On the Velocity in the Lorentz force Law

By A.K. T. Assis and RM. Peixoto

lassical electromagnetism is composed of three distinct parts, namely, C (1) Maxwell's equations; (2) Constitutive relations depending on the medium(1ikeOhm'slawV=RI,D = fE, 1 = (fE,S = )lH , etc.); and (3) the Lorentz force law. This last one states that a point charge q moving in an ar­ bitrary electromagnetic field is acted on by a force

(I)

-> ...... In this equation E E (r, t ) is the electric field at a point r where the charge ~ ~ ~ q is located at the time t, and B = B (r, t) is the magnetic induction at the same point and at the same time. Thc velocity -; that appears in Eg. (1) is the instantaneous velocity of the test charge q. A fundamental question is: Velocity of q relative to what? Of course position, velocity, and acceleration arc not intrinsic properties of any system, and any body can have sillluitaneously different velocities relative to different objects. What is the velocity of a man who is driving a car on a road at 80 km/h? Relative to his own car it is zero, relative to the Earth it is 80 kmfh, relative to another car moving in the opposite direction at 60 km/h it is 140 km/h, relative to the Sun it is approximately 30 km/s, and so on. Andre K.T. Assis is professor 0/ physics at Physically there are many meaningful possibilities: (A) The velocity of the the State University o/Campinas (Institute charge q relative to a fixed ether in space, or relative to an ether at rest in the frame 0/ Physics-DRCC, State University of 1 Campinas, 13081 Campinas, Sp, Brazil) of the "fixed stars" (like the "aether" of Maxwell and Fresnel ); (B) Relative to where he researches Weber s electrodynam­ the laboratory or to the Earth; (C) Relative to an inertial frame of reference; (D) ics, Ampere's force between current ele­ Relative to an arbitrary observer, not necessarily an inertial one; (E) Relative to ments, Mach s principle, and the origin 0/ the macroscopic source of the magnetic field B (a magnet or a wire carrying a inertia. He spelll the year of 1988 as a post­ current /); (F) Relative to an average motion of the microscopic charges which doctoral fellow at the Culham lAboratory (UKAEA, England). From October 1991 to generate 8, the electrons; and (G )Relative to the magnetic field. As a matter of October 1992 he was a visiting scholar at fact, in the dcvelopment of electrodynamics many force laws were proposed with the Center for Electromagnetics Research different quantities being relevant to them. In Weber's electrodynamics, for (Northeastern University, Boston, MA instance, which is the oldest of all these models, only the relative velocities and 02115)_ accelerations between interacting charges were important, so that the force always had the same value for all observcrs.2-9 In Clausius's theory, on the other hand, the force law called for the velocities of the charges relative to an ether.1o

Standard Presentations Curiously, when most textbooks introduce the Lorentz force law they do not state explicitly what the velocity --: in Eg. (I) is relative to. Some examples in well-known works include:

