# Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/137

94.

COMPARISON OF SIMILAR SYSTEMS.

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If one of the bodies be displaced in the first system, and the corresponding body in the second system receive a similar displacement, then these displacements are in the proportion of ${\displaystyle L}$ to ${\displaystyle L'}$, and if the forces acting on the two bodies are as ${\displaystyle F}$ to ${\displaystyle F'}$, then the work done in the two systems will be as ${\displaystyle FL}$ to ${\displaystyle F'L'}$.

But the total electrical energy is half the sum of the quantities of electricity multiplied each by the potential of the electrified body, so that in the similar systems, if ${\displaystyle Q}$ and ${\displaystyle Q'}$ be the total electrical energy,

 ${\displaystyle Q\ :\ Q'\ ::\ EV\ :\ E'V'}$,

and the difference of energy after similar displacements in the two systems will be in the same proportion. Hence, since ${\displaystyle FL}$ is proportional to the electrical work done during the displacement,

 ${\displaystyle FL\ :\ F'L'\ ::\ EV\ :\ E'V'}$.

Combining these proportions, we find that the ratio of the resultant force on any body of the first system to that on the corresponding body of the second system is

 ${\displaystyle F\ :\ F'\ ::\ V^{2}K\ :\ V'^{2}K1}$,

 or${\displaystyle {\color {White}xxxxxxx}}$${\displaystyle F\ :\ F'\ ::\ {\frac {E^{2}}{L^{2}K}}:\ {\frac {E'^{2}}{L'^{2}K'}}}$.

The first of these proportions shews that in similar systems the force is proportional to the square of the electromotive force and to the inductive capacity of the dielectric, but is independent of the actual dimensions of the system.

Hence two conductors placed in a liquid whose inductive capacity is greater than that of air, and electrified to given potentials, will attract each other more than if they had been electrified to the same potentials in air.

The second proportion shews that if the quantity of electricity on each body is given, the forces are proportional to the squares of the electrifications and inversely to the squares of the distances, and also inversely to the inductive capacities of the media.

Hence, if two conductors with given charges are placed in a liquid whose inductive capacity is greater than that of air, they will attract each other less than if they had been surrounded with air and electrified with the same charges of electricity.