Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/109

affected by the presence of other portions, the repulsion between $e'$ units of electricity at $A$ and $e'$ units at $B$ is $ee'$, the distance $AB$ being unity. See Art. 39.

Law of Force between Electrified Bodies.

66.] Coulomb shewed by experiment that the force between electrified bodies whose dimensions are small compared with the distance between them, varies inversely as the square of the distance. Hence the actual repulsion between two such bodies charged with quantities $e$ and $e'$ and placed at a distance $r$ is

${\frac {ee'}{r^{2}}}$

We shall prove in Art. 74 that this law is the only one consistent with the observed fact that a conductor, placed in the inside of a closed hollow conductor and in contact with it, is deprived of all electrical charge. Our conviction of the accuracy of the law of the inverse square of the distance may be considered to rest on experiments of this kind, rather than on the direct measurements of Coulomb.

Resultant Force between Two Bodies.

67.] In order to find the resultant force between two bodies we might divide each of them into its elements of volume, and consider the repulsion between the electricity in each of the elements of the first body and the electricity in each of the elements of the second body. We should thus get a system of forces equal in number to the product of the numbers of the elements into which we have divided each body, and we should have to combine the effects of these forces by the rules of Statics. Thus, to find the component in the direction of $x$ we should have to find the value of the sextuple integral

$\iiint \!\!\!\iiint {\frac {\rho \rho '(x-x')\;dx\;dy\;dz\;dx'\;dy'\;dz'\;}{\{(x-x')^{2}+(y-y')^{2}+(z-z')^{2}\}^{3 \over 2}}}$,

where $x,y,z$ are the coordinates of a point in the first body at which the electrical density is $\rho$, and $x',y',z'$, and $\rho '$ are the corresponding quantities for the second body, and the integration is extended first over the one body and then over the other.

Resultant Force at a Point.

68.] In order to simplify the mathematical process, it is convenient to consider the action of an electrified body, not on another