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 then they tend to polarize. Suppose that when a given neutral molecule is placed in an electric field ${\bf E}$, the centre of charge of its constituent electrons (whose total charge is $q$) is displaced by a distance ${\bf d}$ with respect to the centre of charge of its constituent atomic nucleus. The dipole moment of the molecule is defined ${\bf p} = q {\bf d}$. If there are $N$ such molecules per unit volume then the electric polarization ${\bf P}$ (i.e., the dipole moment per unit volume) is given by ${\bf P} = N {\bf p}$. More generally,

$${\bf P}({\bf r}) = \sum_i N_i \langle {\bf p}_i\rangle,$$ (803)

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

Consider an infinitesimal cube of dielectric material with $x$-coordinates between $x$ and $x+dx$, $y$-coordinates between $y$ and $y+dy$, and $z$-coordinates between $z$ and $z+dz$. Suppose that the dielectric consists of electrically neutral polar molecules, of varying number density $N({\bf r})$, whose electrons, charge $q$, displace a constant distance ${\bf d}$ from the nuclei, charge $-q$. Thus, the dipole moment per unit volume is ${\bf P}({\bf r}) = N({\bf r}) q {\bf d}$. Due to the polarization of the molecules, a net charge $N(x,y,z) q d_x dy dz$ enters the bottom face of the cube, perpendicular to the $x$-axis, whilst a net charge $N(x+dx,y,z) q d_x dy dz$ leaves the top face. Hence, the net charge acquired by the cube due to molecular polarization in the $x$-direction is $dq = -N(x+dx,y,z) q d_x dy dz + N(x,y,z) q d_x dy dz = - [\partial N(x... ...artial x] q d_x dx dy dz = - [\partial P_x(x,y,z)/\partial x] dx dy dz$. There are analogous contributions due to polarization in the $y$- and $z$-directions. Hence, the net charge acquired by the cube due to molecular polarization is $dq = - [\partial P_x(x,y,z)/\partial x+ \partial P_y(x,y,z)/\partial y+\partial P_z(x,y,z)/\partial z] dx dy dz = - (\nabla\!\cdot\!{\bf P}) dx dy dz.$ Thus, it follows that the charge density acquired by the cube due to molecular polarization is simply $-\nabla\!\cdot\!{\bf P}$.

As explained above, it is easily demonstrated that any divergence of the polarization field ${\bf P}({\bf r})$ of a dielectric medium gives rise to an effective charge density $\rho_b$ in the medium, where

$$\rho_b = -\nabla\!\cdot\!{\bf P}.$$ (804)

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

$$\rho = \rho_f + \rho_b.$$ (805)

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

$$\nabla\!\cdot\!{\bf E} = \frac{\rho}{\epsilon_0} = \frac{\rho_f + \rho_b}{\epsilon_0}.$$ (806)

This expression can be rearranged to give

$$\nabla\!\cdot\!{\bf D} = \rho_f,$$ (807)

where

$${\bf D} = \epsilon_0 {\bf E} + {\bf P}$$ (808)

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

$$\oint_S {\bf D}\!\cdot\!d{\bf S} = \int_V \rho_f dV.$$ (809)

In other words, the flux of ${\bf D}$ out of some closed surface $S$ is equal to the total free charge enclosed within that surface. Unlike the electric field ${\bf E}$ (which is the force acting on a unit charge), or the polarization ${\bf P}$ (the dipole moment per unit volume), the electric displacement ${\bf D}$ 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 constitutive relation connecting ${\bf E}$ and ${\bf D}$. It is conventional to assume that the induced polarization ${\bf P}$ is directly proportional to the electric field ${\bf E}$, so that

$${\bf P} = \epsilon_0 \chi_e {\bf E},$$ (810)

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

$${\bf D} = \epsilon_0 \epsilon {\bf E},$$ (811)

where

$$\epsilon= 1 + \chi_e$$ (812)

is termed the dielectric constant or relative permittivity of the medium. (Likewise, $\epsilon_0$ is termed the permittivity of free space.) Note that $\epsilon$ is dimensionless. It follows from Eqs. (807) and (811) that

$$\nabla\!\cdot\!{\bf E} = \frac{\rho_f}{\epsilon_0 \epsilon}.$$ (813)

Thus, the electric fields produced by free charges in a uniform 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 those generated by the free charges. One immediate consequence of this 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. (810)-(813) 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. (811) generalizes to

$${\bf D} = \epsilon_0 \mbox{\boldmath$ \epsilon$}\!\cdot\!{\bf E},$$ (814)

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