(496) |

where is the average dipole moment of the th type of molecule in the vicinity of point , and is the average number of such molecules per unit volume at .

It is easily demonstrated [e.g., by integrating Equation (201) by parts, and then comparing the result with Equation (162)] that any divergence of the polarization field, , gives rise to a charge density, , in the medium. In fact,

This density is attributable to

(498) |

It must be emphasized that both terms on the right-hand side of this equation represent real physical charge. Nevertheless, it is useful to make the distinction between bound and free charges, especially when it comes to working out the energy associated with electric fields in dielectric media.

Gauss' law takes the differential form

(499) |

This expression can be rearranged to give

where

(501) |

is termed the

In other words, the flux of out of some closed surface is equal to the total free charge enclosed within that surface. Unlike the electric field (which is the force acting on unit charge), or the polarization (which is the dipole moment per unit volume), the electric displacement has no clear physical meaning. The only reason for introducing this quantity is that it enables us to calculate electric fields in the presence of dielectric materials without first having to know the distribution of bound charges. However, this is only possible if we have a

where is termed the medium's

where the dimensionless quantity

(505) |

is known as the

Thus, the electric fields produced by free charges in a dielectric medium are analogous to those produced by the same charges in a vacuum, except that they are reduced by a factor . This reduction can be understood in terms of a polarization of the medium's constituent atoms or molecules that produces electric fields in opposition to those of the free charges. One immediate consequence is that the capacitance of a capacitor is increased by a factor if the empty space between the electrodes is filled with a dielectric medium of dielectric constant (assuming that fringing fields can be neglected).

It must be understood that Equations (504)-(507) constitute an approximation that is generally found to hold under terrestrial conditions (provided the electric field-strength does not become too large) when dealing with isotropic media. For anisotropic media (e.g., crystals), Equation (505) generalizes to

(507) |

where