Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/138

The lower magnet exactly neutralizes the effect of terrestrial magnetism on the upper one, and since the magnets are of equal moment, the upper one neutralizes the inductive action of the earth on the lower one.

The value of $M$ is therefore the same in the experiment of vibration as in the experiment of deflexion, and no correction for induction is required.

458.] The most accurate method of ascertaining the intensity of the horizontal magnetic force is that which we have just described. The whole series of experiments, however, cannot be performed with sufficient accuracy in much less than an hour, so that any changes in the intensity which take place in periods of a few minutes would escape observation. Hence a different method is required for observing the intensity of the magnetic force at any instant.

The statical method consists in deflecting the magnet by means of a statical couple acting in a horizontal plane. If $L$ be the moment of this couple, $M$ the magnetic moment of the magnet, $H$ the horizontal component of terrestrial magnetism, and $\theta$ the deflexion,

$MH\sin \theta =L$.

Hence, if $L$ is known in terms of $\theta$, $MH$ can be found.

The couple $L$ may be generated in two ways, by the torsional elasticity of a wire, as in the ordinary torsion balance, or by the weight of the suspended apparatus, as in the bifilar suspension.

In the torsion balance the magnet is fastened to the end of a vertical wire, the upper end of which can be turned round, arid its rotation measured by means of a torsion circle.

We have then

$L=r(a-a_{0})=MH\sin \theta$.

Here $a_{0}$ is the value of the reading of the torsion circle when the axis of the magnet coincides with the magnetic meridian, and a is the actual reading. If the torsion circle is turned so as to bring the magnet nearly perpendicular to the magnetic meridian, so that

$\theta ={\frac {\pi }{2}}-\theta '$, then $\tau (a-a_{0}-{\frac {\pi }{2}}+\theta ')=MH(1-{\tfrac {1}{2}}\theta '^{2})$, or $MH=\tau (1+{\tfrac {1}{2}}\theta '^{2})(a-a_{0}-{\frac {pi}{2}}+\theta ')$.

By observing $\theta '$, the deflexion of the magnet when in equilibrium, we can calculate $MH$ provided we know $\tau$.

If we only wish to know the relative value of $H$ at different times it is not necessary to know either $M$ or $\tau$.

We may easily determine $\tau$ in absolute measure by suspending