The dielectric constant

We divide matter into two categories: conductors and insulators. Free charges in a conductor will respond to exactly cancel an applied field. The charges in an insulator will respond to an applied field in such a way as to partially cancel an applied electric field. The situation in an insulator is more complicated, however, since a molecule in the insulator will also experience a field due to the response of the insulator. There is a reaction field due to the response of the medium to charges on the molecule and there is a local field due to polarization of the solvent in the applied field. These issues are important for relating the molecular polarizability to the bulk polarization. Here we shall demonstrate the role of the dielectric constant (also called the relative permittivity) as a factor that relates the polarization of an insulator to an applied electric field.

The experimental geometry that is most convenient for the purpose of this demonstration is the parallel plate capacitor. We compare a capacitor with vacuum between the charged plates to one with a dielectric. In the case of vacuum the field is

The surface charge density is s = q/A. The vacuum permittivity is e0. The potential is f and the distance between the plates is d. The unit vector z is normal to the capacitor plates. These features are illustrated below.

Note that the potential is just the voltage. Calculating the field is very easy. It is just the voltage/distance. Common units of field are V/cm or V/m. If we now place a dielectric medium inbetween the plates at constant voltage we have

Note that the field is unchanged. It is still the voltage divided by the distance. However, the surface charge density required to attain this field is different (s' instead of s) because the medium has a permittivity e. Note that since the surface charge density increases when the dielectric is present the capacitance also increases.

The relationship between the dielectric constant and the permittivity of vacuum is e = ere0. The relative permittivity er is commonly called the dielectric constant. The dielectric constant is greater than 1 and can be as large 111.0 for formamide (see Table 3.1 in Molecular Spectroscopy by McHale). There is a rough correlation of the dielectric constant with the dipole moment. The larger the dipole moment the greater the tendency of the solvent to respond to an applied field by reorientation of the microscopic dipoles. However, inspection of Table 3.1 shows that there are exceptions and that liquid structure and collective dynamics also play a role.

To see the connection between the dielectric constant and the polarization we perform an experiment. We charge up the capacitor in vacuum. The field is E0 = s/e0. Then we add an insulating medium with dielectric constant er leaving the charges constant. Now the field is E = s/e = s/ere0. The field is reduced in this case because the amount of charge is kept constant. The difference between E and E0 is due to the polarization of the medium.

We call E the macroscopic field. The polarization is proportional to this field.

where ce is the electric susceptibility.