6.8
Summary

Table 6.8.1 is useful both as an outline of this chapter and as a reference. Gauss' theorem is the basis for deriving the surface relations in the right-hand column from the respective volume relations in the left-hand column. By remembering the volume relations, one is able to recall the surface relations.

TABLE 6.8.1 SUMMARY OF POLARIZATION RELATIONS AND LAWS

Polarization Charge Density and Polarization Density
	(6.1.6)		(6.1.7)
Gauss' Law with Polarization
	(6.2.1)		(6.2.3)
	(6.2.15)		(6.2.16)
where
	(6.2.14)
Electrically Linear Polarization
Constitutive Law
	(6.4.2)
	(6.4.3)
Source Distribution, u = 0
	(6.5.9)		(6.5.11

Our first task, in Sec. 6.1, was to introduce the polarization density as a way of representing the polarization charge density. The first volume and surface relations resulted. These are deceptively similar in appearance to Gauss' law and the associated jump condition. However, they are not electric field laws. Rather, they simply relate the volume and surface sources representing the material to the polarization density.

Next we considered the fields due to permanently polarized materials. The polarization density was given. For this purpose, Gauss' law and the associated jump condition were conveniently written as (6.2.2) and (6.2.3), respectively.

With the polarization induced by the field itself, it was convenient to introduce the displacement flux density D and write Gauss' law and the jump condition as (6.2.15) and (6.2.16). In particular, for linear polarization, the equivalent constitutive laws of (6.4.2) and (6.4.3) were introduced.

The theme of this chapter has been the determination of EQS fields when the polarization charge density makes a contribution. In cases where the polarization density is given, this is easy to keep in mind, because the first step in formulating a problem is to evaluate _p from the given P. However, when _p is induced, variables such as D are used and we must be reminded that when all is said and done, _p (or its surface counterpart, _sp) is still responsible for the effect of the material on the field. The expressions for _p and _sp given by the last two relations in the table are useful not only for interpreting the distributions of fields after they have been found but for forming an impression of the fields in complex systems where it would not be worthwhile to find an analytic solution. Remember that these relations hold only in regions where there is no unpaired charge density.

In Chap. 9, we will find that most of this chapter is directly applicable to the description of magnetization. There we will continue to develop insights that will be equally applicable to the polarization phenomena of this chapter.

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