SYSTEMS OF CONDUCTORS.
On the Superposition of Electrical Systems.
84.] Let be a given electrified system of which the potential at a point is , and let be another electrified system of which the potential at the same point would be if did not exist. Then, if and exist together, the potential of the combined system will be .
Hence, if be the potential of an electrified system , if the electrification of every part of be increased in the ratio of to 1 , the potential of the new system will be .
Energy of an Electrified System.
85.] Let the system be divided into parts, , , &c. so small that the potential in each part may be considered constant through out its extent. Let , , &c. be the quantities of electricity in each of these parts, and let , &c. be their potentials.
If now is altered to , to , &c., then the potentials will become , , &c.
Let us consider the effect of changing into in all these expressions. It will be equivalent to charging with a quantity of electricity , with , &c. These charges must be supposed to be brought from a distance at which the electrical action of the system is insensible. The work done in bringing of electricity to , whose potential before the charge is , and after the charge , lf must lie between
In the limit we may neglect the square of , and write the expression