MIT OpenCourseWare http://ocw.mit.edu Electromechanical Dynamics For any use or distribution of this textbook, please cite as follows: Woodson, Herbert H., and James R. Melcher. Electromechanical Dynamics. 3 vols. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons Attribution-NonCommercial-Share Alike For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/termsChapter 10DYNAMICS OFELECTROMECHANICAL CONTINUA10.0 INTRODUCTIONIn Chapter 9 we treated simple examples of mechanical continua to estab-lish the basic techniques of making mathematical models and to illustrate thekinds of dynamic behavior and the mathematical methods needed in analyses.In that chapter simple elastic continua at rest were excited at boundaries sothat the resulting continuum dynamics were determined by mechanicalcharacteristics alone.In this chapter we still restrict our attention to simple elastic continua butwe generalize on the treatment of Chapter 9 to include the effects of distributedforces of electric origin and material motion. By the use of simple models weillustrate the basic phenomena that occur in a wide variety of physical systemsand the analytical techniques used in their mathematical description. In spiteof the diversity of physical situations in which continuum electromechanicalinteractions are important, a unity results from mathematical techniques thatare common to all of the situations. It is our purpose here to illuminate, in thesimplest context possible, these mathematical techniques and the physicalphenomena they describe.As stated earlier, the techniques presented are fundamental to a widevariety of physical situations. It is therefore helpful for the purpose of appre-ciating our objectives to review some of the technical areas concerned withcontinuum electromechanics.Magnetohydrodynamics (MHD) is concerned with the interactions of freecurrents and magnetic fields in fluids (liquids and gases) which have highenough electrical conductivity that a quasi-static magnetic field model isappropriate for describing the electromagnetic part of the system. To reflectmore accurately the nature of the mechanical medium this area is sometimesreferred to as magnetogasdynamics(MGD)or as magnetofluiddynamics(MFD).Areas of application include pumping and levitation of liquids (usuallyDynamics of Electromechanical Continuametals), orientation and confinement of extremely hot ionized gases or plas-mas, as, for example, in thermonuclear fusion experiments,* electric powergeneration from ionized gases produced by combustion of fossil fuels or fromheat produced in a fission reactor,t and space propulsion achieved by electro-magnetic acceleration of ionized gases.$ Scientific interest in this area in-cludes such geophysical and astrophysical topics as the origin of the earth'smagnetic field in its liquid metal core and the dynamics of stellar structurescomposed of highly ionized gases.A similar area isferrohydrodynamics,which is concerned with magnetiza-tion interactions of magnetic fields with a ferromagnetic fluid.§Electrohydrodynamics (EHD) is concerned with interaction between elec-tric fields and free or bound (polarization) charges in fluids. The fluids may beextremely good insulators, slightly conducting, or even highly conducting.The distinguishing feature is that the electromagnetic part of the system isdescribed by a quasi-static electric field model. Applications of EHD includepumping and levitation of liquids and gases, extraction of contaminants fromgases such as smoke,** mixing of liquids, orientation of liquids in near-zero-gravity environments, augmentation of heat transfer, and property measure-ments in fluid systems. EHD interactions also occur in meteorology, in whichcharge distribution in the atmosphere (as in a thunderstorm) is important,and in surface physics, in which the distribution of charges at an interface issignificant, as in frictional electrification.ttThe engineering and scientific applications ofelectron andion beams involvecontinuum electromechanical interactions. Electron beams, confined bymagnetic fields and interacting through electric fields with distributed electriccircuits, are commonly used to generate power at microwave frequencies.1TIn such applications the beam is represented by quasi-static equations, but thedistributed electric circuits support electromagnetic waves and are not amen-able to quasi-static analysis. Electron beams are also used for heating, weld-ing, forming, and purifying metals. Charged particle beams, electrons andions, are used for medical treatment, for measuring collision cross sections,and for heating plasmas.§§* D. J. Rose and M. Clark, Jr., Plasmas and Controlled Fusion, M.I.T. Press and Wiley,New York, 1961.t G. W. Sutton and A. Sherman, Engineering Magnetohydrodynamics, McGraw-Hill, NewYork, 1965.1 Ibid., p. 447.§R. E. Rosensweig, "Magnetic Fluids," Intern. Sci. Technol., 55, 48-56, 90 (July 1966).** H. J. White, IndustrialElectrostatic Precipitation, Addison-Wesley, Reading, Mass.,1963.tt L. B.Loeb, Static Electrification, Springer, Berlin, 1958.++C. C. Johnson, Fieldand Wave Electrodynamics, McGraw-Hill, New York, 1965, p. 275.§§ T. H. Stix, The Theory ofPlasma Waves, McGraw-Hill, New York, 1962, p. 107.IntroductionPlasma dynamics* is concerned with the behavior of gases composed atleast in part of charged particles. Thus continuum electromechanical inter-actions will affect the behavior of a plasma. Probably the most commonexample of a plasma is the ionized gas in a fluorescent lamp. Other examplesare gas-filled rectifiers, flames such as rocket exhausts, and the sun. Thephysical characteristics of ionized gases can assume many forms. In certaincases, such as proposed fusion devices and MHD generators, the plasmabehaves as a highly conducting fluid and its dynamic behavior is describedby a magnetohydrodynamic approximation. In other cases the plasma is onlyslightly ionized and the electrohydrodynamic equations are appropriate. Instill other cases the plasma may be so tenuous that it is best described as acollection of noninteracting particles in imposed magnetic and electric fields.In all of these regimes the plasma exhibits the basic phenomena of wavepropagation and instability, subjects that are treated in this chapter.Electrons and holes in semiconductors, usefully modeled as
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