3.2.5 Summary and Generalization

For all three cases of polarization mechanisms, we had a linear relationship between the electrical field and the dipole moment (for fields that are not excessively large):

Electronic polarization

Ionic polarization

Orientation polarization

m_EP = 4p · e₀ · R³ · E

m_IP =	q² k_IP	· E

m_op =	m² 3kT	· E

It seems on a first glance that we have justified the "law" P = c · E.

However, that is not quite true at this point. In the "law" given by equation above, E refers to the external field, i.e. to the field that would be present in our capacitor without a material inside.

We have E_ex = U / d for our plate capacitor held at a voltage U and a spacing between the plates of d.

On the other hand, the induced dipole moment that we calculated, always referred to the field at the place of the dipole, i.e. the local field E_loc. And if you think about it, you should at least feel a bit uneasy in assuming that the two fields are identical. We will see about this in the next paragraph.

Here we can only define a factor that relates µ and E_loc; it is called the polarizability a. It is rarely used with a number attached, but if you run across it, be careful if e₀ is included or not; in other words what kind of unit system is used.

We now can reformulate the three equations on top of this paragraph into one equation

m = a · E_loc

The polarizability a is a material parameter which depends on the polarization mechanism: For our three paradigmatic cases they are are given by

a_EP	=	4p · e₀ · R³

a_IP	=	q² k_IP

a_OP	=	m² 3kT

This does not add anything new but emphasizes the proportionality to E.

So we almost answered our first basic question about dielectrics - but for a full answer we need a relation between the local field and the external field. This, unfortunately, is not a particularly easy problem

One reason for this is: Whenever we talk about electrical fields, we always have a certain scale in mind - without necessarily being aware of this. Consider: In a metal, as we learn from electrostatics, there is no field at all, but that is only true if we do not look too closely. If we look on an atomic scale, there are tremendous fields between the nucleus and the electrons. At a somewhat larger scale, however, they disappear or perfectly balance each other (e.g. in ionic crystals) to give no field on somewhat larger dimensions.

The scale we need here, however, is the atomic scale. In the electronic polarization mechanism, we actually "looked" inside the atom - so we shouldn't just stay on a "rough" scale and neglect the fine details.

Nevertheless, that is what we are going to do in the next paragraph: Neglect the details. The approach may not be beyond reproach, but it works and gives simple relations.

Questionaire
Multiple Choice questions to all of 3.2