3. The electric field in dielectrics

As already noted earlier, under ordinary conditions dielectrics practically have no free charges, and therefore they are poor conductors of electric current (insulators) (Fig. 3.1). One might think that when a dielectric is placed in an electric field, nothing happens at all. However, this is not so: experiments have shown that the absence of free charges in dielectrics does not at all prevent them from reacting to an external field.

Fig. 3.1. Samples of dielectrics

3.1. Permittivity

If, for example, a plane-parallel dielectric plate is placed in a uniform electric field (as in an ideal parallel-plate capacitor) so that its surfaces form some angle with the direction of the field, and thus do not coincide with its equipotential surfaces, then the value of the field inside this plate will depend on the angle in a rather complicated way, and will be equal to only when . Nor should one think that introducing a dielectric into a field always leads to a decrease in the field strength — it may also increase: it all depends on the "geometry" of the problem. Figure 3.4 below shows that when a thin, long dielectric rod is placed in an electric field parallel to the field lines of the external field, the field strength outside the rod near its ends increases as a result of the appearance of "polarization" charges at the ends of the rod.

Fig. 3.4. Field strength on the axis of a thin dielectric rod

We observed a decrease in the potential difference between the plates and an increase in the capacitance of the capacitor in the problem solved above concerning a spherical capacitor with a metallic shell between the plates. There the reason for the decrease in the potential difference was clear: induced charges appeared on the shell, which compensated the external field from the plates. Accordingly, the electric field existed only in the space not occupied by the shell. If the shell occupied the entire volume of the capacitor, the potential difference between the plates and the field inside it would become zero.

There are no charges in a dielectric that are able to move throughout its entire volume, but the idea that additional charges of some kind arise on its surface (in this case they are called polarization or bound charges) seems attractive because of the possibility of explaining the experimental facts. We therefore adopt a macroscopic model, which of course must subsequently be justified at the microscopic level and verified in practice together with all its consequences. We shall assume that when a dielectric is placed in an electric field, polarization charges with density arise on its surface (Fig. 3.5).

Fig. 3.5. A spherical particle in a uniform electric field of strength E.
The signs "+" and "–" show the bound charges that arise on the surface of the particle upon its polarization.
The electric forces acting on the positive (F₊) and negative (F_–) bound charges are equal

The polarization charges create an additional electric field , directed opposite to the field from the charges on the plates (see Fig. 3.3). This explains the smaller magnitude of the resulting field E compared to the field E₀. Indeed, for the simplest geometry of a parallel-plate capacitor (see the remark above about the shape of the dielectric surface), the change of the field in the dielectric reduces simply to a change in the magnitude of its strength by a factor of

From this we find what part of the resulting field is created by the polarization charges, and what part by the charges on the plates

The negative sign indicates the opposite direction of the field of the polarization charges. Knowing the relation between the surface charge density and the field strength it creates

We find the density of the polarization charges

Note that the case of a conductor corresponds to the limit

Indeed, in that case , and the field inside the material is completely compensated, so we obtain

whence

Values of e for some dielectrics are given in the table (for gases — at standard conditions).

Table

Values of permittivity for some substances

Note that the same substances have different dielectric properties under different conditions. This means that to explain them, a theory must be constructed at the microscopic level, based on the properties of atoms and molecules and taking into account the state of the substance.

3.2. The electric dipole

which goes toward increasing its potential energy. From this we obtain the potential energy of a dipole in an electric field

whence

or

if we set const = 0.

From the figure it is evident that the external electric field tends to rotate the dipole so that the vector of its electric moment p coincides in direction with the vector E. In this case , and, consequently, M = 0 as well. On the other hand, at the potential energy of the dipole in the external field takes its minimum value , which corresponds to a position of stable equilibrium. If the dipole is displaced from this position, a mechanical torque again arises that returns the dipole to its original position. Another equilibrium position, in which the dipole moment is directed against the field is unstable. In this case the potential energy takes its maximum value , and for small deviations from this position the resulting forces do not return the dipole but deflect it even further.

