Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/193

Let us next consider a portion of the first surface whose area is $S$, taken so that no part of $S$ is near the boundary of the surface.

The quantity of electricity on this surface is $E_{1}=S\sigma$, and, by Art. 79, the force acting on every unit of electricity is ${\tfrac {1}{2}}R$, so that the whole force acting on the area $S$, and attracting it towards the other plane, is

$F={\frac {1}{2}}RS\sigma ={\frac {1}{8\pi }}R^{2}S={\frac {S}{8\pi }}{\frac {(B-A)^{2}}{c^{2}}}.$

Here the attraction is expressed in terms of the area $S$, the difference of potentials of the two surfaces ($A-B$), and the distance between them $c$. The attraction, expressed in terms of the charge $E$, on the area $S$, is

$F={\frac {2\pi }{S}}E_{1}^{2}$.

The electrical energy due to the distribution of electricity on the area $S$, and that on an area $S'$ on the surface $B$ denned by projecting $S$ on the surface $B$ by a system of lines of force, which in this case are normals to the planes, is

${\begin{array}{ll}Q&={\frac {1}{2}}\left(E_{1}A+E_{2}B\right),\\\\&={\frac {1}{2}}\left({\frac {S}{4\pi }}{\frac {(A-B)^{2}}{c}}\right),\\\\&={\frac {R^{2}}{8\pi }}Sc,\\\\&={\frac {2\pi }{S}}E_{1}^{2}c,\\\\&=Fc.\end{array}}$

The first of these expressions is the general expression of electrical energy.

The second gives the energy in terms of the area, the distance, and the difference of potentials.

The third gives it in terms of the resultant force $R$, and the volume $Sc$ included between the areas $S$ and $S'$, and shews that the energy in unit of volume is $p$ where $8\pi p=R^{2}$.

The attraction between the planes is $pS$, or in other words, there is an electrical tension (or negative pressure) equal to $p$ on every unit of area.

The fourth expression gives the energy in terms of the charge.

The fifth shews that the electrical energy is equal to the work which would be done by the electric force if the two surfaces were to be brought together, moving parallel to themselves, with their electric charges constant.