Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/128

  CHAPTER III.

SYSTEMS OF CONDUCTORS.

On the Superposition of Electrical Systems.

84.] Let $$E_l$$ be a given electrified system of which the potential at a point $$P$$ is $$V_1$$, and let $$E_2$$ be another electrified system of which  the potential at the same point would be $$V_2$$ if $$E_l$$ did not exist. Then, if $$E_1$$ and $$E_2$$ exist together, the potential of the combined system will be $$V_1+V_2$$.

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, $$A_1$$, $$A_2$$, &c. so small that the potential in each part may be considered constant through  out its extent. Let $$e_l$$ ,$$e_2$$, &c. be the quantities of electricity in each of these parts, and let $$V_1$$, $$V_2$$  &c. be their potentials.

If now $$e_1$$ is altered to $$ne_1$$, $$e_2$$ to $$ne_2$$, &c., then the potentials will become $$nV_1$$, $$nV_2$$, &c.

Let us consider the effect of changing $$n$$ into $$n + dn$$ in all these expressions. It will be equivalent to charging $$A_1$$ with a quantity of electricity $$e_l dn$$, $$A_2$$ with $$e_2 dn$$, &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 $$e_1 dn$$ of electricity to $$A_1$$, whose potential before the charge is $$nV_1$$,  and after  the charge $$(n + dn)V_1$$, lf must lie between

$$nV_1e_1\,dn\,\!$$ and $$(n+dn)V_1e_1\,dn\,\!$$.

In the limit we may neglect the square of $$dn$$, and write the expression