MIT OpenCourseWare http://ocw.mit.edu Haus, Hermann A., and James R. Melcher. Electromagnetic Fields and Energy. Englewood Cliffs, NJ: Prentice-Hall, 1989. ISBN: 9780132490207. Please use the following citation format: Haus, Hermann A., and James R. Melcher, Electromagnetic Fields and Energy. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed [Date]). License: Creative Commons Attribution-NonCommercial-Share Alike. Also available from Prentice-Hall: Englewood Cliffs, NJ, 1989. ISBN: 9780132490207. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms7 CONDUCTION AND ELECTROQUASISTATIC CHARGE RELAXATION 7.0 INTRODUCTION This is the last in the sequence of chapters concerned largely with electrostatic and electroquasistatic fields. The electric field E is still irrotational and can therefore be represented in terms of the electric potential Φ. � × E = 0 ⇔ E = −�Φ (1) The source of E is the charge density. In Chap. 4, we began our exploration of EQS fields by treating the distribution of this source as prescribed. By the end of Chap. 4, we identified solutions to boundary value problems, where equipotential surfaces were replaced by perfectly conducting metallic electrodes. There, and throughout Chap. 5, the sources residing on the surfaces of electrodes as surface charge densities were made self-consistent with the field. However, in the volume, the charge density was still prescribed. In Chap. 6, the first of two steps were taken toward a self-consistent description of the charge density in the volume. In relating E to its sources through Gauss’ law, we recognized the existence of two types of charge densities, ρu and ρp, which, respectively, represented unpaired and paired charges. The paired charges were related to the polarization density P with the result that Gauss’ law could be written as (6.2.15) (2)� · D = ρu where D ≡ �oE + P. Throughout Chap. 6, the volume was assumed to be perfectly insulating. Thus, ρp was either zero or a given distribution. 12 Conduction and Electroquasistatic Charge Relaxation Chapter 7 Fig. 7.0.1 EQS distributions of potential and current density are analogous to those of voltage and current in a network of resistors and capacitors. (a) Systems of perfect dielectrics and perfect conductors are analogous to capaci-tive networks. (b) Conduction effects considered in this chapter are analogous to those introduced by adding resistors to the network. The second step toward a self-consistent description of the volume charge density is taken by adding to (1) and (2) an equation expressing conservation of the unpaired charges, (2.3.3). ∂ρu � · Ju + ∂t = 0 (3) That the charge appearing in this equation is indeed the unpaired charge den-sity follows by taking the divergence of Amp`ere’s law expressed with polarization, (6.2.17), and using Gauss’ law as given by (2) to eliminate D. To make use of these three differential laws, it is necessary to specify P and J. In Chap. 6, we learned that the former was usually accomplished by either specifying the polarization density P or by intro ducing a polarization constitutive law relating P to E. In this chapter, we will almost always be concerned with linear dielectrics, where D = �E. A new constitutive law is required to relate Ju to the electric field intensity. The first of the following sections is therefore devoted to the constitutive law of conduction. With the completion of Sec. 7.1, we have before us the differential laws that are the theme of this chapter. To anticipate the developments that follow, it is helpful to make an analogy to circuit theory. If the previous two chapters are regarded as describing circuits consisting of interconnected capacitors, as shown in Fig. 7.0.1a, then this chapter adds resistors to the circuit, as in Fig. 7.0.1b. Suppose that the voltage source is a step function. As the circuit is composed of resistors and capacitors, the distribution of currents and voltages in the circuit is finally determined by the resistors alone. That is, as t → ∞, the capacitors cease charging and are equivalent to open circuits. The distribution of voltages is then determined by the steady flow of current through the resistors. In this long-time limit, the charge on the capacitors is determined from the voltages already specified by the resistive network. The steady current flow is analogous to the field situation where ∂ρu/∂t →in the conservation of charge expression, (3). We will find that (1) and (3), the latter written with Ju represented by the conduction constitutive law, then fully determine the distribution of potential, of E, and hence of Ju. Just as the charges 03 Sec. 7.1 Conduction Constitutive Laws on the capacitors in the circuit of Fig. 7.0.1b are then specified by the already determined voltage distribution, the charge distribution can be found in an after-the-fact fashion from the already determined field distribution by using Gauss’ law, (2). After considering the physical basis for common conduction constitutive laws in Sec. 7.1, Secs. 7.2–7.6 are devoted to steady conduction phenomena. In the circuit of Fig. 7.0.1b, the distribution of voltages an instant after the voltage step is applied is determined by the capacitors without regard for the re-sistors. From a field theory point of view, this is the physical situation described in Chaps. 4 and 5. It is the objective of Secs. 7.7–7.9 to form an appreciation for how this initial distribution of the fields and sources relaxes to the steady condition, already studied in Secs. 7.2–7.6, that prevails when t → ∞. In Chaps. 3–5 we invoked the “perfect conductivity” model for a conductor.
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