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Polarization

We have seen Maxwell's equations in vacuum and now we wish to know what is going to happen to them in a material. Let us consider a dielectric medium! A dielectric is basically an electric insulator, like glass or water. We can represent such a material as a collection of positive and negative charges. In the simplest case, these are the nucleus and the electrons of an atom. As you probably know, the net charge of this atomic system is zero, but what if an electromagnetic wave comes along?

Well, under the influence of the electric field, the negative and positive charges are going to split. In turn, this is going to influence the electric field and that influence will show up in Maxwell's equations. We can now proceed and identify the new quantities.

The electric dipole moment is defined as the product of charge and separation. The charges themselves could be protons and electrons of an atom stretched in an external electric field,  the poles of a polar molecule, or two oppositely charged metal spheres.

The two charges making up the individual dipole are now bound charges. This is often indicated with a small b written in subscript, however, in the following, I will omit this notation since it is clear that all the charges we will be dealing with are bound charges. The units are Coulomb meter (C.m). The dipole moment vector points from negative to positive charge, see Fig. 8.

 Fig. 8 Illustration of the charge separation in an atom due to an applied electric field.

The polarization of a region is defined as the dipole moment per unit volume (V). It is therefore a measure for the density of dipole moments:

If all the atoms have the same dipole moment, or if p designates the average dipole moment, equation 8.2 becomes:

where N is the number of molecules and n is the number of molecules per unit volume.

Let us try to visualize this. In figure 9, you can see a unit volume inside a nonpolar dielectric, which encloses several atoms. At the maximum of the electric field (E) applied to this dielectric, the positive and negative charges are separated by a distance s. Note that when the electric field is applied to the right side of figure 9, s is actually equivalent to the vector r in Eq. 8.1!

 Fig. 9 The unit volume V inside a nonpolar dielectric is the product of an element of area dA and of the maximum distance between the electrical charges (s) under the influence on an external electric field E. The vector dA is perpendicular to the area dA.

When E increases, N⁺ positive charges and N^ negative charges cross the small element of area dA. The net charge that crosses dA is then:

But of course, this is also the number of molecules within the unit volume V = s.dA. Then remembering that n is the number of molecules per unit volume and that s = r,

There we are, the net charge that crosses an element of surface dA in a dielectric, under the influence of an electric field, is proportional to the vector product of the polarization by the vector dA. This relationship is at the heart of the light-matter interaction, since its consequences (see below) will show up in Maxwell's equations.

8.1 First consequence of Eq 8.5

For instance, let us now consider the volume v, which consists of the volume V enclosed behind dA. Of course, v = V/2. Equation 8.5 tells us that the net charge that flows out of v, through the surface dA is P.dA. We can rephrase this and say that the net charge that flows out of the closed surface A which encloses the volume v is:

but then, it is clear that the charge that remains within v must be -Q_out!

By the way, another way of looking at the charge within v is to consider the volume density of charge over the volume v. Let this volume density of charge be r_b. It follows then that:

where the divergence theorem was used in order to obtain the last integral from the previous. Hence we have:

This means that there are sources of polarization.

8.1 Second consequence of Eq 8.5

Under the influence of a time dependent electric field (like, say, the one in a light wave), by definition, the current flowing through a surface A  is related to the current density J by the formula:

where we applied again the divergence theorem. From Eq. 8.7, we then obtain:

And therefore:

This means that the motion of bound charges results in a polarization current density.

Note that from equation 8.10 we can also deduce that:

This is called the continuity equation.

Polarization is proportional to the electric field, depending on the dielectric material. Because the units of the dipole moment are C.m, the units of polarization are C.m² which is the same unit as a surface charge density.

In anisotropic materials P and E are not parallel and are not simply related. In ferroelectric materials, this relationship may be nonlinear and could also depend on the past history of the sample, i.e. there can be a hysteresis.

© V. K. Valev