Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/81

39.] It is sometimes more convenient to use a torsion-balance in which a horizontal arm is suspended by a fine wire or fibre, so as to be capable of vibrating about the vertical wire as an axis, and the body is attached to one end of the arm and acted on by the force in the tangential direction, so as to turn the arm round the vertical axis, and so twist the suspension wire through a certain angle. The torsional rigidity of the wire is found by observing the time of oscillation of the arm, the moment of inertia of the arm being otherwise known, and from the angle of torsion and the torsional rigidity the force of attraction or repulsion can be deduced. The torsion-balance was devised by Michell for the determination of the force of gravitation between small bodies, and was used by Cavendish for this purpose. Coulomb,working independently of these philosophers, reinvented it, and successfully applied it to discover the laws of electric and magnetic forces; and the torsion-balance has ever since been used in all researches where small forces have to be measured. See Art. 215.

39.] Let us suppose that by either of these methods we can measure the force between two electrified bodies. We shall suppose the dimensions of the bodies small compared with the distance between them, so that the result may not be much altered by any inequality of distribution of the electrification on either body, and we shall suppose that both bodies are so suspended in air as to be at a considerable distance from other bodies on which they might induce electrification.

It is then found that if the bodies are placed at a fixed distance and charged respectively with $$e$$ and $$e'$$ of our provisional units of electricity, they will repel each other with a force proportional to the product of $$e$$ and $$e'$$. If either $$e$$ or $$e'$$ is negative, that is, if one of the charges is vitreous and the other resinous, the force will be attractive, but if both $$e$$ and $$e'$$ are negative the force is again repulsive.

We may suppose the first body, $$A$$, charged with $$m$$ units of vitreous and $$n$$ units of resinous electricity, which may be conceived separately placed within the body, as in Experiment V.

Let the second body, $$B$$, be charged with $$m'$$ units of positive and $$n'$$ units of negative electricity.

Then each of the $$m$$ positive units in $$A$$ will repel each of the $$m'$$ positive units in $$B$$ with a certain force, say $$f$$, making a total effect equal to $$mm'f$$.

Since the effect of negative electricity is exactly equal and