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

$nV_{1}e_{1}\,dn\,\!$ and $(n+dn)V_{1}e_{1}\,dn\,\!$.

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

$V_{1}e_{1}n\,dn\,\!$