Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/136

96 potentials are maintained constant, they tend to move so that the energy of the system is increased, and the work done by the  electrical forces during the displacement is equal to the increment  of the energy of the system. The energy spent by the batteries is equal to double of either of these quantities, and is spent half  in mechanical, and half in electrical work.

On the Comparison of Similar Electrified Systems.

94.] If two electrified systems are similar in a geometrical sense., so that the lengths of corresponding lines in the two systems  are as $$L$$ to $$L'$$, then if the dielectric which separates the conducting  bodies is the same in both systems, the coefficients of induction  and of capacity will be in the proportion of $$L$$ to $$L'$$. For if we consider corresponding portions, $$A$$ and $$A'$$, of the two systems, and  suppose the quantity of electricity on $$A$$ to be $$E$$, and that on $$A'$$ to be $$E'$$, then the potentials $$V$$ and $$V'$$ at corresponding points  $$B$$ and $$B'$$, due to this electrification, will be But $$AB$$ is to $$A'B'$$ as $$L$$ to $$L'$$, so that we must have But if the inductive capacity of the dielectric is different in the two systems, being $$K$$ in the first and $$K'$$ in the second, then if the  potential at any point of the first system is to that at the corresponding point of the second as $$V$$ to $$V'$$ and if the quantities  of electricity on corresponding parts are as $$E$$ to $$E'$$,  we shall have

By this proportion we may find the relation between the total electrification of corresponding parts of two systems, which are  in the first place geometrically similar, in the second place composed of dielectric media of which the dielectric inductive capacity  at corresponding points is in the proportion of $$K$$ to $$K'$$ and in  the third place so electrified that the potentials of corresponding  points are as $$V$$ to $$V'$$.

From this it appears that if $$q$$ be any coefficient of capacity or induction in the first system, and $$q'$$ the corresponding one in the second,

and if $$p$$ and $$p'$$ denote corresponding coefficients of potential in the two systems,