Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/82

opposite to that of positive electricity, each of the $m$ positive units in $A$ will attract each of the $n'$ negative units in $B$ with the same force $f$, making a total effect equal to $mn'f$.

Similarly the $n$ negative units in $A$ will attract the $m'$ positive units in $B$ with a force $nm'f$, and will repel the $n'$ negative units in $B$ with a force $nn'f.$

The total repulsion will therefore be $(mm'+nn')f$; and the total attraction will be $(mn'+m'n)f.$

The resultant repulsion will be

$(mm'+nn'-mn'-nm')for(m-n)(m'-n')f\,\!$.

Now $m-n=e$ is the algebraical value of the charge on $A$, and $m'-n'=e'$ is that of the charge on $B$, so that the resultant repulsion may be written $ee'f$, the quantities $e$ and $e'$ being always understood to be taken with their proper signs.

Variation of the Force with the Distance.

40.] Having established the law of force at a fixed distance, we may measure the force between bodies charged in a constant manner and placed at different distances. It is found by direct measurement that the force, whether of attraction or repulsion, varies inversely as the square of the distance, so that if $f$ is the repulsion between two units at unit distance, the repulsion at distance $r$ will be $fr^{-2}.$and the general expression for the repulsion between $e$ units and $e'$ units at distance $r$ will be

$fee'r^{-2}\,\!$.

Definition of the Electrostatic Unit of Electricity.

41.] We have hitherto used a wholly arbitrary standard for our unit of electricity, namely, the electrification of a certain piece of glass as it happened to be electrified at the commencement of our experiments. We are now able to select a unit on a definite principle, and in order that this unit may belong to a general system we define it so that $f$ may be unity, or in other words—

The electrostatic unit of electricity is that quantity of electricity which, when placed at unit of distance from an equal quantity, repels it with unit of force.

This unit is called the Electrostatic unit to distinguish it from the Electromagnetic unit, to be afterwards defined.

We may now write the general law of electrical action in the simple form

$F=ee'r^{-2};{\color {White}xxxx}$ or,