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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/terms10 MAGNETOQUASISTATIC RELAXATION AND DIFFUSION 10.0 INTRODUCTION In the MQS approximation, Amp`ere’s law relates the magnetic ﬁeld intensity H to the current density J. � × H = J (1) Augmented by the requirement that H have no divergence, this law was the theme of Chap. 8. Two types of physical situations were considered. Either the current density was imposed, or it existed in perfect conductors. In both cases, we were able to determine H without being concerned about the details of the electric ﬁeld distribution. In Chap. 9, the eﬀects of magnetizable materials were represented by the magnetization density M, and the magnetic ﬂux density, deﬁned as B≡ µo(H+M), was found to have no divergence. � · B = 0 (2) Provided that M is either given or instantaneously determined by H (as was the case throughout most of Chap. 9), and that J is either given or subsumed by the boundary conditions on perfect conductors, these two magnetoquasistatic laws determine H throughout the volume. In this chapter, our ﬁrst objective will be to determine the distribution of E around perfect conductors. Then we shall broaden our physical domain to include ﬁnite conductors, especially in situations where currents are caused by an E that 1� 2 Magnetoquasistatic Relaxation and Diﬀusion Chapter 10 is induced by the time rate of change of B. In both cases, we make explicit use of Faraday’s law. ∂B = (3)� × E − ∂t In the EQS systems considered in Chaps. 4–7, the curl of H generated by the time rate of change of the displacement ﬂux density was not of interest. Amp`ere’s law was adequately incorporated by the continuity law. However, in MQS systems, the curl of E generated by the magnetic induction on the right in (1) is often of primary importance. We had ﬁelds that depended on time rates of change in Chap. 7. We have already seen the consequences of Faraday’s law in Sec. 8.4, where MQS systems of perfect conductors were considered. The electric ﬁeld intensity E inside a perfect conductor must be zero, and hence B has to vanish inside the perfect conductor if B varies with time. This leads to n B = 0 on the surface of a · perfect conductor. Currents induced in the surface of perfect conductors assure the proper discontinuity of n× H from a ﬁnite value outside to zero inside. Faraday’s law was in evidence in Sec. 8.4 and accounted for the voltage at terminals connected to each other by perfect conductors. Faraday’s law makes it possible to have a voltage at terminals connected to each other by a perfect “short.” A simple experiment brings out some of the subtlety of the voltage deﬁnition in MQS systems. Its description is followed by an overview of the chapter. Demonstration 10.0.1. Nonuniqueness of Voltage in an MQS System A magnetic ﬂux is created in the toroidal magnetizable core shown in Fig. 10.0.1 by driving the winding with a sinsuoidal current. Because it is highly p ermeable (a ferrite), the core guides a magnetic ﬂux density B that is much greater than that in the surrounding air. Looped in series around the core are two resistors of unequal value, R1 = R2. Thus, the terminals of these resistors are connected together to form a pair of “no des.” One of these nodes is grounded. The other is connected to high-impedance voltmeters through two leads that follow the diﬀerent paths shown in Fig. 10.0.1. A dual-trace oscilloscope is convenient for displaying the voltages. The voltages observed with the leads connected to the same node not only diﬀer in magnitude but are 180 degrees out of phase. Faraday’s integral law explains what is observed. A cross-section of the core, showing the pair of resistors and voltmeter leads, is shown in Fig. 10.0.2. The scope resistances are very large compared to R1 and R2, so the current carried by the voltmeter leads is negligible. This means that if there is a current i through one of the series resistors, it must be the same as that through the other. The contour Cc follows the closed circuit formed by the series resistors. Fara-day’s integral law is now applied to this contour. The ﬂux passing through the surface Sc spanning Cc is deﬁned as Φλ. Thus, � dΦλE ds = · − dtC1 = i(R1 + R2) (4) where �Φλ ≡ B da· (5) Sc Cite as: Haus, Hermann A., and James R. Melcher. Electromagnetic Fields and Energy. Englewood Cliffs, NJ: Prentice-Hall, 1989. ISBN: 9780132490207. Also available online at MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].� � 3 Sec. 10.0 Introduction Fig. 10.0.1 A pair of unequal resistors are connected in series around a magnetic circuit. Voltages measured between the terminals of the re-sistors by connecting the nodes to the dual-trace oscilloscope, as shown, diﬀer in magnitude and are 180 degrees out of phase. Fig. 10.0.2 Schematic of circuit for experiment of Fig. 10.0.1, showing contours used with Faraday’s law to predict the diﬀering voltages v1 and v2. Given the magnetic ﬂux, (4) can be solved for the current i that must circulate around the loop formed by the

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