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

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

$V={\frac {E}{AB}}$,${\color {White}xxx}$and ${\color {White}xxx}$$V={\frac {E'}{A'B'}}$

But $AB$ is to $A'B'$ as $L$ to $L'$, so that we must have

$E\ :E'\ ::LV\ :\ L'V'\,\!$

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

$E\ :E'\ ::LVK\ :\ L'V'K'\,\!$

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,

$q\ :q'\ ::LK\ :\ L'K'\,\!$;

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

$p\ :p'\ ::{\frac {1}{LK}}\ :\ {\frac {1}{L'K'}}\,\!$.