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In this chapter we are going to discuss why it is that materials are dielectric. We said in the last chapter that we could understand the properties of electrical systems with dielectrics once we appreciated that when an electric field is applied to a dielectric it induces a dipole moment in the atoms. We have already discussed how this equation is applied; now we have to discuss the mechanism by which polarization arises when there is an electric field inside a material. We begin with the simplest possible example—the polarization of gases. But even gases already have complications: there are two types.

The molecules of some gases, like oxygen, which has a symmetric pair of atoms in each molecule, have no inherent dipole moment. But the molecules of others, like water vapor which has a nonsymmetric arrangement of hydrogen and oxygen atoms carry a permanent electric dipole moment.

Since the center of gravity of the negative charge and the center of gravity of the positive charge do not coincide, the total charge distribution of the molecule has a dipole moment. Such a molecule is called a polar molecule. In oxygen, because of the symmetry of the molecule, the centers of gravity of the positive and negative charges are the same, so it is a nonpolar molecule. It does, however, become a dipole when placed in an electric field. The forms of the two types of molecules are sketched in Fig.

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We will first discuss the polarization of non polar molecules. We can start with the simplest case of a monatomic gas for instance, helium. When an atom of such a gas is in an electric field, the electrons are pulled one way by the field while the nucleus is pulled the other way, as shown in Fig. Although the atoms are very stiff with respect to the electrical forces we can apply experimentally, there is a slight net displacement of the centers of charge, and a dipole moment is induced. For small fields, the amount of displacement, and so also the dipole moment, is proportional to the electric field.

The displacement of the electron distribution which produces this kind of induced dipole moment is called electronic polarization.


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I, when we were dealing with the theory of the index of refraction. If you think about it for a moment, you will see that what we must do now is exactly the same as we did then. But now we need worry only about fields that do not vary with time, while the index of refraction depended on time-varying fields. For our purposes, however, we are interested only in the case of constant fields, i.

It is a measure of how easy it is to induce a moment in an atom with an electric field. Comparing Putting From Eq. Our formula is, of course, only a very rough approximation, because in Eq. For example, we have assumed that an atom has only one resonant frequency, when it really has many.

Suppose we try hydrogen. We should not expect any better, because the measurements were, of course, made with normal hydrogen gas, which has diatomic molecules, not single atoms. We should not be surprised if the polarization of the atoms in a molecule is not quite the same as that of the separate atoms. The molecular effect, however, is not really that large. In any case, it is clear that our model of a dielectric is fairly good.

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Another check on our theory is to try Eq. So, from So we have understood the dielectric constant of nonpolar gas, but only qualitatively, because we have not yet used a correct atomic theory of the motions of the atomic electrons. With no electric field, the individual dipoles point in random directions, so the net moment per unit volume is zero. But when an electric field is applied, two things happen: First, there is an extra dipole moment induced because of the forces on the electrons; this part gives just the same kind of electronic polarizability we found for a nonpolar molecule.

For very accurate work, this effect should, of course, be included, but we will neglect it for the moment.

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It can always be added in at the end. Second, the electric field tends to line up the individual dipoles to produce a net moment per unit volume. If all the dipoles in a gas were to line up, there would be a very large polarization, but that does not happen. At ordinary temperatures and electric fields the collisions of the molecules in their thermal motion keep them from lining up very much. But there is some net alignment, and so some polarization see Fig.

To use this method we need to know the energy of a dipole in an electric field. As we would expect, the energy is lower when the dipoles are lined up with the field. We now find out how much lining up occurs by using the methods of statistical mechanics.

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The same arguments would say that using Eq. We can evaluate the sum by integrating over the angular distribution. Also, as we expect, the polarization depends inversely on the temperature, because at higher temperatures there is more disalignment by collisions. We should now try to see how well Eq. The dielectric constant has been measured at several different pressures and temperatures, chosen such that the number of molecules in a unit volume remained fixed. There is another characteristic of the dielectric constant of polar molecules—its variation with the frequency of the applied field.

Due to the moment of inertia of the molecules, it takes a certain amount of time for the heavy molecules to turn toward the direction of the field.


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    4. So if we apply frequencies in the high microwave region or above, the polar contribution to the dielectric constant begins to fall away because the molecules cannot follow. In contrast to this, the electronic polarizability still remains the same up to optical frequencies, because of the smaller inertia in the electrons.

    We now turn to an interesting but complicated question—the problem of the dielectric constant in dense materials. Suppose that we take liquid helium or liquid argon or some other nonpolar material. We still expect electronic polarization.

    The question is, what electric field acts on the individual atom? Imagine that the liquid is put between the plates of a condenser. If the plates are charged they will produce an electric field in the liquid. This true electric field varies very, very rapidly from point to point in the liquid. It is very high inside the atoms—particularly right next to the nucleus—and relatively small between the atoms. The potential difference between the plates is the line integral of this total field.

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    This is the field we were using in the last chapter. We should think of this field as the average over a space containing many atoms.