# 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

 $V_1e_1n\,dn\,\!$