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MIT 6 001 - LECTURE NOTES

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MIT OpenCourseWare http://ocw.mit.edu Continuum Electromechanics For any use or distribution of this textbook, please cite as follows: Melcher, James R. Continuum Electromechanics. Cambridge, MA: MIT Press, 1981. Copyright Massachusetts Institute of Technology. ISBN: 9780262131650. Also available online from MIT OpenCourseWare at http://ocw.mit.edu (accessed MM DD, YYYY) under Creative Commons license Attribution-NonCommercial-Share Alike. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.5Charge Migration, Convectionand Relaxation5.1 IntroductionIn Chap. 4, the subject is electromechanical kinematics. Field sources are physically constrainedto have predetermined spatial distributions and the relative motion is prescribed. As a result, in atypical example, the electromechanical dynamics can be incapsulated in a lumped-parameter model. Inthis and the next chapter, the mechanics remain kinematic, in that the material deformations are againprescribed. However, now material may be suffering relative deformations, represented by a given ve-locity field v(r,t). More important, in this and the next chapter, electrodes and wires are no longerused to constrain the "free" field sources. Rather, the distribution of free charge and current is nowdetermined by.the field laws themselves, augmented by conservation laws and constitutive relations.The physical situations now considered are electroquasistatic and the sources are therefore chargedensities. In Chap. 6, magnetoquasistatic systems are of interest, the relevant sources are the freecurrent density and magnetization density, and the subject is magnetic diffusion in the face of materialconvection.In the next section, equations are deduced that represent the fate of each species of charge.Throughout this chapter, the charge carriers are dominated in their motions by collisions with neutralparticles and with each other. On the average, collisions are so frequent that the inertia of eachcarrier can be ignored. Such collision-dominated carrier motions are introduced in Sees. 3.2 and 3.3,where the observation is made that it is only if the particle inertia is ignorable that the electricalforce on the carrier can be taken as instantaneously transmitted to the media through which it moves.If the carrier inertia is important, the carrier densities constitute mechanical continua in their ownright. Such examples are the electron beam in vacuum and the ions and electrons that constitute a "cold"plasma. These models are therefore appropriately included in Chaps. 7 and 8, where fluids and fluid-like continua are studied.The conservation of charge equations, together with the electroquasistatic field laws and thespecified material deformation, constitute a description of the way in which the fields and theirsources self-consistently evolve. Whether to gain insights concerning the implications of these equa-tions, or to solve these equations in a specific situation, characteristic coordinates are valuable.Thus, the characteristic approach to partial differential equations is introduced in the context ofcharge-charrier migration, relaxation and convection. The method of characteristics will be used ex-tensively to describe other phenomena involving propagation in later sections and chapters.Examples treated in Secs. 5.4 and 5.5, which illustrate "imposed field and flow" dynamics ofsystems of carriers, involve a space charge due to the charge carriers that is ignorable in its con-tribution to the field. The impact charging of macroscopic particles treated in Sec. 5.5 results in amodel widely used in atmospheric sciences, macroscopic particle physics and air-pollution control.When space-charge effects are significant, it is necessary to be more specialized in the treat-ment. In Sec. 5.6 only one species of charge carrier is presumed to be significant. The unipolarcarriers might be ions injected by a corona discharge into a neutral gas or into a highly insulatingliquid. They might also be charged macroscopic particles carrying a constant charge per particle andmigrating through a gas or liquid. Section 5.7 considers steady-flow one-dimensional unipolar con-duction and its relation to the d-c family of energy converters.Bipolar conduction, discussed in Secs. 5.8 and 5.9, has as a limiting model ohmic conduction;These sectiona have two major objectives, to illustrate charge migration and convection phenomenawith more than one species of carrier, and to put the ohmic conduction model in perspective. InSec. 5.10, charge relaxation is described in general terms by again resorting to the method of charac-teristics. The remaining sections are based on the ohmic conduction model.The transfer relations for regions of uniform conductivity are discussed in Sec. 5.12 and appliedto important illustrative physical situations in Secs. 5.13 and 5.14. These case studies are profit-ably contrasted with their magnetic counterparts developed in Secs. 6.4 and 6.5.Temporal transients, initiated from spatially periodic initial conditions, are considered inSec. 5.15. Just as the natural modes are closely related to the driven response of lumped-parameterlinear systems, the natural modes of the continuum systems discussed in terms of their responses tospatially periodic drives in Secs. 5.13 and 5.14 are found to be closely related to the natural modesfor distributed systems. This section, which is the first to illustrate the third category of responsefor linear systems that are uniform in at least one direction, as presaged in Sec. 1.2, also illus-trates how heterogeneous systems of uniform ohmic conductors (which support a charge relaxation processin each bulk region) can display charge diffusion in the system taken as a whole. This type of dif-fusion should be discriminated from diffusion at the carrier (microscopic) level. Diffusion in thelatter sense is included in Sec. 5.2 so that the domain of validity of migration and convection proc-Sec. 5.1esses in which diffusion is neglected can be appreciated. Molecular diffusion and its effect on chargeevolution, introduced in Sec. 5.2, is largely delayed until Chap. 10.Finally, in.Sec. 5.16, the response of an Ohmic moving sheet is used to introduce the fourth typeof continuum linear response eluded to in Sec. 1.2, a spatially transient response to a drive that istemporarily in the sinusoidal steady state.5.2 Charge Conservation with Material ConvectionWith the


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