## Dielectrics

J. C. Daly
October 10, 2012

 Figure 1   A parallel plate capacitor is shown. The insulating dielectric between the plates is polarizes under the influence of the electric field produced by the free charge, Qf , on the plates. Figure 2   When a dielectric is uniformly polarized, a net polarization charge, Qb exists only on the surface of the dielectric.
Figure 1 shows a parallel plate capacitor. Positive free charge, Qf , on the upper plate produces an electric field. The electric field causes the dielectric material between the plates to polarize generating a polarization charge, Qb. Within the dielectric where the polarization is uniform, the positive and negative polarization charges cancel leaving no net charge.

The component of the electric field produced by the free charge, Qf , is Eo.

Recall Gauss' law.

 (1)
where Qf   is the free charge enclosed within an arbitrary surface s. It is the source of Eo.

The bound charge, Qb , reduces the electric field. The source of the total electric field, E , is both the bound and the free charge,

 (2)
where E is the total electric field.

For linear materials the bound charge is proportional to the free charge. Also, the total electric field is proportional to the electric field produced by the free charge.

 (3)
k is the dielectric constant. It is greater than 1.

Define the electric displacement vector, D

 (4)
where .   is the permittivity of the material. is the permittivity of free space.

Combining equations,

 (5)
The source of the D field is only the free charge Qf enclosed within the surface s.
 (6)
where v is the volume within the surface s. r   is the density of the free charge, coulombs per meter.

Gauss' law is usually written as,

 (7)
In solving problems where a dielectric is present partially or completely filling the area, the dielectric can be ignored and Gauss' law (equation 7) can be used to find D . Then the electric field can be calculated using equation 4.