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

86.] Similarly the work required to increase the charge of $$A_2$$ is $$V_2e_2n dn$$, so that the whole work done in increasing the charge  of the system is

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 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,

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

Now let $$A_1$$ be discharged, and $$A_2$$ charged with unit of electricity, and let the potentials of $$A_1, A 2 ,$$ ... $$A_ n$$ be $$p_{21}$$,$$p_{22}$$,...$$p_{2n}$$ potentials due to $$E_2$$ on $$A_2$$ will be

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 $$A_s$$,