2.19: Polarization of Moving Media

Effects of polarization and magnetization are included in the formulation of electrodynamics postulated in Sec. 2.2. In this and the next section a review is made of the underlying models.

Consider the electroquasistatic systems, where the dominant field source is the charge density. Not all of this charge is externally accessible, in the sense that it cannot all be brought to some position through a conduction process. If an initially neutral dielectric medium is stressed by an electric field, the constituent molecules and domains become polarized. Even though the material retains its charge neutrality, there can be a local accrual or loss of charge because of the polarization. The first order of business is to deduce the relation of such polarization charge to the polarization density.

For conceptual purposes, the polarization of a material is pictured as shown in Figure 2.8.1.

Figure 2.8.1. Model for dipoles fixed to deformable material. The model pictures the negative charges as fixed to the material, and then the positive halves of the dipoles fixed to the negative charges through internal constraints.

Figure 2.8.2 Polarization results in net charges passing through a surface.

The molecules or domains are represented by dipoles composed of positive and negative charges \(\pm q\), separated by the vector distance \(\overrightarrow{d}\). The dipole moment is then \(\overrightarrow{p} = q \overrightarrow{d}\), and if the particles have a number density \(n\), the polarization density is defined as

\[ \overrightarrow{P} = nq\overrightarrow{d} \label{1} \]

In the most common dielectrics, the polarization results because of the application of an external electric field. In that case, the internal constraints (represented by the springs in Figure 2.8.1) make the charges essentially coincident in the absence of an electric field, so that, on the average, the material is (macroscopically) neutral. Then,.with the application of the electric field, there is a separation of the charges in some direction which might be coincident with the applied electric field intensity. The effect of the dipoles on the average electric field distribution is equivalent to that of the medium they model.

To see how the polarization charge density is related to the polarization density, consider the motion of charges through the arbitrary surface \(S\) shown in Figure 2.8.2. For the moment, consider the surface as being closed, so that the contour enclosing the surface shown is shrunk to zero. Because polarization results in motion of the positive charge, leaving behind the negative image charge, the net polarization charge within the volume \(V\) enclosed by the surface \(S\) is equal to the negative of the net charge having left the volume across the surface \(S\). Thus,

\[ \int \rho_p dV = - \oint_{S} nq \overrightarrow{d} \cdot \overrightarrow{n} da = - \oint_{S} \overrightarrow{P} \cdot \overrightarrow{n} da \label{2} \]

Gauss' theorem, Equation 2.6.2, converts the surface integral to one over the arbitrary volume \(V\). It follows that the integrand must vanish so that

\[ \rho_p = - \nabla \cdot \overrightarrow{P} \label{3} \]

This polarization charge density is now added to the free charge density as a source of the electric field intensity in Gauss' law:

\[ \nabla \cdot \varepsilon_o \overrightarrow{E} = \rho_f + \rho_p \label{4} \]

and Eqs. \ref{3} and \ref{4} comprise the postulated form of Gauss' law, Equation 2.3.23a

By definition, polarization charge is conserved, independent of the free charge. Hence, the polarization current \(\overrightarrow{J_p}\) is defined such that it satisfies the conservation equation

\[ \nabla \cdot \overrightarrow{J_p} + \frac{\partial{\rho_p}}{\partial{t}} = 0 \label{5} \]

To establish the way in which \(\overrightarrow{J_p}\) transforms between inertial reference frames, observe that in a primed frame of reference, by dint of Equation 2.5.2c, the conservation of polarization charge equation becomes

\[ \nabla \cdot \big[ \overrightarrow{J_p^{'}} + \overrightarrow{u}\rho_p^{'} \big] + \frac{\partial{\rho_p^{'}}}{\partial{t}} = 0 \label{6} \]

It has been shown that \(\overrightarrow{P}\), and hence \(\rho_p\), are the same in both frames (Equation 2.5.10a). It follows that the required transformation law is

\[ \overrightarrow{J_p^{'}} = \overrightarrow{J_p} - \overrightarrow{u}\rho_p \label{7} \]

If the dipoles are attached to a moving medium, so that the negative charges move with the same velocity \(\overrightarrow{v}\) as the moving material, the motion gives rise to a current which should be included in Ampere's law as a source of magnetic field. Even if the material is fixed, but the applied field is time-varying so as to induce a time-varying polarization density, a given surface is crossed by a net charge and there is a current caused by a time-varying polarization density. The following steps determine the current density \(\overrightarrow{J_p}\) in terms of the polarization density and the material velocity.

The starting point is the statement

\[ \int_{S} \overrightarrow{J_p^{'}} \cdot \overrightarrow{n} da = \frac{d}{dt} \int_{S} \overrightarrow{P} \cdot \overrightarrow{n} da \label{8} \]

The surface \(S\), depicted by Figure 2.8.2, is attached to the material itself. It moves with the negative charges of the dipoles. Integrated over this deforming surface of fixed identity, the polarization current density evaluated in the frame of reference of the material is equal to the rate of change with respect to time of the net charge penetrating that surface.

With the surface velocity identified with the material velocity, Equation 2.6.4 and Equation \ref{3} convert Equation \ref{8} to

\[ \int_{S} \overrightarrow{J_p^{'}} \cdot \overrightarrow{n} da = \int_{S} (\frac{\partial{\overrightarrow{P}}}{\partial{t}} - \rho_p \overrightarrow{v}) \cdot \overrightarrow{n} da + \oint_{C} \overrightarrow{P} \times \overrightarrow{v} \cdot \overrightarrow{d} l \label{9} \]

On the left, \(\overrightarrow{J_p^{'}}\) is replaced by Equation \ref{7} evaluated with \(\overrightarrow{u} = \overrightarrow{v}\), while on the right Stokes's theorem, Equation 2.6.3, is used to convert the line integral to a surface integral. The result is an equation in surface integrals alone. Although fixed to the deforming material, the surface \(S\) is otherwise arbitrary and so it follows that the required relation between \(\overrightarrow{J_p}\) and \(\overrightarrow{P}\) for the moving material is

\[ \overrightarrow{J_p} = \frac{\partial{\overrightarrow{P}}}{\partial{t}} + \nabla \times (\overrightarrow{P} \times \overrightarrow{v}) \label{10} \]

It is this current density that has been added to the right-hand side of Ampere's law, Equation 2.3.26a, to complete the formulation of polarization effects in the electroquasistatic system.