On the Superposition of Electrical Systems

84.] Let E1 be a given electrified system of which the potential at a point P is V1, and let E2 be another electrified system of which the potential at the same point would be V2 if E1 did not exist. Then, if E1 and E2 exist together, the potential of the combined system will be V1 + V2.

Hence, if V be the potential of an electrified system E, if the electrification of every part of E be increased in the ratio of n to 1, the potential of the new system nE will be nV.

Energy of an Electrified System

85.] Let the system be divided into parts, A1, A2, &c. so small that the potential in each part may be considered constant throughout its extent. Let e1, e2, &c. be the quantities of electricity in each of these parts, and let V1, V2, &c. be their potentials.

If now e1 is altered to ne1, e2 to ne2, &c. then the potentials will become nV1, nV2, &c.

Let us consider the effect of changing n into n + dn in all these expressions. It will be equivalent to charging A1 with a quantity of electricity e1dn, A2 with e2dn, &c. These charges must be supposed to be brought from a distance at which the electrical action of the system is insensible.