Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/413

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779.]

condenser combined with coil.

381

Finding the values of A and B by substitution in the equation (3), we obtain

x=MCn{\frac {RCn\cos \theta -(1-CLn^{2})\sin \theta }{R^{2}C^{2}n^{2}+(1-CLn^{2})^{2}}}

.

(6)

The moment of the force with which the magnet acts on the coil $L'$ , in which the current ${\dot {x}}$ is flowing, is

\Theta ={\dot {x}}{\frac {d}{d\theta }}(M\cos \theta )=M\sin \theta {\frac {dx}{dt}}

.

(7)

Integrating this expression with respect to $t$ , and dividing by $t$ , we find, for the mean value of $\Theta$ ,

{\overline {\Theta }}={\tfrac {1}{2}}{\frac {M^{2}RC^{2}n^{3}}{R^{2}C^{2}n^{2}+(1-CLn^{2})^{2}}}

.

(8)

If the coil has a considerable moment of inertia, its forced vibrations will be very small, and its mean deflexion will be proportional to ${\overline {\Theta }}$ .

Let $D_{1},D_{2},D_{3}$ be the observed deflexions corresponding to angular velocities $n_{1},n_{2},n_{3}$ of the magnet, then in general

P{\frac {n}{D}}=\left({\frac {1}{n}}-CLn\right)^{2}+R^{2}C^{2}

,

(9)

where $P$ is a constant.

Eliminating $P$ and $R$ from three equations of this form, we find

C^{2}L^{2}={\frac {1}{n_{1}^{2}n_{2}^{2}n_{3}^{2}}}{\frac {{\frac {n_{1}^{3}}{D_{1}}}(n_{2}^{2}-n_{3}^{2})+{\frac {n_{2}^{2}}{D_{2}}}(n_{3}^{2}-n_{1}^{2})+{\frac {n_{3}^{2}}{D_{3}}}(n_{1}^{2}-n_{2}^{2})}{{\frac {n_{1}}{D_{1}}}(n_{2}^{2}-n_{3}^{2})+{\frac {n_{2}}{D_{2}}}(n_{3}^{2}-n_{1}^{2})+{\frac {n_{3}}{D_{3}}}(n_{1}^{2}-n_{2}^{2})}}

.

(10)

If $n_{2}$ is such that $CLn_{2}^{2}=1$ , the value of ${\frac {n}{D}}$ will be a minimum for this value of n. The other values of n should be taken, one greater, and the other less, than $n_{2}$ .

The value of $CL$ , determined from this equation, is of the dimensions of the square of a time. Let us call it $\tau ^{2}$ .

If $C_{s}$ be the electrostatic measure of the capacity of the condenser, and $L_{m}$ the electromagnetic measure of the self-induction of the coil, both $C_{s}$ and $L_{m}$ are lines, and the product

C_{s}L_{m}=v^{2}C_{s}L_{s}=v^{2}C_{m}L_{m}=v^{2}\tau ^{2}

(11)

and

v^{2}={\frac {C_{s}L_{m}}{\tau ^{2}}}

(1s)

where $\tau ^{2}$ is the value of $C^{2}L^{2}$ , determined by this experiment. The experiment here suggested as a method of determining $v$ is of the same nature as one described by Sir W. R. Grove, Phil. Mag.,

Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/413

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