Dielectric materials

Up to this point, we have examined the behavior of electric fields in a vac-uum. The results were correct but we may now be wondering what the effects of applying the electric field in a material would be. The wearing of rubber gloves seems to have some desirable protective features when one is close to touching a high voltage line. Manufacturers of capacitors or integrated circuits usually insert an oxide layer between the two metal surfaces in order to keep the top conductor from falling down and touching the bottom conductor. How do these materials affect the electric field? Some answers will be provided here.

Figure 2-19. A material is placed between two electrodes that are separated by a distance L. An electric field is applied between the two electrodes. (a) Random orientation of the atoms before the application of the electric field.

(b) Reorientation of the atoms after the application of the electric field.

As noted earlier, materials consist of atoms and in a simple model, these atoms can be considered to be a large collection of randomly oriented small electric dipoles as shown in Figure 2-19. Certain molecules, called polar molecules normally have a permanent displacement between the positively charged nucleus at the center of the atom and the negatively charged electron at the edge. This distance is of the order of 10^-10 meters. This distance is also

+ - + - + -+ - + - + -+ - + - +

-+ -+

-+

-+ -+ -- +

+

-+ -+

-(a) (b)

E

equal to 1 Å in honor of the scientist Anders Jonas Ångstrom. Each pair of charges acts as an electric dipole. If an electric field is externally applied to this material, then the dipoles may reorient themselves. If the field is strong enough, there will actually be an additional displacement of the positive and negative charges. A nonpolar molecule does not have this dipole arrangement of charges unless an external electric field is applied. The positive and negative charges separate by a certain distance after the application of the electric field.

In some materials, the dipoles may reorient themselves such that a large number or even all of the atoms will realign themselves causing the electric field created by the dipoles to add to the applied electric field. In other materials, the reorientation may cause the dipole electric field to subtract from the applied field.

This dipole field created by the atoms will be examined here.

After the application of the electric field between the two electrodes in Figure 2-19b, the atoms have been reoriented. Since the atomic distances depicted in this figure, it is possible to regroup the electric dipoles and suggest that the positive charge of one atom could unite with the negative charge of the adjacent atom in order to form a new distribution of electric dipoles as depicted in Figure 2-20. This regrouping of the electric dipoles will leave a thin layer of charge of the opposite sign at either edge of the material. This charge which is due to the application of the electric field is called the polarization charge. The polarization charge cannot be found in a vacuum and it does not come out of the battery. It is only due to the fact that the atoms had been reoriented due to the application of the electric field. We will define a polarization charge density using the symbol Up as being the polarization charge per unit volume.

Figure 2-20. The reorientation of the atoms in a material due to the application of an electric field creates polarization charge at the two edges whose density is UP. This polarization charge creates a polarization field P

In the region between the two dashed lines, a positive nucleus of one atom "pairs" with an electron of the adjacent atom. The positive and negative charge centers overlap. However, in the region between the left electrode and the dashed line, there are more positively charged particles. In the region be-tween the second dashed line and the right electrode, there are more negatively charged particles. This effectively states that there is a very narrow region of charge of one sign that has migrated to that edge of the dielectric while there is a narrow region of charge of the opposite sign that has migrated to the other edge of the dielectric. Between these two edges, a charge-neutral region exists. This displaced charge cannot be removed from the material, it is bound to the material. It is given the name of a polarization charge. Herein, we will just describe the polarization charge at the surfaces that is called the surface polarization charge. The density of this polarization charge has the symbol ȡ^P and it is shown in Figure 2-19b. This bound charge will set up a field that is called the polarization field P and it is defined as the dipole moment per unit volume. It is written via the relation

+

-+

+ + +

-P

¿¾

½

¯®

­ ' o

'

¦

v j ₁

1 0 v

lim

pj

P (2.86)

where p_j = Qdu_d is the dipole moment of an individual dipole. The units are (C-m) / m³ = (C / m²). Within the volume ǻv, there are N atoms. With the notation given in (2.86), we see that the polarization field depends on position since we have let the differential volume ¨v shrink to zero. In Figure 2-20, this would imply that the distance separating the two thin layers of polarization charge shrinks to zero. In analogy with Gauss's law, we can relate the polarization charge ȡp to an electric field. This field is called the polarization P and we write

UP = -’ x P (2.87)

Let us add the polarization charge density ȡp to the real charge density ȡv. The real charge density could come from a battery or from the ground. This will dramatically influence the resulting electric field that we calculated from (2.26)

o P v

H U

 x U

’ E (2.88)

Replacing the polarization charge density in (2.88) with (2.87), we finally obtain

’ x D = Uv (2.89)

where

D = HoE + P (2.90)

is called the electric flux density or the displacement flux density. The unit of this quantity is also (C / m²). The total electric flux Ȍe that passes through a surface equals the surface integral of the electric flux density integrated over the surface 's

³

Ȍe D ds (2.91)

Note that the displacement flux density has a significant meaning only when materials that can be polarized are discussed. In a vacuum, it is just equal to a constant İr times the electric field.

Gauss’s law which was used to compute the electric field in a vacuum can be employed to calculate the displacement flux density with the same restrictive limitations of symmetry requirements that were encountered previously. The procedure to develop this equation follows directly from an integration of (2.89) over the same volume. The volume integration of the divergence of the displacement flex density can be converted to a closed surface integral using the divergence theorem. The result of this is

Qenc

³

Therefore, the total dielectric flux emanating from or terminating on a closed surface 's is equal to the total charge that is enclosed within this surface.

A dielectric material is susceptible to being polarized. In many materials, this polarization is linearly proportional to the applied electric field if the electric field remains small. In these cases, we can write that P = İoȤe E where Ȥe is the electric susceptibility. Finally, we obtain

D = İo (1 + Ȥ_e)E = İoİr E = İ E. (2.93) The term İr is the relative dielectric constant for a material. Tabulated values of İr for various materials are given in Appendix 3. In a vacuum, Ȥ_e = 0 and İ_r = 1 by definition.

The expression (2.93) applies only for linear and isotropic materials. It is not difficult to create a material that does not satisfy this criterion. For example, the application of an external magnetic field to an ionized gas will make it anisotropic. Large amplitude signals that are applied to a material may cause the material to have a nonlinear response. This case could occur if the relative dielectric constant changed, say due to the dielectric being modified where the modification was proportional to the square of the magnitude of the applied electric field |E|2. Such nonlinear materials do exist and are currently under

active investigation. In what follows, we will restrict our discussion to linear materials.

Example 2-17. A dielectric slab is placed between two parallel plates. A battery is connected to one plate and the other plate is grounded. The area of each plate is equal to A and the charge on each plate is r . The separation of the plates Q is d. Sketch the following quantities between the plates:

a) surface charge density ȡs, b) displacement flux density D, c) electric field E,

d) polarization P, and

e) the bound surface polarization charge density ȡps.

Answer: a) The real charge Q can come from the battery or from ground. It will be distributed on the surface of the metal plates creating a surface charge density Us =

A Q.

b) The displacement flux density D will be determined by the real charge from the battery or from the ground. It will not depend on whether a dielectric or a vacuum exists between the plates. It follows from Gauss's law that D = Us.

c) The electric field E = D/(İoİr). Hence the electric field will be decreased within the dielectric below its value in the vacuum since İr > 1.

d) The polarization field P will exist in the dielectric. Its value will be de-termined from (2.90).

e) The bound surface polarization charge density ȡps can be evaluated from

A 1 Q A

A

Q _r

r p

PS

¸¸¹

¨¨ ·

©

§ H

H



 x

U P A

.

surface charge