Fig. 3.11 shows an experiment illustrating the appearance of a torque of electric forces acting on a dielectric in an electric field. A torque acts on an elongated dielectric sample positioned at some angle to the field lines of the electrostatic field, tending to turn this sample so as to align it with the field. A dielectric rod, suspended at its midpoint inside a parallel-plate capacitor, turns perpendicular to its plates after a high voltage from an electrostatic machine is applied to them. The appearance of the torque is due to the interaction of the polarized rod with the electric field of the capacitor.

Fig. 3.11. Torque of electric forces acting on a dielectric in an electric field

In the case of a nonuniform field, the dipole under consideration will also be acted on by a resultant force F_res, tending to displace it. We shall consider here a particular case. Let us direct the x-axis along the field E. Suppose that under the action of the field the dipole has already turned to align with the field line, so that the negative charge is located at the point with coordinate x, and the positive charge is located at the point with coordinate x + l. Let us imagine that the magnitude of the field strength depends on the coordinate x. Then the resultant force F_res is equal to

The same result can be obtained from the general relation

where the energy Π is defined in (3.8). If E increases with increasing x, then

and the projection of the resultant force is positive. This means that it tends to draw the dipole into the region where the field strength is greater. This explains the well-known effect whereby neutral pieces of paper are attracted to an electrified comb. In a parallel-plate capacitor with a uniform field they would remain motionless.

Let us consider several experiments illustrating the appearance of a force acting on a dielectric placed in a nonuniform electric field.

Fig. 3.12 shows the dielectric being drawn into the space between the plates of a parallel-plate capacitor. In a nonuniform electrostatic field, forces act on the dielectric that draw it into the region of the stronger field.

Fig. 3.12. Pulling a liquid dielectric into a parallel-plate capacitor

This is demonstrated using a transparent vessel containing a parallel-plate capacitor with a certain amount of liquid dielectric — kerosene — poured into it (Fig. 3.13). The capacitor is connected to a high-voltage power source — an electrostatic machine. When it operates, at the lower edge of the capacitor, in the region of the nonuniform field, a force acts on the kerosene, pulling it into the space between the plates. Therefore, the level of kerosene inside the capacitor settles higher than outside. After the field is switched off, the level of kerosene between the plates drops back to its level in the vessel.

Fig. 3.13. Pulling kerosene into the space between the plates of a parallel-plate capacitor

Video 3.1. Experiment on pulling a liquid dielectric into a capacitor.

In real substances, dipoles formed by only two charges are rarely encountered. Usually we are dealing with more complex systems. But the concept of the electric dipole moment applies to systems with many charges as well. In this case the dipole moment is defined as

where , — the magnitude of the charge numbered i and the radius vector determining its location, respectively. In the case of two charges we arrive at the earlier expression

Let our system of charges be electrically neutral. It contains positive charges, whose magnitudes and locations we denote with the index «+». With the index «–» we denote the absolute magnitudes of the negative charges and their radius vectors. Then expression (3.10) can be written as

In (3.11) the first term is summed over all positive charges, and the second — over all negative charges of the system.

The electrical neutrality of the system means that the total positive charge equals the sum of the absolute magnitudes of all negative charges

Let us now introduce the concept of the «center of charges» — positive R⁺ and negative R^–

Expressions (3.13) are analogous to the formulas for the center of mass in mechanics, and hence we call them the centers of positive and negative charges, respectively. With these notations and taking into account relation (3.12), we write the electric dipole moment (3.11) of the system of charges as

where l is the vector drawn from the center of negative charges to the center of positive charges. The point of our exercise is to demonstrate that any electrically neutral system of charges can be represented as some equivalent dipole.

3.3. Polarization of dielectrics

In accordance with formula (3.8), the potential energy of a dipole in an external field E depends on the orientation of the dipole

According to the statistical Boltzmann law (Fig. 3.19), which describes the distribution of particles over energies in an external field under conditions of thermodynamic equilibrium, the number of molecules whose dipole moment is oriented at an angle , to the external field, is determined as

(3.17)

Here C — is a normalization constant, whose value we will find later, T — is the absolute temperature, and the Boltzmann constant — k_B = 1.38·10^–23 J/K. Owing to the smallness of the dipole moment of the molecules, for ordinary (not too low) temperatures the exponent is small, and we can expand the exponential in a Taylor series, keeping the first two terms

(3.18)

Fig. 3.19. L. Boltzmann (1844–1906) — Austrian physicist

We emphasize that the use of the approximate expression (3.18) and of all the conclusions following from it is justified at not too low temperatures, when . An exact calculation using (3.17) instead of the approximate (3.18) presents no difficulty, and the reader may carry it out independently.

The integral over the full solid angle must give the total number N of molecules in the system. Since the average value of the cosine is zero, only the first term in (3.18) is integrated. Since the value of the full solid angle is , we obtain

Now we know the constant C and can write expression (3.18) as

(3.19)

We need to determine the value of the projection of the total dipole moment onto the direction of the field (the other projections are obviously zero owing to the axial symmetry of the problem). The projection of the dipole moment of a single molecule equals pcosa, hence the total dipole moment P of all molecules in a unit volume equals

(3.20)

The integral over equals , and the integral over is evaluated using a change of variable

We then find

(3.21)

From (3.21) it follows that in the case of dipolar orientational polarization of a substance as well, the polarization is proportional to the strength of the electric field. Moreover, we have found the dependence of the polarization on temperature. This is Curie's law, which is confirmed experimentally (Fig. 3.20).

Fig. 3.20. Dependence of the polarization of a polar dielectric on temperature (exact solution)

Summarizing this section, we briefly repeat the main conclusions. An external electric field either creates dipole moments oriented along the field, or orients the dipole moments of individual molecules, and the dielectric acquires a certain macroscopic dipole moment. The vector is called the polarization of the dielectric. It is proportional to the strength of the external electric field, and this relation can be represented as

(3.22)

where — is the proportionality coefficient (it is called the dielectric susceptibility). The coefficient is proportional to the concentration of particles in the dielectric and, in the case of a polar dielectric, depends on its temperature. Since the dimension of the dipole moment in SI is

the polarization vector in SI is measured in C/m². Its dimension coincides with the dimension of surface charge density. This suggests that the polarization vector is related to the density of polarization charges arising on the surface and in the volume of a dielectric placed in an external field (Fig. 3.21).

Fig. 3.21. Polarization vector and density of polarization charges

The proportionality between the polarization P and the strength E of the external electrostatic field is explained, in the case of electronic and ionic polarization, by the fact that as E increases, the dipole moments of individual atoms p_i grow. In dipolar polarization, the degree of orientation of the vectors p_i increases in proportion to the increase in the strength of the external electrostatic field. Above we found general formulas for the dielectric susceptibility for different types of polarization. It should be emphasized that they are valid for gases: we did not take into account the influence of molecules on one another, which is permissible for systems where the particles are not too densely packed. But the general conclusion remains valid for condensed media (liquids and solids) as well: under the action of an external electric field a unit volume of dielectric acquires a dipole moment P; in the simplest cases there is a linear dependence

All three mechanisms considered contribute to the total dielectric susceptibility of a dielectric:

It is usually rare for all the contributions to the dielectric susceptibility to be equally large. For example, in ionic crystals the dipolar part is absent altogether. Experimentally, the contribution of each part can be found by measuring the dielectric permittivities at different frequencies of the electromagnetic wave. At low frequencies (the static field, which we are now dealing with), all three contributions to the dielectric susceptibility make a contribution (Fig. 3.22).

Fig. 3.22. Dependence of the total dielectric susceptibility of a dielectric
on the frequency of the electromagnetic wave. The frequency ranges shown are:
I — the radio and microwave region, II — the infrared region, III — the ultraviolet region

As the frequency increases, the contribution of the dipolar part disappears first: the molecules will not manage to turn fast enough to follow the rapidly changing electric field of the wave. The transition to the new regime usually takes place at radio-range frequencies. With a further increase in frequency, the contribution of the ionic part disappears: ions are more inertial than electrons. In the optical frequency range the electronic part of the polarization dominates. On going to still higher frequencies — beyond the ultraviolet region — even the electron clouds will not manage to follow the changes of the electric field, and the polarizability of the dielectric vanishes.

Let us give an example: for common salt NaCl the dielectric permittivity in a static field equals 5.62, while in the field of an electromagnetic wave in the optical range it is only 2.25. Dipolar polarizability is absent in such crystals, and the difference should be attributed to ionic polarizability.

