have been observed, then we find the logarithmic decrement from the equation

${\frac {\Sigma (d)-\Sigma (b)}{\Sigma (a)-\Sigma (c)}}=e^{-\lambda }$,

(31)

and $Q$ from the equation

$KQ(1+e^{-\lambda })(2n-1)$ ${}=\Sigma _{n}(a-b-c+d)(1+e^{-2\lambda })-(a_{1}-b_{1})-(d_{n}-c_{n})e^{-2\lambda }$.

(32)

Fig. 60.

The motion of the magnet in the method of recoil is graphically represented in Fig. 60, where the abscissa represents the time, and the ordinate the deflexion of the magnet at that time. See Art. 760.

Method of Multiplication.

751.] If we make the transient current pass every time that the magnet passes through the zero point, and always so as to increase the velocity of the magnet, then, if $\theta _{1}$, $\theta _{2}$, &c. are the successive elongations,

		$\theta _{2}=-KQ-e^{-\lambda }\theta _{1}$,		(33)
		$\theta _{3}=-KQ-e^{-\lambda }\theta _{2}$.		(34)

The ultimate value to which the elongation tends after a great many vibrations is found by putting $\theta _{n}=-\theta _{n-1}$, whence we find

$\theta =\pm {\frac {1}{1-e^{-\lambda }}}KQ$.

(35)

If $\lambda$ is small, the value of the ultimate elongation may be large, but since this involves a long continued experiment, and a careful determination of $\lambda$, and since a small error in $\lambda$ introduces a large error in the determination of $Q$, this method is rarely useful for