John W. Dooley, Physics Department, Millersville University
PERTURBATION/RESPONSE
Our framework for studying the deformation of a solid was called a "perturbation/response" picture: The applied stress is the perturbation, and the resulting strain is the response. The elastic constants describe the relation between perturbation and response.
Recall that, while this cause/effect picture is useful for developing our understanding, it is less than the whole picture. The field picture of stress and strain and of fluid flow does not hold cause/effect information, even though it was derived from F=ma cause/effect ideas. The same will hold true for electric field perturbation and electric polarization response.
This time the perturbation is an electrical field, and the response is the (small scale) motion of charges in an insulating sample which experiences the field. As with strain, there is a particular measure of the response that works best. The name of this measure is the polarization.
To understand the problem, we begin with experimental results. We will find evidence that the polarization is proportional to the applied electric field. The constant of proportionality is called the polarizability. The most general mathematical representation of the polarizability is a polarizability tensor. Consideration of energy conservation in the polarization of a dielectric leads to the fact that the polarizability tensor is "symmetric" and also sets the stage for a famous problem called the Eigenvalue Problem.
AN EXPERIMENT
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| Insertion of plastic decreases capacitor potential difference. |
The figure illustrates a useful thought experiment: A parallel plate capacitor is charged so that the voltage difference from one plate to another is 
. The battery is removed. The capacitor is perfect, and so is the voltmeter, so that the charge placed on the plates never changes after we remove the battery.
A piece of (uncharged) plastic is slipped between the plates. When the plastic is inserted, the voltage across the capacitor decreases.
Insertion of plastic decreases capacitor potential difference
Subsequent experiments confirm that the charge on the capacitor did not change. (No charged hopped on or off the plastic.)
We will try to understand this in terms of the energy changes that occur. To begin, recall the way in which the voltage difference between plates is calculated. The magnitude of this difference is equal to the work per unit charge done on a small test charge, 
, in order to move it from one plate to the other. As usual, the work is
where 
is the x component of the electric field, 
is the separation of the capacitor plates, and is the (constant) test charge on which the work is done. Then the change in potential difference is
.
The electric field just at the inside surface of each metal plate is determined solely by the electrical charge, (of magnitude 
) on each plate:
where
.
The only way for the voltage to change is for the electric field to change. Since the field in the gap near the metal is determined by the charge distribution on the plate, that field does not change. It must be the field inside the plastic which changes.
The only easy to produce a static electric field is to use a distribution of electric charges. Since the plastic was uncharged when we put it in, and since there are no other sources of charge, we conclude that the required charge distribution is created by rearranging charges which are already in the plastic. When the charges are rearranged like this, we say that the plastic is polarized, and we say that the polarization was induced by the applied electric field.
DIELECTRIC POLARIZATION
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| A charge distribution which accounts for the reduction |
| in electric field within the plastic. |
Since the plastic reduced the voltage, the electric field in the space now occupied by plastic must be smaller than before the plastic was pushed in. Thus the polarization produces an electric field in the direction opposite to the original electric field in the capacitor.
The simplest charge distribution which produces such a field is sketched in the figure. If the edges of the capacitor are ignored, this distribution not only the simplest; it is correct.
A charge distribution which accounts for the reduction in electric field by the plastic.
The charge is distributed on the surface of the plastic, and the sign of the charge is opposite to the charge on the nearest metal plate. The plastic acts like a second capacitor, inside the original one. The main difference is that the second electric field points opposite to the original one.
The net effect is that the original field induces charge motion and a second electric field which points "backwards."
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| Dividing the plastic does not change the potential |
| difference across the capacitor. |
We now extend the experiment: Slice the plastic into pieces as shown in the figure. Experimentally, it is observed that the voltage on the capacitor does not change. The effect of the separate pieces in the same as if they were one piece. We conclude that
1) The polarization effect is local: Each bit of plastic is polarized when it is put into the electric field, and each bit acts to reduce the field inside of it.
2) The polarization effects are additive, so that the bits of material separately have the same effect as one large chunk.
Dividing the plastic does not change the potential difference across the capacitor.
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| Dipole moment of a small volume, induced by |
| an applied electric field. |
A small chunk of the plastic is shown in the figure. A useful way to characterize the charge distribution is to measure its dipole moment,
where 
has unit magnitude and points from the -Q towards the +Q.
As in an ordinary capacitor, the induced "backwards" field is proportional to the charge magnitude, Q1. Thus the backwards field is proportional to the induced dipole moment.
The dipole moment of a small volume induced by an applied electric field
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If we think of this experiment from the point of view of a piece of the plastic, the electric field (from the charge distribution on the metal plates of the capacitor) is the perturbation. The redistribution of charge in the dielectric material is the response. One measure of the size of the response is the size of the induced dipole moment in the chunk.
For simplicity, we imagine that the dipole moment is induced by moving positive charge from the right to the left of the chunk, leaving an equal amount of unbalanced negative charge behind.
The size of the dipole moment depends on how much charge is moved (and how far) by the perturbing electric field. This in turn depends on how free the charges are to move. The materials that give the largest response are metals. In metals, a large number of charges act as though they are free to move anywhere within the sample. For a metal, the charges move until the electric field inside the sample goes all the way to zero.
For insulators, there are charges that are relatively free to move short distances, but there are no charges that can easily be moved over macroscopic distances. When an insulator is polarized, we imagine that short-distance charge motion does the job throughout the sample, with the net effect of an excess positive charge on the right and negative charge on the left.
Because they tend to cancel the electric field inside them, these insulators are called dielectric materials. The fact that metals are perfect dielectric materials is usually not emphasized. When you hear someone speak of a dielectric, that person is almost certainly thinking of an insulating dielectric.
Return to index.Return to survey.
Go forward to polarizability discussion.