3.4. The electric displacement vector

Having dealt with the behavior of a dielectric at the microscopic level, let us return to the parallel-plate capacitor shown in Fig. 3.3. Where, then, did the polarization charges on the surface of the dielectric slab between the plates come from?

We now know that in the external field created by the plates, a unit volume of dielectric acquires a dipole moment P. Say, the positive charges shift in the direction of the field (upward in Fig. 3.3), and the negative ones — downward. With complete uniformity of the field and of the dielectric, no uncompensated volume charges appear inside the dielectric. But such a shift leads to the appearance of uncompensated charges on the surface of the dielectric slab. The dipole moment of the slab equals VP, where V = Sd — is its volume. On the other hand, the total surface charge on the slab equals

and the distance between the centers of positive and negative charges equals d (see Fig. 3.3). Therefore the dipole moment of the slab can also be written as

Comparing these two expressions, we find the relation between the surface density of polarization charges and the polarization vector

The strength E of the total field inside the dielectric is smaller than the strength of the field E₀, created by the plates. It is precisely the field E that acts on the molecules of the dielectric, it is precisely this field that they «feel», and therefore relation (3.22) holds for it

Using relation (3.3) between the field strength E ' of the polarization charges and the total field E

we find the relation between the dielectric permittivity and the dielectric susceptibility

In the general case the polarization vector P is not parallel to the strength vector of the total field E: in anisotropic dielectrics the polarization vector may rotate relative to the field strength. However, we can always write the relation

In the particular case of a linear dependence of the polarization on the field strength

the electric displacement vector equals

where — is the dielectric permittivity of the medium. The relation

holds for isotropic dielectrics. In the general case the vector D is not parallel to E. The field of the vector D can be represented graphically by lines of electric displacement, which are defined in the same way as the lines of electric field strength (Figs. 3.23 and 3.24).

Fig. 3.23. Conditions at a plane boundary between two dielectrics for the field strength and the electric displacement

Fig. 3.24. Lines of field strength and electric displacement of the electric field
from a point charge located at the interface between two dielectrics

3.5. Electrostatics of dielectrics

Let us apply the Ostrogradsky–Gauss theorem to the electric field in a dielectric. The flux of the strength vector through a closed surface is proportional to the algebraic sum of the charges (free and polarization) located inside the volume bounded by this surface

where q_i — are the free, and q '_i — the polarization charges. This expression is inconvenient, since it includes polarization charges, which, in turn, depend on the strength of the electric field at the given point of the dielectric.

Let us now consider the flux of the electric displacement vector

Since the strength of the field of the polarization charges can be written as

then

Consequently,

from which

where q_i — are the free charges. It should be emphasized that the lines of the vector D can begin and end on free charges, but not on polarization charges.

Attention should be drawn to the absence, in the right-hand side, of the factor , which is present in the analogous expression for the flux of the strength vector in vacuum.

From the Ostrogradsky–Gauss theorem for a point charge q inside a dielectric it follows that

The vector D does not determine the force acting on a charge from the external electric field. The force characteristic, as before, is , that is . For a linear dependence of on , to calculate the force one should use the relation

from which

Let us now obtain Coulomb's law for such dielectrics. A free charge q₂ creates in the dielectric an electric displacement

from which follows the expression for the force of interaction with another free charge q₁

Accordingly, the expression for the potential created by a free charge q changes

and, as a consequence, so do the formulas for the work of moving a free charge in a field and for the energy of interaction of free charges. We note that, compared with the analogous formulas for systems of charges in vacuum, for dielectrics one must make the substitution Since the expressions given served as the primary source for all the other relations we derived for vacuum, we immediately obtain, for example, the expressions for the capacitances of a parallel-plate (2.12), cylindrical (2.14), and spherical (2.17) capacitor filled with a homogeneous dielectric (Figs. 3.25, 3.26, 3.27, 3.28)

Fig. 3.25. The basis of a capacitor's construction — two conducting plates with a dielectric between them

Fig. 3.26. A parallel-plate capacitor with a dielectric

Fig. 3.27. A cylindrical capacitor with a dielectric

Fig. 3.28. A spherical capacitor with a dielectric

For the energy density of the electric field (2.57) we can now write the expression

which can be represented in vector form: