Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/150

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magnetic measurements.

[464.

force. The most convenient instrument for continuous observations is the vertical force magneto meter, which is simply a magnet balanced on knife edges so as to be in stable equilibrium with its magnetic axis nearly horizontal.

If $Z$ is the vertical component of the magnetic force, $M$ the magnetic moment, and $\theta$ the small angle which the magnetic axis makes with the horizon

MZ=mga\cos(\alpha -\theta )

,

where $m$ is the mass of the magnet, $g$ the force of gravity, $a$ the distance of the centre of gravity from the axis of suspension, and $\alpha$ the angle which the plane through the axis and the centre of gravity makes with the magnetic axis.

Hence, for the small variation of vertical force $\delta Z$ , there will be a variation of the angular position of the magnet $\delta \theta$ such that

M\delta Z=mga\sin(\alpha -\theta )\delta \theta

.

In practice this instrument is not used to determine the absolute value of the vertical force, but only to register its small variations.

For this purpose it is sufficient to know the absolute value of $Z$ when $\theta =0$ , and the value of ${\frac {dZ}{d\theta }}$ .

The value of $Z$ , when the horizontal force and the dip are known, is found from the equation $Z=H\tan \theta _{0}$ , where $\theta _{0}$ is the dip and $H$ the horizontal force.

To find the deflexion due to a given variation of $Z$ , take a magnet and place it with its axis east and west, and with its centre at a known distance $r_{1}$ east or west from the declinometer, as in experiments on deflexion, and let the tangent of deflexion be $D_{1}$ .

Then place it with its axis vertical and with its centre at a distance $r_{2}$ above or below the centre of the vertical force magnetometer, and let the tangent of the deflexion produced in the magnetometer be $D_{2}$ . Then, if the moment of the deflecting magnet is $M$ ,

M=Hr_{1}^{3}D_{1}={\frac {dZ}{d\theta }}r_{2}^{3}D_{2}

.

Hence

{\frac {dZ}{d\theta }}=H{\frac {r_{1}^{3}}{r_{2}^{3}}}{\frac {D_{1}}{D_{2}}}

.

The actual value of the vertical force at any instant is

Z=Z_{0}+\theta {\frac {dZ}{d\theta }}

,

where $Z_{0}$ is the value of $Z$ when $\theta =0$ .

For continuous observations of the variations of magnetic force