10.8
Summary

Before tackling the concepts in this chapter, we had studied MQS fields in two limiting situations:

In the first, currents in Ohmic conductors were essentially stationary, with distributions governed by the steady conduction laws investigated in Secs. 7.2-7.6. The associated magnetic fields were then found by using these current distributions as sources. In the absence of magnetizable material, the Biot-Savart law of Sec. 8.2 could be used for this purpose. With or without magnetizable material, the boundary value approaches of Secs. 8.5 and 9.6 were applicable.

In the second extreme, where fields were so rapidly varying that conductors were "perfect," the effect on the magnetic field of currents induced in accordance with the laws of Faraday, Ampère, and Ohm was to nullify the magnetic flux density normal to conducting surfaces. The boundary value approach used to find self-consistent fields and surface currents in this limit was the subject of Secs. 8.4 and 8.6.

In this chapter, the interplay of the laws of Faraday, Ohm, and Ampère has again been used to find self-consistent MQS fields and currents. However, in this chapter, the conductivity has been finite. This has made it possible to explore the dynamics of fields with source currents that were neither distributed throughout the volumes of conductors in accordance with the laws of steady conduction nor confined to the surfaces of perfect conductors.

In dealing with perfect conductors in Chaps. 8 and 9, the all-important role of E could be placed in the background. Left for a study of this chapter was the electric field induced by a time-varying magnetic induction. So, we began in Sec. 10.1 by picturing the electric field in systems of perfect conductors. The approach was familiar from solving EQS (Chap. 5) and MQS (Chap. 8) boundary value problems involving Poisson's equation. The electric field intensity was represented by the superposition of a particular part having a curl that balanced - B/ t at each point in the volume, and an irrotational part that served to make the total field tangential to the surfaces of the perfect conductors.

Having developed some insight into the rotational electric fields induced by magnetic induction, we then undertook case studies aimed at forming an appreciation for spatial and temporal distributions of currents and fields in finite conductors. By considering the effects of finite conductivity, we could answer questions left over from the previous two chapters.

Under what conditions are distributions of current and field quasistationary in the sense of being essentially snapshots of a sequence of static fields?

Under what conditions do they consist of surface currents and fields having negligible normal components at the surfaces of conductors?

We now know that the answer comes in terms of characteristic magnetic diffusion (or relaxation) times that depend on the electrical conductivity, the permeability, and the product of lengths.

The lengths in this expression make it clear that the size and topology of the conductors plays an important role. This has been illustrated by the thin-sheet models of Secs. 10.3 and 10.4 and one-dimensional magnetic diffusion into the bulk of conductors in Secs. 10.6 and 10.7. In each of these classes of configurations, the role played by has been illustrated by the step response and by the sinusoidal steady state response. For the former, the answer to the question, "When is a conductor perfect?" was literal. The conductor tended to be perfect for times that were short compared to a properly defined . For the latter, the answer came in the form of a condition on the frequency. If 1, the conductor tended to be perfect.

In the sinusoidal state, a magnetic field impressed at the surface of a conductor penetrates a distance into the conductor that is the skin depth and is given by setting = ² = 2 and solving for .

It is true that conductors will act as perfect conductors if this skin depth is much shorter than all other dimensions of interest. However, the thin sheet model of Sec. 10.4 teaches the important lesson that the skin depth may be larger than the conductor thickness and yet the conductor can still act to shield out the normal flux density. Indeed, in Sec. 10.4 it was assumed that the current was uniform over the conductor cross-section and hence that the skin depth was large, not small, compared to the conductor thickness. Demonstration 8.6.1, where current passes through a cylindrical conductor at a distance l above a conducting ground plane, is an example. It would be found in that demonstration that if l is large compared to the conductor thickness, the surface current in the ground plane would distribute itself in accordance with the perfectly conducting model even if the frequency is so low that the skin depth is somewhat larger than the thickness of the ground plane. If is the ground plane thickness, we would expect the normal flux density to be small so long as _m = l 1. Typical of such situations is that the electrical dissipation due to conduction is confined to thin conductors and the magnetic energy storage occupies relatively larger regions that are free of dissipation. Energy storage and power dissipation are subjects taken up in the next chapter.