This chapter can be divided into three parts. In the first, Sec. 7.1, conduction constitutive laws are related to the average motions of microscopic charge carriers. Ohm's law, as it relates the current density Ju to the electric field intensity E
is found to describe conduction in certain materials which are constituted of at least one positive and one negative species of charge carrier. As a reminder that the current density can be related to field variables in many ways other than Ohm's law, the unipolar conduction law is also derived in Sec. 7.1, (7.1.8). But in this chapter and those to follow, the conduction law (1) is used almost exclusively.
The second part of this chapter, Secs. 7.2-7.6, is concerned with "steady" conduction. A summary of the differential laws and corresponding continuity conditions is given in Table 7.10.1. Under steady conditions, the unpaired charge density is determined from the last expressions in the table after the first two have been used to determine the electric potential and field intensity.
In the third part of this chapter, Secs. 7.7-7.9, the dynamics of EQS systems is developed and exemplified. The laws used to determine the electric potential and field intensity, given by the first two lines in Table 7.10.2, are valid for frequencies and characteristic times that are arbitrary relative to electrical relaxation times, provided those times are themselves long compared to times required for an electromagnetic wave to propagate through the system. The last expressions identify how the unpaired charge density is relaxing under dynamic conditions.
In EQS systems, the magnetic induction makes a negligible contribution and the electric field intensity is essentially irrotational. Thus, E is represented by -grad (
) in both Table 7.10.1 and Table 7.10.2. In the EQS approximation, neglecting the magnetic induction is tantamount to ignoring the finite transit time effects of electromagnetic waves. This we saw in Chap. 3 and will see again in Chaps. 14 and 15.
TABLE 7.10.1 SUMMARY OF LAWS FOR STEADY STATE OHMIC CONDUCTION
TABLE 7.10.2 SUMMARY OF EQS LAWS FOR INHOMOGENEOUS OHMIC MEDIA
In MQS systems, fields may be varying so slowly that the effect of magnetic induction on the current flow is again ignorable. In that case, the laws of Table 7.10.1 are once again applicable. So it is that the second part of this chapter is a logical base from which to begin the next chapter. At least under steady conditions we already know how to predict the distribution of the current density, the source of the magnetic field intensity. How rapidly can MQS fields vary without having the magnetic induction come into play? We will answer this question in Chap. 10.
