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 8FIELD DESCRIPTIONOF MAGNETIC ANDELECTRIC FORCES8.0 INTRODUCTIONChapter 7 is restricted to the effects of mechanical motion on magneticand electric fields. In general, electromechanical interactions involve effectson the mechanical system from the electromagnetic fields as well. Thesearise from the mechanical forces of electrical origin.In Chapters 3 through 6 we were concerned with total forces acting onrigid bodies. In systems in which the mechanical medium must be representedby a deformable continuum the details of the force distribution must beknown. Hence in continuum electromechanics we are concerned withmagnetic or electric force densities, which are, in general, functions of spaceand time.Electromagnetic fields are defined by forces composed of two parts:those exerted on free charges by electric fields and those exerted on freecurrents (moving free charges) by magnetic fields. The relative importance ofthese forces depends on the type of system being considered. In magnetic fieldsystems, as defined in Section 1.1, the important field excitation is providedby the free current density J,. Hence for magnetic field systems the onlyimportant forces arise from the interactions of the free current density J,with magnetic fields. Similarly, the only forces of significance in electric fieldsystems, as defined in Section 1.1, are the interactions of free charge densityp, with electric fields. The validity of these assumptions is checked in particularproblems. Following the pattern established in earlier sections, we treatforces in magnetic field and electric field systems separately. Our object is todescribe electromagnetic forces mathematically in alternative forms that willprove useful in work with continuum electromechanical systems.Forces in Magnetic-Field SystemsTwo other technically important electromagnetic forces are those resultingfrom the interactions of polarization density P with electric fields andmagnetization density M with magnetic fields. In Chapters 3 to 5 we calculatetotal forces on polarizable and magnetizable bodies by using an energy method.We extend this method to account for force densities in polarized or magne-tized media that are electrically linear, isotropic, and homogeneous. Thislimitation in our discussion of polarization and magnetization forces isimposed because use of an energy method requires a knowledge of themechanical and thermodynamic properties of the material.8.1 FORCES IN MAGNETIC-FIELD SYSTEMSConsider first the force resulting from the interaction of moving freecharge (i.e., J,) and a magnetic field. The Lorentz force (1.1.28) gives thetotal magnetic force on a charge q moving with velocity v asf = qv x B. (8.1.1)The force density F (newtons per cubic meter) can be obtained from thisexpression by writingIf f qvv, x BiF = lim ---= lim , (8.1.2)av-.o0V av-o 6Vwhere f4, qg, and vi refer to all the particles in 6V and Bi is the flux densityexperienced by qj. If we can say that all particles within 6V experience thesame flux density B, we can use the definition of free current density (seeSection B.1.2)* to write (8.1.2) asF = J, x B. (8.1.3)The general definition of (8.1.2) requires the averaging of products, whereasthe result of (8.1.3) is the product of averages. It is not, in general, true forvariables x and y that[zY]av = [1]av~Ylav.The force density expressed by (8.1.3) however, agrees, to a high degree ofaccuracy, with all experimental results obtained with common conductors.The relation (8.1.3) is valid because the volume 6V can be made smallenough to enclose a region of essentially constant magnetic flux density,although still including many free charges.In fact, we could have used (8.1.3) rather than (8.1.1) as the definition ofB, for the original experiments of Biot and Savart and later Amphret con-cerned themselves with relating the force density to the free current density* J = lim [( qv) 6Vt J. D. Jackson, Classical Electrodynamics, Wiley, New York 1962, p. 133.Field Description of Magnetic and Electric ForcesJ,. Some writers start with (8.1.3) as the basic definition ofthe magnetic forceon moving free charge.* However, the averaging process used to make(8.1.2) and (8.1.3) consistent is then inherent to the definition.It is important to remember that (8.1.3) represents the average of forceson the charges. This is equivalent to the force on a medium if there is somemechanism by which each charge transmits the Lorentz force to the material.For example, in a conductor, the charges can be thought of as particlesmoving through a viscous material-in which case the force that acts oneach charge is transmitted to the medium by the viscous retarding force and(8.1.3) is the force density experienced by the medium.There are situations in which the charges do not interact individually withthe medium. For example, in a polarized medium, pairs of charges (dipoles)transmit a force to the medium--each pair being connected through thestructure of an atom or molecule. For these cases it is the dipoles rather thanthe charges that transmit a force to the medium. Then it is appropriate toconsider the average of the forces on individual dipoles as equivalent to theforce density on the medium. This class of forces is developed in Section 8.5.The force density given in (8.1.3) is expressed in terms of source and fieldquantities. It is useful to have the force expressed as a function of fieldquantities alone because we often solve field problems without calculatingthe free current density. We find it useful to define the Maxwell stress tensoras a function of the field quantities from which the force density can beobtained by space differentiation. The Maxwell stress tensor is particularlyuseful for finding electromechanical boundary conditions in a concise form.It is useful also for finding the total electromagnetic force on a body.A tensor has particular properties that are useful in this and the chapterswhich follow. We therefore devote Section 8.2 to a
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