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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 , the centre of charge of its
constituent electrons (whose total
charge is ) is displaced by a distance with respect
to the centre of charge of its constituent
atomic nucleus. The dipole moment
of the molecule is defined
. If there are
such molecules per unit volume then the electric polarization
(i.e., the dipole moment per unit volume) is
given by
. More generally,

(803) 
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 .
Consider an infinitesimal cube of dielectric material with coordinates between and , coordinates between and , and coordinates
between and . Suppose that the dielectric consists of electrically neutral polar
molecules, of varying number density , whose electrons, charge
, displace a constant distance from the nuclei, charge . Thus, the
dipole moment per unit volume is
.
Due to the polarization of the molecules, a net charge
enters the bottom face of the cube, perpendicular to the axis,
whilst a net charge
leaves the top face.
Hence, the net charge acquired by the cube due to molecular polarization in the direction
is
. There are analogous contributions due to polarization in the
 and directions. Hence, the net charge acquired by the cube due to molecular polarization is
Thus, it follows that the
charge density acquired by the cube due to molecular polarization is simply
.
As explained above, it is easily demonstrated that any divergence of the polarization
field
of a dielectric medium gives rise to an effective charge density
in the medium, where

(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 due to
free charges, which represents a net surplus or
deficit of electrons in the medium. Thus, the total
charge density in the medium is

(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

(806) 
This expression can be rearranged to give

(807) 
where

(808) 
is termed the electric displacement, and has the same dimensions
as (dipole moment
per unit volume). Gauss' theorem tells us that

(809) 
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 a unit charge),
or the polarization (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 constitutive
relation connecting and . It is conventional
to assume that the induced polarization is
directly proportional to the electric field , so that

(810) 
where is termed the electric susceptibility of the medium. It follows that

(811) 
where

(812) 
is termed the dielectric constant or relative permittivity of the medium.
(Likewise, is termed the permittivity of free space.)
Note that is dimensionless.
It follows from Eqs. (807) and (811)
that

(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 . 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
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 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

(814) 
where is a secondrank tensor known as the
dielectric tensor. For strong electric fields,
ceases to vary linearly with . Indeed, for sufficiently
strong electric fields, neutral molecules are disrupted, and the whole
concept of a dielectric medium becomes meaningless.
Next: Boundary conditions for and
Up: Dielectric and magnetic media
Previous: Introduction
Richard Fitzpatrick
20060202