Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/174

the total induction through a closed surface is equal to the total quantity of electricity within the surface multiplied by 4$\pi$. For what we have called the induction through the surface is simply the electric displacement multiplied by 4$\pi$, and the total displacement outwards is necessarily equal to the total electrification within the surface.

The theory also accounts for the impossibility of communicating an absolute charge to matter. For every particle of the dielectric is electrified with equal and opposite charges on its opposite sides, if it would not be more correct to say that these electrifications are only the manifestations of a single phenomenon, which we may call Electric Polarization.

A dielectric medium, when thus polarized, is the seat of electrical energy, and the energy in unit of volume of the medium is numerically equal to the electric tension on unit of area, both quantities being equal to half the product of the displacement and the resultant electromotive force, or

$p={\frac {1}{2}}{\mathfrak {DE}}={\frac {1}{8\pi }}K{\mathfrak {E}}^{2}={\frac {2\pi }{K}}{\mathfrak {D}}^{2},$

where $p$ is the electric tension, ${\mathfrak {D}}$ the displacement, ${\mathfrak {E}}$ the electro motive force, and $K$ the specific inductive capacity.

If the medium is not a perfect insulator, the state of constraint, which we call electric polarization, is continually giving way. The medium yields to the electromotive force, the electric stress is relaxed, and the potential energy of the state of constraint is converted into heat. The rate at which this decay of the state of polarization takes place depends on the nature of the medium. In some kinds of glass, days or years may elapse before the polarization sinks to half its original value. In copper, this change may occupy less than the billionth of a second.

We have supposed the medium after being polarized to be simply left to itself. In the phenomenon called the electric current the constant passage of electricity through the medium tends to restore the state of polarization as fast as the conductivity of the medium allows it to decay. Thus the external agency which maintains the current is always doing work in restoring the polarization of the medium, which is continually becoming relaxed, and the potential energy of this polarization is continually becoming transformed into heat, so that the final result of the energy expended in maintaining the current is to raise the temperature of the conductor.