The electrostatic equations with dielectrics
Now let’s combine the above result with our theory of electrostatics. The fundamental equation is \begin{equation} \label{Eq:II:10:17} \FLPdiv{\FLPE}=\frac{\rho}{\epsO}. \end{equation}  The $\rho$ here is the density of all electric charges. Since it is not easy to keep track of the polarization charges, it is convenient to separate $\rho$ into two parts. Again we call $\rho_{\text{pol}}$ the charges due to nonuniform polarizations, and call $\rho_{\text{free}}$ all the rest.  Usually $\rho_{\text{free}}$ is the charge we put on conductors, or at known places in space. Equation (10.17) then becomes \begin{equation} \FLPdiv{\FLPE}=\frac{\rho_{\text{free}}+\rho_{\text{pol}}}{\epsO}= \frac{\rho_{\text{free}}-\FLPdiv{\FLPP}}{\epsO}\notag \end{equation}  or \begin{equation} \label{Eq:II:10:18} \FLPdiv{\biggl(\FLPE+\frac{\FLPP}{\epsO}\biggr)}= \frac{\rho_{\text{free}}}{\epsO}. \end{equation}  Of course, the equation for the curl of $\FLPE$ is unchanged: \begin{equation} \label{Eq:II:10:19} \FLPcurl{\FLPE}=\FLPzero. \end{equation}  Taking $\FLPP$ from Eq. (10.8), we get the simpler equation \begin{equation} \label{Eq:II:10:20} \FLPdiv{[(1+\chi)\FLPE]}=\FLPdiv{(\kappa\FLPE)}= \frac{\rho_{\text{free}}}{\epsO}. \end{equation}  These are the equations of electrostatics when there are dielectrics. They don’t, of course, say anything new, but they are in a form which is more convenient for computation in cases where $\rho_{\text{free}}$ is known and the polarization $\FLPP$ is proportional to $\FLPE$. Notice that we have not taken the dielectric “constant,” $\kappa$, out of the divergence. That is because it may not be the same everywhere. If it has everywhere the same value, it can be factored out and the equations are just those of electrostatics with the charge density $\rho_{\text{free}}$ divided by $\kappa$. In the form we have given, the equations apply to the general case where different dielectrics may be in different places in the field. Then the equations may be quite difficult to solve.  There is a matter of some historical importance which should be mentioned here. In the early days of electricity, the atomic mechanism of polarization was not known and the existence of $\rho_{\text{pol}}$ was not appreciated. The charge $\rho_{\text{free}}$ was considered to be the entire charge density. In order to write Maxwell’s equations in a simple form, a new vector $\FLPD$ was defined to be equal to a linear combination of $\FLPE$ and $\FLPP$: \begin{equation} \label{Eq:II:10:21} \FLPD=\epsO\FLPE+\FLPP. \end{equation}  As a result, Eqs. (10.18) and (10.19) were written in an apparently very simple form: \begin{equation} \label{Eq:II:10:22} \FLPdiv{\FLPD}=\rho_{\text{free}},\quad\FLPcurl{\FLPE}=\FLPzero. \end{equation}  Can one solve these? Only if a third equation is given for the relationship between $\FLPD$ and $\FLPE$. When Eq. (10.8) holds, this relationship is \begin{equation} \label{Eq:II:10:23} \FLPD=\epsO(1+\chi)\FLPE=\kappa\epsO\FLPE. \end{equation}  This equation was usually written \begin{equation} \label{Eq:II:10:24} \FLPD=\epsilon\FLPE, \end{equation}  where $\epsilon$ is still another constant for describing the dielectric property of materials. It is called the “permittivity.” (Now you see why we have $\epsilon_0$ in our equations, it is the “permittivity of empty space.”) Evidently, \begin{equation} \label{Eq:II:10:25} \epsilon=\kappa\epsO=(1+\chi)\epsO. \end{equation}   Today we look upon these matters from another point of view, namely, that we have simpler equations in a vacuum, and if we exhibit in every case all the charges, whatever their origin, the equations are always correct. If we separate some of the charges away for convenience, or because we do not want to discuss what is going on in detail, then we can, if we wish, write our equations in any other form that may be convenient. One more point should be emphasized. An equation like $\FLPD=\epsilon\FLPE$ is an attempt to describe a property of matter. But matter is extremely complicated, and such an equation is in fact not correct. For instance, if $\FLPE$ gets too large, then $\FLPD$ is no longer proportional to $\FLPE$. For some substances, the proportionality breaks down even with relatively small fields. Also, the “constant” of proportionality may depend on how fast $\FLPE$ changes with time. Therefore this kind of equation is a kind of approximation, like Hooke’s law. It cannot be a deep and fundamental equation.  On the other hand, our fundamental equations for $\FLPE$, (10.17) and (10.19), represent our deepest and most complete understanding of electrostatics. 