Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/129

Similarly the work required to increase the charge of $A_{2}$ is $V_{2}e_{2}ndn$, so that the whole work done in increasing the charge of the system is

$(v_{1}e_{1}+V_{2}e_{2}+etc.)\,n\,dn\,\!$.

If we suppose this process repeated an indefinitely great number of times, each charge being indefinitely small, till the total effect becomes sensible, the work done will be

$\sum (Ve)\int n\,dn={1 \over 2}\sum (Ve)(n_{1}^{2}-n_{0}^{2})$;

where $\sum (Ve)$ means the sum of all the products of the potential of each element into the quantity of electricity in that element when $n=1$, and $n_{0}$ is the initial and $n_{1}$ the final value of $n$.

If we make $n_{0}=0$ and $n_{1}=1$, we find for the work required to charge an unelectrified system so that the electricity is $e$ and the potential $V$ in each element,

$Q={1 \over 2}\sum (Ve)$.

General Theory of a System of Conductors.

86.] Let $A_{1},A_{2},\ldots A_{N}$ be any number of conductors of any form. Let the charge or total quantity of electricity on each of these be $E_{1},E_{2},\ldots E_{N}$ and let their potentials be $V_{1},V_{2},\ldots V_{N}$ respectively.

Let us suppose the conductors to be all insulated and originally free of charge, and at potential zero.

Now let $A_{1}$ be charged with unit of electricity, the other bodies being without charge. The effect of this charge on $A_{1}$ will be to raise the potential of $A_{1}$ to $p_{11}$, that of $A_{2}$ to $p_{12}$, and that of $A-n$ to $p_{1n}$ , where $p_{11}$, &c. are quantities depending on the form and relative position of the conductors. The quantity $p_{11}$ may be called the Potential Coefficient of $A_{l}$ on itself, and $p_{12}$ may be called the Potential Coefficient of $A_{1}$ on $A_{2}$ , and so on.

If the charge upon $A_{1}$ is now made $E_{l}$ , then, by the principle of superposition, we shall have

$V_{1}=p_{11}E_{1}\ldots \ldots v_{n}=p_{1n}E_{1}\,\!$.

Now let $A_{1}$ be discharged, and $A_{2}$ charged with unit of electricity, and let the potentials of $A_{1},A2,$ ... $A_{n}$ be $p_{21}$,$p_{22}$,...$p_{2n}$ potentials due to $E_{2}$ on $A_{2}$ will be

$V_{1}=p_{21}E_{2}\ldots \ldots V_{n}=p_{2n}E_{2}$.

Similarly let us denote the potential of $A_{s}$ due to a unit charge on $A_{r}$ by $p_{rs}$ , and let us call $p_{rs}$ the Potential Coefficient of $A_{r}$ on $A_{s}$,