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.2Electrodynamic Laws,Approximations and Relations:142.1 DefinitionsContinuum electromechanics brings together several disciplines, and so it is useful to summarizethe definitions of electrodynamic variables and their units. Rationalized MKS units are used not onlyin connection with electrodynamics, but also in dealing with subjects such as fluid mechanics and heattransfer, which are often treated in English units. Unless otherwise given, basic units of meters (m),kilograms (kg), seconds (sec), and Coulombs (C) can be assumed.Table 2.1.1. Summary of electrodynamic nomenclature.Name Symbol UnitsDiscrete VariablesVoltage or potential difference v [V] = volts = m2 kg/C sec2Charge q [C] = Coulombs = CCurrent i [A] = Amperes = C/secMagnetic flux X [Wb] = Weber = m2 kg/C secCapacitance C [F] = Farad C2 sec2/m2 kgInductance L [H] = Henry = m2 kg/C2Force f [N] = Newtons = kg m/sec2Field SourcesFree charge density Pf C/m3Free surface charge density •f C/m2Free current density 4f A/m2Free surface current density Kf A/mFields (name in quotes is often used for convenience)"Electric field" intensity V/m"Magnetic field" intensity A/mElectric displacement C/m2Magnetic flux density Wb/m2 (tesla)Polarization density C/m2Magnetization density M A/mForce density F N/m3Physical ConstantsPermittivity of free space 6o = 8.854 x 1012 F/mPermeability of free space 1o = 4r x 10- 7 H/mAlthough terms involving moving magnetized and polarized media may not be familiar, Maxwell'sequations are summarized without prelude in the next section. The physical significance of the un-familiar terms can best be discussed in Secs. 2.8 and 2.9 after the general laws are reduced to theirquasistatic forms, and this is the objective of Sec. 2.3. Except for introducing concepts concernedwith the description of continua, including integral theorems, in Secs. 2.4 and 2.6, and the dis-cussion of Fourier amplitudes in Sec. 2.15, the remainder of the chapter is a parallel development ofthe consequences of these quasistatic laws. That the field transformations (Sec. 2.5), integral laws(Sec. 2.7), splicing conditions (Sec. 2.10), and energy storages are derived from the fundamental quasi-static laws, illustrates the important dictum that internal consistency be maintained within the frame-work of the quasistatic approximation.The results of the sections on energy storage are used in Chap. 3 for deducing the electric andmagnetic force densities on macroscopic media. The transfer relations of the last sections are animportant resource throughout all of the following chapters, and give the opportunity to explore thephysical significance of the quasistatic limits.2.2 Differential Laws of ElectrodynamicsIn the Chu formulation,l with material effects on the fields accounted for by the magnetizationdensity M and the polarization density P and with the material velocity denoted by v, the laws ofelectrodynamics are:Faraday's law4+ 3H P-•at o M +(o Sto Bt1. P. Penfield, Jr., and H. A. Haus, Electrodynamics of Moving Media, The M.I.T. Press, Cambridge,Massachusetts, 1967, pp. 35-40.Ampere's lawV x H = E + + V x (P x v) + J (2)ot t fGauss' lawV*E = -V*P + Pf (3)divergence law for magnetic fieldsoV.H = -ioV *M (4)and conservation of free chargeV'Jf + •t = 0 (5)This last expression is imbedded in Ampere's and Gauss' laws, as can be seen by taking the diver-gence of÷-Eq. 2 and exploiting Eq. 3. In this formulation the electric displacement and magnetic fluxdensity B are defined fields:D = E + P (6)o4- -B = o(H + M) (7)2.3 Quasistatic Laws and the Time-Rate ExpansionWith a quasistatic model, it is recognized that relevant time rates of change are sufficientlylow that contributions due to a particular dynamical process are ignorable. The objective in thissection is to give some formal structure to the reasoning used to deduce the quasistatic field equa-tions from the more general Maxwell's equations. Here, quasistatics specifically means that timesof interest are long compared to the time, Tem, for an electromagnetic wave to propagate through thesystem.Generally, given a dynamical process characterized by some time determined by the parameters ofthe system, a quasistatic model can be used to exploit the comparatively long time scale for proc-esses of interest. In this broad sense, quasistatic models abound and will be encountered in manyother contexts in the chapters that follow. Specific examples are:(a) processes slow compared to wave transit times in general; acoustic waves and the model isone of incompressible flow, Alfvyn and other electromechanical waves and the model is less standard;(b) processes slow compared to diffusion (instantaneous diffusion models). What diffuses canbe magnetic field, viscous stresses, heat, molecules or hybrid electromechanical effects;(c) processes slow compared to relaxation of continua (instantaneous relaxation or constant-potential models). Charge relaxation is an important example.The point of making a quasistatic approximation is often to focus attention on significantdynamical processes. A quasistatic model is by no means static. Because more than one rate processis often imbedded in a given physical system, it is important to agree upon the one with respect towhich the dynamics are quasistatic.Rate processes other than those due to the transit time of electromagnetic waves enter throughthe dependence of the field sources on the fields and material motion. To have in view the additionalcharacteristic times typically brought in by the field sources, in this section the free currentdensity is postulated to have the dependenceJf = G(r)E + Jv(v,pf,H) (i)In the absence of motion, Jv is zero. Thus, for media at rest the conduction model is ohmic, with theel-ctrical conductivity a in general a funqtion Qf position. Examples of Jv are a convection currentpfv, or an ohmic motion-induced current a(v x 0oH). With an underbar used to denote a
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