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 terms Chapter 8 FIELD DESCRIPTION OF MAGNETIC AND ELECTRIC FORCES 8 0 INTRODUCTION Chapter 7 is restricted to the effects of mechanical motion on magnetic and electric fields In general electromechanical interactions involve effects on the mechanical system from the electromagnetic fields as well These arise from the mechanical forces of electrical origin In Chapters 3 through 6 we were concerned with total forces acting on rigid bodies In systems in which the mechanical medium must be represented by a deformable continuum the details of the force distribution must be known Hence in continuum electromechanics we are concerned with magnetic or electric force densities which are in general functions of space and time Electromagnetic fields are defined by forces composed of two parts those exerted on free charges by electric fields and those exerted on free currents moving free charges by magnetic fields The relative importance of these forces depends on the type of system being considered In magnetic field systems as defined in Section 1 1 the important field excitation is provided by the free current density J Hence for magnetic field systems the only important forces arise from the interactions of the free current density J with magnetic fields Similarly the only forces of significance in electric field systems as defined in Section 1 1 are the interactions of free charge density p with electric fields The validity of these assumptions is checked in particular problems Following the pattern established in earlier sections we treat forces in magnetic field and electric field systems separately Our object is to describe electromagnetic forces mathematically in alternative forms that will prove useful in work with continuum electromechanical systems Forces in Magnetic Field Systems Two other technically important electromagnetic forces are those resulting from the interactions of polarization density P with electric fields and magnetization density M with magnetic fields In Chapters 3 to 5 we calculate total forces on polarizable and magnetizable bodies by using an energy method We extend this method to account for force densities in polarized or magnetized media that are electrically linear isotropic and homogeneous This limitation in our discussion of polarization and magnetization forces is imposed because use of an energy method requires a knowledge of the mechanical and thermodynamic properties of the material 8 1 FORCES IN MAGNETIC FIELD SYSTEMS Consider first the force resulting from the interaction of moving free charge i e J and a magnetic field The Lorentz force 1 1 28 gives the total magnetic force on a charge q moving with velocity v as f qv x B 8 1 1 The force density F newtons per cubic meter can be obtained from this expression by writing If f qvv x Bi F lim lim av o0V av o 6V 8 1 2 where f4 qg and vi refer to all the particles in 6V and Bi is the flux density experienced by qj If we can say that all particles within 6V experience the same flux density B we can use the definition of free current density see Section B 1 2 to write 8 1 2 as F J x B 8 1 3 The general definition of 8 1 2 requires the averaging of products whereas the result of 8 1 3 is the product of averages It is not in general true for variables x and y that zY av 1 av Ylav The force density expressed by 8 1 3 however agrees to a high degree of accuracy with all experimental results obtained with common conductors The relation 8 1 3 is valid because the volume 6V can be made small enough 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 of B for the original experiments of Biot and Savart and later Amphret concerned themselves with relating the force density to the free current density J lim qv 6V t J D Jackson Classical Electrodynamics Wiley New York 1962 p 133 Field Description of Magnetic and Electric Forces J Some writers start with 8 1 3 as the basic definition ofthe magnetic force on 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 forces on the charges This is equivalent to the force on a medium if there is some mechanism by which each charge transmits the Lorentz force to the material For example in a conductor the charges can be thought of as particles moving through a viscous material in which case the force that acts on each 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 with the medium For example in a polarized medium pairs of charges dipoles transmit a force to the medium each pair being connected through the structure of an atom or molecule For these cases it is the dipoles rather than the charges that transmit a force to the medium Then it is appropriate to consider the average of the forces on individual dipoles as equivalent to the force 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 field quantities It is useful to have the force expressed as a function of field quantities alone because we often solve field problems without calculating the free current density We find it useful to define the Maxwell stress tensor as a function of the field quantities from which the force density can be obtained by space differentiation The Maxwell stress tensor is particularly useful 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 chapters which follow We therefore devote Section 8 2 to a discussion of the stress tensor using magnetic field stresses as an example We can write 8 1 3 in terms of the
View Full Document
Unlocking...