# 4.7 Dielectrics

The fundamental difference between a conductor and a dielectric is that electrons in the outermost atomic shells of a conductor are only weakly tied to atoms and hence can freely migrate through the material, whereas in a dielectric they are strongly bound to the atom. In the absence of an electric field, the electrons in nonpolar molecules form a symmetrical cloud around the nucleus, with the center of the cloud coinciding with the nucleus (Fig. 4-18(a)). The electric field generated by the positively charged nucleus attracts and holds the electron cloud around it, and the mutual repulsion of the electron clouds of adjacent atoms shapes its form. When a conductor is subjected to an externally applied electric field, the most loosely bound electrons in each atom can jump from one atom to the next, thereby setting up an electric current. In a dielectric, however, an externally applied electric field E cannot effect mass migration of charges since none are able to move freely. Instead, E will polarize the atoms or molecules in the material by moving the center of the electron cloud away from the nucleus (Fig. 4-18(b)). The polarized atom or molecule may be represented by an electric dipole consisting of charges +q in the nucleus and −q at the center of the electron cloud (Fig. 4-18(c)). Each such dipole sets up a small electric field pointing from the positively charged nucleus to the center of the equally but negatively charged electron cloud. This induced electric field, called a polarization field, generally is weaker than and opposite in direction to E. Consequently, the net electric field present in the dielectric material is smaller than E. At the microscopic level, each dipole exhibits a dipole moment similar to that described in Example 4-9. Within a block of dielectric material subject to a uniform external field, the dipoles align themselves linearly, as shown in Fig. 4-19. Along the upper and lower edges of the material, the dipole arrangement exhibits positive and negative surface charge densities, respectively.

# Figure 4-18 In the absence of an external electric field E, the center of the electron cloud is co-located with the center of the nucleus, but when a field is applied, the two centers are separated by a distance d.

(a) External Eext = 0

(b) External Eext ≠ 0

Figure 4 18-b Full Alternative Text

(c) Electric dipole

# Figure 4-19 A dielectric medium polarized by an external electric field E.

Figure 4 19 Full Alternative Text

It is important to stress that this description applies to only nonpolar molecules that do not have permanent dipole moments. Nonpolar molecules become polarized only when an external electric field is applied; when the field is removed, the molecules return to their original unpolarized state.

In polar materials such as water, the molecules possess built-in permanent dipole moments that are randomly oriented in the absence of an applied electric field, and owing to their random orientations, the dipoles of polar materials produce no net macroscopic dipole moment (at the macroscopic scale, each point in the material represents a small volume containing thousands of molecules). Under the influence of an applied field, the permanent dipoles tend to align themselves along the direction of the electric field in a manner similar to that shown in Fig. 4-19 for nonpolar materials.

# 4.7.1 Polarization Field

In free space D=ε 0 E, the presence of microscopic dipoles in a dielectric material alters that relationship to

D=ε 0 E+P,(4.93)

where P, called the electric polarization field, accounts for the polarization properties of the material. The polarization field is produced by the electric field E and depends on the material properties. A dielectric medium is said to be linear if the magnitude of the induced polarization field P is directly proportional to the magnitude of E, and isotropic if P and E are in the same direction. Some crystals allow more polarization to take place along certain directions, such as the crystal axes, than along others. In such anisotropic dielectrics, E and P may have different directions. A medium is said to be homogeneous if its constitutive parameters (ε, μ, and σ) are constant throughout the medium. Our present treatment will be limited to media that are linear, isotropic, and homogeneous. For such media, P is directly proportional to E and is expressed as

P=ε 0 χ e E,(4.94)

where χ e is called the electric susceptibility of the material. Inserting Eq. (4.94) into Eq. (4.93), we have

D=ε 0 E+ε 0 χ e E=ε 0(1+χ e)E=ε E,(4.95)

which defines the permittivity ε of the material as

ε=ε 0(1+χ e).(4.96)

It is often convenient to characterize the permittivity of a material relative to that of free space, ε 0; this is accommodated by the relative permittivity ε r=ε/ε 0. Values of ε r are listed in Table 4-2 for a few common materials, and a longer list is given in Appendix B. In free space ε r=1, and for most conductors, ε r≈1. The dielectric constant of air is approximately 1.0006 at sea level and decreases toward unity with increasing altitude. Except in some special circumstances, such as when calculating electromagnetic wave refraction (bending) through the atmosphere over long distances, air can be treated as if it were free space.

# Table 4-2 Relative permittivity (dielectric constant) and dielectric strength of common materials.

Note: ε = ε rε0 and ε0 = 8.854 × 10−12 F/m.

Table 4 2 Full Alternative Text

# 4.7.2 Dielectric Breakdown

The preceding dielectric–polarization model presumes that the magnitude of E does not exceed a certain critical value, which is known as the dielectric strength E d s of the material. Beyond this, electrons will detach from the molecules and accelerate through the material in the form of a conduction current. When this happens, sparking can occur, and the dielectric material can sustain permanent damage due to electron collisions with the molecular structure. This abrupt change in behavior is called dielectric breakdown.

Dielectric breakdown can occur in gases, liquids, and solids. The dielectric strength E d s depends on the material composition, as well as other factors such as temperature and humidity. For air, E d s is roughly 3 (MV/m); for glass, 25 to 40 (MV/m); and for mica, 200 (MV/m) (see Table 4-2).

A charged thundercloud at electric potential V relative to the ground induces an electric field E=V/d in the air beneath it, where d is the height of the cloud base above the ground. If V is sufficiently large so that E exceeds the dielectric strength of air, ionization occurs and a lightning discharge follows. The breakdown voltage V b r of a parallel-plate capacitor is discussed in Example 4-12.

# Example 4-12: Dielectric Breakdown

In a parallel-plate capacitor with a separation d between the conducting plates, the electric field E in the dielectric material separating the two plates is related to the voltage V between the two plates by

E=V d.

The breakdown voltage V br corresponds to the value of V at which E=E d s, where E d s is the dielectric strength of the material contained between the plates. That is,

V br=E d s d.

If V exceeds V br, the electric charges will “spark” their way between the two plates.

A thin capacitor filled with quartz operates at 60 V. If d=0.01 mm, what is the breakdown voltage, and how does it compare with the operating voltage?

Solution: From Table 4-2, E d s=30×10 6 V/m for quartz. Hence, the breakdown voltage is

V br=E d s d=30×10 6×10−5=300 V,

which is much higher than the operating voltage of 60 V. Therefore, the capacitor should experience no issues with dielectric breakdown.

Concept Question 4-19: What is a polar material? A nonpolar material?

Concept Question 4-20: Do D and E always point in the same direction? If not, when do they not?

Concept Question 4-21: What happens when dielectric breakdown occurs?