Polarization

The terrestrial environment is characterized by dielectric media (e.g., air, water) which are, for the most part, electrically neutral, since they are made up of neutral atoms and molecules. However, if these atoms and molecules are placed in an electric field they tend to polarize. Suppose that when a given neutral molecule is placed in an electric field ${\bfm E}$ the centre of charge of its constituent electrons (whose total charge is $-q$) is displaced by a distance $-{\bfm r}$ with respect to the centre of charge of its constituent atomic nuclei. The dipole moment of the molecule is defined ${\bfm p} = q \,{\bfm r}$. If there are $N$ such molecules per unit volume then the electric polarization ${\bfm P}$ (i.e., the dipole moment per unit volume) is given by ${\bfm P} = N {\bfm p}$. More generally,

$${\bfm P}({\bfm r}) = \sum_i N_i \langle {\bfm p}_i\rangle,$$ (417)

where $\langle {\bfm p}_i\rangle$ is the average dipole moment of the $i$th type of molecule in the vicinity of point ${\bfm r}$, and $N_i$ is the average number of such molecules per unit volume at ${\bfm r}$.

It is easily demonstrated that any divergence of the polarization field ${\bfm P}({\bfm r})$ gives rise to an effective charge density $\rho_b$ in the medium. In fact,

$$\rho_b = -\nabla\!\cdot\!{\bfm P}.$$ (418)

This charge density is attributable to bound charges (i.e., charges which arise from the polarization of neutral atoms), and is usually distinguished from the charge density $\rho_f$ due to free charges, which represents a net surplus or deficit of electrons in the medium. Thus, the total charge density $\rho$ in the medium is

$$\rho = \rho_f + \rho_b.$$ (419)

It must be emphasized that both terms in 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

$$\nabla\!\cdot\!{\bfm E} = \frac{\rho}{\epsilon_0} = \frac{\rho_f + \rho_b}{\epsilon_0}.$$ (420)

This expression can be rearranged to give

$$\nabla\!\cdot\!{\bfm D} = \rho_f,$$ (421)

where

$${\bfm D} = \epsilon_0 {\bfm E} + {\bfm P}$$ (422)

is termed the electric displacement, and has the same dimensions as ${\bfm P}$ (dipole moment per unit volume). The divergence theorem tells us that

$$\oint_S {\bfm D}\!\cdot\!d{\bfm S} = \int_V \rho_f\,dV.$$ (423)

In other words, the flux of ${\bfm D}$ out of some closed surface $S$ is equal to the total free charge enclosed within that surface. Unlike the electric field ${\bfm E}$ (which is the force acting on unit charge) or the polarization ${\bfm P}$ (the dipole moment per unit volume), the electric displacement ${\bfm D}$ has no clear physical meaning. The only reason for introducing it is that it enables us to calculate fields in the presence of dielectric materials without first having to know the distribution of polarized charges. However, this is only possible if we have a constitutive relation connecting ${\bfm E}$ and ${\bfm D}$. It is conventional to assume that the induced polarization ${\bfm P}$ is directly proportional to the electric field ${\bfm E}$, so that

$${\bfm P} = \epsilon_0\chi_e {\bfm E},$$ (424)

where $\chi_e$ is termed the electric susceptibility of the medium. It follows that

$${\bfm D} = \epsilon_0\epsilon{\bfm E},$$ (425)

where

$$\epsilon= 1 + \chi_e$$ (426)

is termed the dielectric constant or relative permittivity of the medium. (Likewise, $\epsilon_0$ is termed the permittivity of free space.) It follows from Eqs. (3.5) and (3.9) that

$$\nabla\!\cdot\!{\bfm E} = \frac{\rho_f}{\epsilon_0\epsilon}.$$ (427)

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 $\epsilon$. This reduction can be understood in terms of a polarization of the atoms or molecules of the dielectric medium that produces electric fields in opposition to that of given charge. One immediate consequence is that the capacitance of a capacitor is increased by a factor $\epsilon$ if the empty space between the electrodes is filled with a dielectric medium of dielectric constant $\epsilon$ (assuming that fringing fields can be neglected).

It must be understood that Eqs. (3.8)-(3.11) are just an approximation which is generally found to hold under terrestrial conditions (provided that the fields are not too large) for isotropic media. For anisotropic media (e.g., crystals) Eq. (3.9) generalizes to

$${\bfm D} = \epsilon_0 \,{\bfm \epsilon}\!\cdot\!{\bfm E},$$ (428)

where ${\bfm\epsilon}$ is a second rank tensor known as the dielectric tensor. For strong electric fields ${\bfm D}$ ceases to vary linearly with ${\bfm E}$. Indeed, for sufficiently strong electric fields neutral molecules are disrupted and the whole concept of a dielectric medium becomes meaningless.