Page 229 - Electromagnetics

dipole moment is proportional to the local electric field strength E :

p = αE , (4.94)
where α is called the polarizability of the elementary constituent. Each of the polarization
effects listed above may have its own polarizability: α e for electronic polarization, α a for
atomic polarization, and α d for dipole polarization. The total polarizability is merely the
sum α = α e + α a + α d .
In a rarefied gas the particles are so far apart that their interaction can be neglected.

Here the localized field E is the same as the applied field E. In liquids and solids where
particles are tightly packed, E depends on the manner in which the material is polarized

and may differ from E. We therefore proceed to determine a relationship between E
and P.

The Clausius–Mosotti equation. We seek the local field at an observation point
within a polarized material. Let us first assume that the fields are static. We surround
the observation point with an artificial spherical surface of radius a and write the field at
the observation point as a superposition of the field E applied, the field E 2 of the polarized
molecules external to the sphere, and the field E 3 of the polarized molecules within the
sphere. We take a large enough that we may describe the molecules outside the sphere in
terms of the macroscopic dipole moment density P, but small enough to assume that P
is uniform over the surface of the sphere. We also assume that the major contribution to
E 2 comes from the dipoles nearest the observation point. We then approximate E 2 using
the electrostatic potential produced by the equivalent polarization surface charge on the
sphere ρ Ps = ˆ n · P (where ˆ n points toward the center of the sphere). Placing the origin
of coordinates at the observation point and orienting the z-axis with the polarization P
so that P = P 0 ˆ z, we find that ˆ n · P =− cos θ and thus the electrostatic potential at any
point r within the sphere is merely

1 P 0 cos θ

(r) =− dS .

4π 0 S |r − r |
This integral has been computed in § 3.2.7 with the result given by (3.103) Hence
P 0 P 0
(r) =− r cos θ =− z
3 0 3 0
and therefore
P
E 2 = . (4.95)
3 0
Note that this is uniform and independent of a.
The assumption that the localized field varies spatially as the electrostatic field, even
when P may depend on frequency, is quite good. In Chapter 5 we will find that for a
frequency-dependent source (or, equivalently, a time-varying source), the fields very near
the source have a spatial dependence nearly identical to that of the electrostatic case.
We now have the seemingly more difficult task of determining the field E 3 produced
by the dipoles within the sphere. This would seem difficult since the field produced by
dipoles near the observation point should be highly-dependent on the particular dipole
arrangement. As mentioned above, there are various mechanisms for polarization, and
the distribution of charge near any particular point depends on the molecular arrange-
ment. However, Lorentz showed [115] that for crystalline solids with cubical symmetry,

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