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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.3Electromagnetic Forces, ForceDensities and Stress Tensors72://A//e e3.1 Macroscopic versus Microscopic ForcesMost important in this chapter is the distinction between forces on fundamental particles andforces on macroscopic media. It is common to speak of the "force on a charge" or the "force on a current"even though what is meant is the force on ponderable material. Interest might actually be in electricand magnetic forces acting on collections of fundamental charge carriers. (Motions of electron beams invacuum are an example. The charged particles in that case constitute the continuum, in the sense thatit is the electron inertia that enters into the equation of motion.) But, more commonly, the chargedparticles are imbedded in media, and it is the resulting force on the material that is of interest.Examples are as obvious as the electrical force of attraction between the capacitor plates of an electro-static voltmeter or the magnetic torque exerted on current-carrying conductors in a meter movement.Section 3.2 develops a specific model to illustrate how momentum imparted to charged particles bythe fields is transferred to the neutral media that support those particles. That macroscopic forcesare more than simply an average over the forces on fundamental charges is further emphasized by consider-ing the practical cases of polarization and magnetization forces. Force densities of engineering signifi-cance exist even in regions where the free charge and free current (and for that matter polarizationcharge or magnetization charge) are absent. Such forces can be associated with a microscopic picture,discussed in Sec. 3.6, in which electrical forces on dipoles are transferred to the media.Although the dipole model is useful for forming a microscopic picture of electric polarizationforces, it is restricted to cases where the dipoles do not significantly interact. In the pursuit ofa less restricted force density, developments in Secs. 3.7-3.8 are based on such measured macroscopicparameters as the permittivity and permeability. It is the business of thermodynamics to convert thatinformation into the desired force densities. In its own way, the line of reasoning presented inSecs. 3.5, 3.7 and 3.8 exemplifies a more basic point of view than one geared to a particular microscopicmodel. Thermodynamic concepts provide a means for replacing detailed and specialized derivations bycarefully defined physical measurements.The stress-tensor representation of electromagnetic forces which concludes this chapter will seecontinual application in the following chapters. The tensor concept itself, introduced in Sec. 3.9,will also be applied to the formulation of continuum mechanical and electromechanical equations.3.2 The Lorentz Force DensityAlthough macroscopic forces were the first measured in the development of electricity and mag-netism, it is now normally accepted that the fundamental force is that on a "test" charge. This chargemight be a jingle electron in free space. If the charged particle has a total charge q and moves witha velocity vp, then the Lorentz force acting on the particle supporting the charge is= qE + qvp x o H (1)This statement, like the electrodynamic laws summarized in Chap. 2, is an empirical one. In most of theareas of continuum electromechanics, it is forces due to many charges that are of interest, and it istherefore appropriate to sum the individual forces of Eq. 1 over the charges within a given unit ofvolume to arrive at the Lorentz force densityF = pfE + Jf x oH (2)Incremental volumes of interest have dimensions much greater than the characteristic distances betweenparticles. But also, for the average electrical field to have meaning, it must be primarily dueto sources external to the differential volume of interest. This ensures that, over an incrementalvolume, each particle experiences essentially the same electric field. The contribution to the fieldof the charges within the differential volume is negligible. Similar arguments apply to the magneticfield intensity, which must be produced over a given differential volume largely by currents outsidethe volume.Equation 2 represents the force density acting on a ponderable medium if means are available forthe force on the particles to be transmitted to the medium. The mechanisms by which this happens arediverse, and implicit to the conduction process. Whether the fundamental carriers are electrons in ametal, holes and electrons in a semiconductor or ions in a liquid or gas, the average motions offundamental charge carriers are superimposed on random motions. The flights of fundamental carriersare interrupted by collisions with lattice molecules (in a solid) or molecules that are themselves ina Brownian equilibrium (in a liquid or gas) with a frequency that is usually extremely high comparedto reciprocal times of interest. These collisions transfer momentum from the fundamental chargecarriers to the ponderable medium.Secs. 3.1 & 3.2To more fully appreciate the transition from the force acting on fundamental carriers, Eq. 1, tothat on a material, Eq. 2, it is helpful to make a formal derivation. Although the discussion leadsto rather general conclusions, only two families of carriers are now considered, one positive withcharge per particle q~and number density n+ and the other negative with a magnitude of charge q_ andnumber density n_. The average Lorentz force, Eq. 1, is in equilibrium with an average force repre-senting the effect of collisions on the net migration of the particles:qE + q_(v_ + v) x U°H .m__q -q(v + v) x mVvThe retarding forces on the right are much as would be conceived for a swarm of macroscopic particlesmoving through a viscous liquid. The average carrier velocities -+ are measured relative to the mediumwhich itself has the velocity V. Hence, on the right it is relative velocities of particles and mediumthat appear, while in the Lorentz force


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