The dielectric constant
Here we begin to discuss another of the peculiar properties of matter under the influence of the electric field. In an earlier chapter we considered the behavior of conductors, in which the charges move freely in response to an electric field to such points that there is no field left inside a conductor. Now we will discuss insulators, materials which do not conduct electricity. One might at first believe that there should be no effect whatsoever. However, using a simple electroscope and a parallel-plate capacitor, Faraday discovered that this was not so. His experiments showed that the capacitance of such a capacitor is increased when an insulator is put between the plates. If the insulator completely fills the space between the plates, the capacitance is increased by a factor $\kappa$ which depends only on the nature of the insulating material. Insulating materials are also called dielectrics; the factor $\kappa$ is then a property of the dielectric, and is called the dielectric constant. The dielectric constant of a vacuum is, of course, unity. Our problem now is to explain why there is any electrical effect if the insulators are indeed insulators and do not conduct electricity. We begin with the experimental fact that the capacitance is increased and try to reason out what might be going on. Consider a parallel-plate capacitor with some charges on the surfaces of the conductors, let us say negative charge on the top plate and positive charge on the bottom plate. Suppose that the spacing between the plates is $d$ and the area of each plate is $A$. As we have proved earlier, the capacitance is \begin{equation} \label{Eq:II:10:1} C=\frac{\epsO A}{d}, \end{equation}  and the charge and voltage on the capacitor are related by \begin{equation} \label{Eq:II:10:2} Q=CV. \end{equation}  Now the experimental fact is that if we put a piece of insulating material like lucite or glass between the plates, we find that the capacitance is larger. That means, of course, that the voltage is lower for the same charge. But the voltage difference is the integral of the electric field across the capacitor; so we must conclude that inside the capacitor, the electric field is reduced even though the charges on the plates remain unchanged.  Now how can that be? We have a law due to Gauss that tells us that the flux of the electric field is directly related to the enclosed charge. Consider the Gaussian surface $S$ shown by broken lines in Fig. 10–1. Since the electric field is reduced with the dielectric present, we conclude that the net charge inside the surface must be lower than it would be without the material. There is only one possible conclusion, and that is that there must be positive charges on the surface of the dielectric.  Since the field is reduced but is not zero, we would expect this positive charge to be smaller than the negative charge on the conductor. So the phenomena can be explained if we could understand in some way that when a dielectric material is placed in an electric field there is positive charge induced on one surface and negative charge induced on the other. We would expect that to happen for a conductor. For example, suppose that we had a capacitor with a plate spacing $d$, and we put between the plates a neutral conductor whose thickness is $b$, as in Fig. 10–2. The electric field induces a positive charge on the upper surface and a negative charge on the lower surface, so there is no field inside the conductor. The field in the rest of the space is the same as it was without the conductor, because it is the surface density of charge divided by $\epsO$; but the distance over which we have to integrate to get the voltage (the potential difference) is reduced. The voltage is \begin{equation*} V=\frac{\sigma}{\epsO}\,(d-b). \end{equation*}  The resulting equation for the capacitance is like Eq. (10.1), with $(d-b)$ substituted for $d$: \begin{equation} \label{Eq:II:10:3} C=\frac{\epsO A}{d[1-(b/d)]}. \end{equation}  The capacitance is increased by a factor which depends upon $(b/d)$, the proportion of the volume which is occupied by the conductor. This gives us an obvious model for what happens with dielectrics—that inside the material there are many little sheets of conducting material. The trouble with such a model is that it has a specific axis, the normal to the sheets, whereas most dielectrics have no such axis. However, this difficulty can be eliminated if we assume that all insulating materials contain small conducting spheres separated from each other by insulation, as shown in Fig. 10–3. The phenomenon of the dielectric constant is explained by the effect of the charges which would be induced on each sphere. This is one of the earliest physical models of dielectrics used to explain the phenomenon that Faraday observed. More specifically, it was assumed that each of the atoms of a material was a perfect conductor, but insulated from the others. The dielectric constant $\kappa$ would depend on the proportion of space which was occupied by the conducting spheres. This is not, however, the model that is used today.