PERTURBATION/RESPONSE PARAMETERS

John W. Dooley, Physics Department, Millersville University

Polarization

The description of a dielectric material responding to an applied electric field is analogous to the description of an elastic solid in which the stress was the perturbation and the displacement field was the response.

As with the solid, there are several ways to measure the response. In the solid, the strain was preferred because with strain we could write a generalized form of Hooke's law.

For the electric field perturbation, the preferred measure of response is the induced dipole moment of a unit volume of the sample. The dipole moment per unit volume is called the polarization. It is a vector, and we can use the symbol , even though we have used the scalar P for pressure. The polarization is a vector field, a function of position in space:

The value of changes from point to point in space. If the volume, , of a sample is small enough, the dipole moment of the sample volume is

More generally, allowing for the spatial variation of , the dipole moment of a sample volume is

where is a volume element.

Since the units of electric dipole moment are Coulomb – meter, the units of electric polarization must be

While the polarization vector is generally a vector field whose value depends on position in space, we have chosen an idealized capacitor, in which the field is the same everywhere as long as we stay between the two capacitor plates. We will continue with this simple case because it is illustrative of the general case. To go further, we must first return to experimental results.

ISOTROPIC POLARIZIBILITY

First we fill a capacitor completely with a dielectric so that we have only one field to calculate. Then repeat the first polarization experiment, moving the (plastic) dielectric slab in and out of the gap between the plates. As before, for a given charge, the filled capacitor has a smaller voltage difference than the empty capacitor.

Now repeat the experiment for a variety of empty-capacitor voltages; that is for a variety of charge quantities on one plate of the capacitor.

The ratio of empty to full capacitor potential difference

is the same no matter what charge is on the capacior.

The experimental plot of potential difference versus charge is shown in the figure. Also shown in the observed ratio of the two measured voltages. The ratio is constant. The filled capacitor voltage is proportional to the empty capacitor voltage (for the same charge on each).

This means that the electric field due to the polarization of the dielectric is proportional to the applied electric field.

The ratio of empty to full capacitor potential difference is the same, no matter what charge is on the capacitor.

In the simplest case, the dielectric is isotropic, so that the constant of proportionality is independent of direction:

or, in subscript notation

The constant of proportionality, , is called the dielectric polarizability.

The electric field used in this equation could be either the field of the empty capacitor or of the full capacitor (since the two are proportional to each other). We follow tradition and say that is the field of the full capacitor; the field which exists after the charges in the dielectric have responded to the perturbation.

This equation works experimentally for many substances. Note that unlike the stress/strain situation, there is no side effect in an isotropic material. The side effect in strain reflected a tendency for the material to maintain a constant volume, even for isotropic materials. Evidently there is no analogous tendency in the polarization of isotropic dielectrics.

ANISOTROPIC POLARIZABLITY TENSOR

As with stress/strain, the situation is more complicated for anisotropic materials. However, since the perturbation and the response are only vectors, the description is less complicated than the second order tensors which describe stress and strain.

In the most general case each polarization component is related by a (different) constant to each electric field component. The net polarization is the sum of all the polarization components caused by each component of the electric field. This collection of terms is nicely collected in terms of the components of the polarization: