Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/395

 

To determine the motion of the magnet, we have to combine this equation with (7) and eliminate $\gamma$. The result is

$\left(R+L{\frac {d}{dt}}\right)\left(A{\frac {d^{2}}{dt^{2}}}+B{\frac {d}{dt}}+C\right)\phi +G^{2}m^{2}{\frac {d\phi }{dt}}=0$,

(10)

a linear differential equation of the third order.

We have no occasion, however, to solve this equation, because the data of the problem are the observed elements of the motion of the magnet, and from these we have to determine the value of $R$.

Let $\alpha _{0}$ and $\omega _{0}$ be the values of $\alpha$ and $\omega$ in equation (2) when the circuit is broken. In this case $R$ is infinite, and the equation is reduced to the form (8). We thus find

$B=2A\alpha _{0}$,⁠$C=A(\alpha _{0}^{2}+\omega _{0}^{2})$.

(11)

Solving equation (10) for $R$, and writing

${\frac {d}{dt}}=-(\alpha +i\omega )$,⁠where⁠$i={\sqrt {-1}}$,

(12)

we find

$R={\frac {G^{2}m^{2}}{A}}{\frac {\alpha +i\omega }{\alpha ^{2}-\omega ^{2}+2i\alpha \omega -2\alpha _{0}(\alpha +i\omega )+\alpha _{0}^{2}+\omega _{0}^{2}}}+L(\alpha +i\omega )$.

(13)

Since the value of $\omega$ is in general much greater than that of $\alpha$, the best value of $R$ is found by equating the terms in $i\omega$,

$R={\frac {G^{2}m^{2}}{2A(\alpha -\alpha _{0})}}+{\frac {1}{2}}L\left(3\alpha -\alpha _{0}-{\frac {\omega ^{2}-\omega _{0}^{2}}{\alpha -alpha_{0}}}\right)$.

(14)

We may also obtain a value of $R$ by equating the terms not involving $i$, but as these terms are small, the equation is useful only as a means of testing the accuracy of the observations. From these equations we find the following testing equation,

��-a> 2 ) 2 }. (15)

Since $LA\omega ^{2}$ is very small compared with $G^{2}m^{2}$, this equation gives

$\omega ^{2}-\omega _{0}^{2}=\alpha _{0}^{2}-\alpha ^{2}$;

(16)

and equation (14) may be written

$R={\frac {G^{2}m^{2}}{2A(\alpha -\alpha _{0})}}+2L\alpha$.

(17)

In this expression $G$ may be determined either from the linear measurement of the galvanometer coil, or better, by comparison with a standard coil, according to the method of Art. 753. $A$ is the moment of inertia of the magnet and its suspended apparatus, which is to be found by the proper dynamical method. $\omega$, $\omega _{0}$, $\alpha$ and $\alpha _{0}$, are given by observation.