Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/306

established the fundamental laws of electricity were made by measuring the force between two small spheres charged with electricity, one of which was fixed while the other was held in equilibrium by two forces, the electrical action between the spheres, and the torsional elasticity of a glass fibre or metal wire. See Art. 38.

The balance of torsion consists of a horizontal arm of gum-lac, suspended by a fine wire or glass fibre, and carrying at one end a little sphere of elder pith, smoothly gilt. The suspension wire is fastened above to the vertical axis of an arm which can be moved round a horizontal graduated circle, so as to twist the upper end of the wire about its own axis any number of degrees.

The whole of this apparatus is enclosed in a case. Another little sphere is so mounted on an insulating stem that it can be charged and introduced into the case through a hole, and brought so that its centre coincides with a definite point in the horizontal circle described by the suspended sphere. The position of the suspended sphere is ascertained by means of a graduated circle engraved on the cylindrical glass case of the instrument.

Now suppose both spheres charged, and the suspended sphere in equilibrium in a known position such that the torsion-arm makes an angle $\theta$ with the radius through the centre of the fixed sphere. The distance of the centres is then $2a\sin {\tfrac {1}{2}}\theta$, where $a$ is the radius of the torsion-arm, and if $F$ is the force between the spheres the moment of this force about the axis of torsion is $Fa\cos {\tfrac {1}{2}}\theta$.

Let both spheres be completely discharged, and let the torsion-arm now be in equilibrium at an angle $\phi$ with the radius through the fixed sphere.

Then the angle through which the electrical force twisted the torsion-arm must have been $\theta -\phi$, and if $M$ is the moment of the torsional elasticity of the fibre, we shall have the equation

$Fa\cos {\frac {1}{2}}\theta =M(\theta -\phi )$

Hence, if we can ascertain $M$, we can determine $F$, the actual force between the spheres at the distance $2a\sin {\frac {1}{2}}\theta$.

To find $M$, the moment of torsion, let $I$ be the moment of inertia of the torsion-arm, and $T$ the time of a double vibration of the arm under the action of the torsional elasticity, then

$M={\frac {1}{4\pi ^{2}}}IT^{2}$.

In all electrometers it is of the greatest importance to know what force we are measuring. The force acting on the suspended