Maxwell’s equations in a dielectric
The existence of polarization in matter means that there are polarization charges and currents inside of the material, and these must be put into the complete Maxwell equations in order to find the fields. We are going to solve Maxwell’s equations this time in a situation in which the charges and currents are not zero, as in a vacuum, but are given implicitly by the polarization vector. Our first step is to find explicitly the charge density $\rho$ and current density $\FLPj$, averaged over a small volume of the same size we had in mind when we defined $\FLPP$. Then the $\rho$ and $\FLPj$ we need can be obtained from the polarization. We have seen in Chapter 10 that when the polarization $\FLPP$ varies from place to place, there is a charge density given by \begin{equation} \label{Eq:II:32:9} \rho_{\text{pol}}=-\FLPdiv{\FLPP}. \end{equation}  At that time, we were dealing with static fields, but the same formula is valid also for time-varying fields. However, when $\FLPP$ varies with time, there are charges in motion, so there is also a polarization current. Each of the oscillating charges contributes a current equal to its charge $q_e$, times its velocity $v$. With $N$ such charges per unit volume, the current density $\FLPj$ is \begin{equation*} \FLPj=Nq_e\FLPv. \end{equation*}  Since we know that $v=dx/dt$, then $j=Nq_e(dx/dt)$, which is just $dP/dt$. Therefore the current density from the varying polarization is \begin{equation} \label{Eq:II:32:10} \FLPj_{\text{pol}}=\ddp{\FLPP}{t}. \end{equation}  Our problem is now direct and simple. We write Maxwell’s equations with the charge density and current density expressed in terms of $\FLPP$, using Eqs. (32.9) and (32.10). (We assume that there are no other currents and charges in the material.) We then relate $\FLPP$ to $\FLPE$ with Eq. (32.8), and we solve the equation for $\FLPE$ and $\FLPB$—looking for the wave solutions. Before we do this, we would like to make an historical note. Maxwell originally wrote his equations in a form which was different from the one we have been using. Because the equations were written in this different form for many years—and are still written that way by many people—we will explain the difference. In the early days, the mechanism of the dielectric constant was not fully and clearly appreciated. The nature of atoms was not understood, nor that there was a polarization of the material. So people did not appreciate that there was a contribution to the charge density $\rho$ from $\FLPdiv{\FLPP}$. They thought only in terms of charges that were not bound to atoms (such as the charges that flow in wires or are rubbed off surfaces). Today, we prefer to let $\rho$ represent the total charge density, including the part from the bound atomic charges. If we call that part $\rho_{\text{pol}}$, we can write \begin{equation*} \rho=\rho_{\text{pol}}+\rho_{\text{other}}, \end{equation*}  where $\rho_{\text{other}}$ is the charge density considered by Maxwell and refers to the charges not bound to individual atoms. We would then write \begin{equation*} \FLPdiv{\FLPE}=\frac{\rho_{\text{pol}}+\rho_{\text{other}}}{\epsO}. \end{equation*}  Substituting $\rho_{\text{pol}}$ from Eq. (32.9), \begin{equation} \FLPdiv{\FLPE}=\frac{\rho_{\text{other}}}{\epsO}- \frac{1}{\epsO}\,\FLPdiv{\FLPP}\notag \end{equation}  or \begin{equation} \label{Eq:II:32:11} \FLPdiv{(\epsO\FLPE+\FLPP)}=\rho_{\text{other}}. \end{equation}  The current density in the Maxwell equations for $\FLPcurl{\FLPB}$ also has, in general, contributions from bound atomic currents. We can therefore write \begin{equation*} \FLPj=\FLPj_{\text{pol}}+\FLPj_{\text{other}}, \end{equation*}  and the Maxwell equation becomes \begin{equation} \label{Eq:II:32:12} c^2\FLPcurl{\FLPB}=\frac{\FLPj_{\text{other}}}{\epsO}+ \frac{\FLPj_{\text{pol}}}{\epsO}+\ddp{\FLPE}{t}. \end{equation}  Using Eq. (32.10), we get \begin{equation} \label{Eq:II:32:13} \epsO c^2\FLPcurl{\FLPB}=\FLPj_{\text{other}}+ \ddp{}{t}(\epsO\FLPE+\FLPP). \end{equation}   Now you can see that if we were to define a new vector $\FLPD$ by \begin{equation} \label{Eq:II:32:14} \FLPD=\epsO\FLPE+\FLPP, \end{equation}  the two field equations would become \begin{equation} \label{Eq:II:32:15} \FLPdiv{\FLPD}=\rho_{\text{other}} \end{equation}  and \begin{equation} \label{Eq:II:32:16} \epsO c^2\FLPcurl{\FLPB}=\FLPj_{\text{other}}+\ddp{\FLPD}{t}. \end{equation}  These are actually the forms that Maxwell used for dielectrics. His two remaining equations were \begin{equation*} \FLPcurl{\FLPE}=-\ddp{\FLPB}{t}, \end{equation*}  and \begin{equation*} \FLPdiv{\FLPB}=0, \end{equation*}  which are the same as we have been using. Maxwell and the other early workers also had a problem with magnetic materials (which we will take up soon). Because they did not know about the circulating currents responsible for atomic magnetism, they used a current density that was missing still another part. Instead of Eq. (32.16), they actually wrote \begin{equation} \label{Eq:II:32:17} \FLPcurl{\FLPH}=\FLPj'+\ddp{\FLPD}{t}, \end{equation}  where $\FLPH$ differs from $\epsO c^2\FLPB$ because it includes the effects of atomic currents. (Then $\FLPj'$ represents what is left of the currents.) So Maxwell had four field vectors—$\FLPE$, $\FLPD$, $\FLPB$, and $\FLPH$—the $\FLPD$ and $\FLPH$ were hidden ways of not paying attention to what was going on inside the material. You will find the equations written this way in many places. To solve the equations, it is necessary to relate $\FLPD$ and $\FLPH$ to the other fields, and people used to write \begin{equation} \label{Eq:II:32:18} \FLPD=\epsilon\FLPE\quad\text{and}\quad \FLPB=\mu\FLPH. \end{equation}  However, these relations are only approximately true for some materials and even then only if the fields are not changing rapidly with time. (For sinusoidally varying fields one often can write the equations this way by making $\epsilon$ and $\mu$ complex functions of the frequency, but not for an arbitrary time variation of the fields.) So there used to be all kinds of cheating in solving the equations. We think the right way is to keep the equations in terms of the fundamental quantities as we now understand them—and that’s how we have done it.