480 THE PHYSICS TEACHER VOL. 30, NOV. 1992 "On the Velocity in the Lnrentz Force Law·' • (I) Symon II: "The force exerted by a magnetic ficld on a charged particle ~ ~ at a point r depends on the velocity v of the particle, and is given in terms -I- -I- -I- -1- ...... ofthemagneticinductionB (r )bytheequationF = qv xB Ic." • Fcynman 12: "We can write the force 1 on a charge q moving with velocity ...... --'> -I- -+ ...... v as F = q (E + v x B)." "The force on an electric charge depends not only on where it is, but also on how fast it is moving ... It is possible ... to write the magnetic force as q-; x B." • Jackson13: "Also essential for consideration of charged particle motion is the Lorentz force eguation, 1 = qE + q-; x B, that gives the force acting on a point charge q in the presence of electromagnetic fields." "The total elec­ tromagnetic force on a charge particle is 1= qE + q--; x B." • Reitz and Milfordl4: "For the purpose of defining the magnetic induction it is convenient to define the magnetic force (frequently called the Lorentz 1,m force), as that part of the force exerted on a moving charge which is neither electrostatic nor mechanical. Thc magnetic induction, B, is then defined as the vector which satisfies = x for all velocities." 1m q--; B, J5 • Sears : "Force on a moving charge .... A positive charge q, moving with velocity v perpendicular to the direction of the induction, is found to experience a force F in the direction shown, pcrpendicular to its velocity v and to the induction B. The magnitude of this force is given by F = qvB.... Evidently, then, the force 1 on a charge q moving with velocity --; in a --'> -+ ->-1- magnetic field of flux density B is in vector notation, F = qv x B ." • Purcell l6: "We say that an electric current has associated with it a magnetic field which pervades the surrounding space. Somc other current, or any moving charged particle which finds itself in this field experiences a force proportional to the strength of the magnetic field in that locality. The force is always perpendicular to the velocity, for a charged particle. The entirc --'> --'> ...... -> forceona particle carrying chargeq isgivcn by F = qE + qv x B where B is the magnetic field. We shall take Eg. (1) as the definition of B." • Panofsky and Phillips17: "According to Lorentz' electron theory, howevcr, the only force which has physical significance is a resultant force which arises from the space-time forces acting on material charges and currents, namely, those obtained by averaging = q (i+ -; x ~ 1 II ). Fabio M. Peixoto is currently fillishing his After reading all these passages, a curiolls student could ask quite naturally and ulldergraduate courses ill physics at the with complete reason, "But velocity of the charge q relative to what?" State Ulliversity of Campillos (Institute of In ouropinion this lack of a clear initial statement on thc meaning of the velocity Physics-DRCC, State University of Cam­ in Eq. (I) is the reason for the confusion of students on this essential aspect of thc pillos, 13081 Campinas, SF, Brazil). Heper­ theory. Vlhen we asked students who had taken courses of electromagnetism formed research ill Weber s electrodynamics and also has all interest in Mach s principle. (undergraduate or graduatc) to explain the meaning of--; in Eg. (1) we received all sorts of answers from (A) to (G) above. At the conclusion of this article we present a short discussion of the origins and meanings of the expression q--; x B, where it is shown that even historically therc have been different interpretations.

The Meaning o/the li?locity Of course when we study any of the books citcd, especially the sections dealing with the special theory of relativity, we grasp the c~ect answer according to the standard electromagnctic theory, namely, in Eg. (1) v is the velocity of the charge q relative to an observer or frame of reference. If we apply the Lorcntz force law together with Newton's second law of motion (1= 11/-::) or with ils analogous --'>...... -+ -+; 22 relativistic generalization (F = dp Idt, withp = mav I I-v Ie ),thenthe observer or reference frame needs to be an inertial one. If this were stated more

··On the V~locity in the Lorentz Force Law·· VOL. 30, NOV. 1992 IHE PHYSICS TEACHER 481 clearly when the Lorentz force law is presented, students would be Fm q~ x B",' Let us suppose now that the frame could understand much more easily the interdependence and ~ ~ mutual transfonnation of electric and magnetic fields. This S' is moving relative to the frame S with a velocity v = vq , so that in this frame S' the charge q would be seen at rest, interconnection of E and B appears only due to this meaning ~ ~ ~ of the velocity -; in Eq. (1). Since this velocity is understood namely, v 'q = O. Moreover v '", = -vq , so that the magnetic as a velocity relative to an inertial observer, fields E and B force in this frame S' would be, according to our last expres­ must also be understood like that, so their values must be sion, F'm= q' [0 - (-~)] x B'm. Since q' = q and different in different inertial frames. We can see the necessity Bm = if~ we then would get F'm = F · of this if we imagine an inertial frame S in which there is no m electric field but only a magnetic field, and that a charge q This indicates clearly that the transfonnation properties of moves relative to this frame S. According to the Lorentz force E and i into E' and B', and vice versa, arise (are necessary) ~ ~ ~ law, it should feel a force given by F '" qv x B. Relative only because -; in the Lorentz force law has different values to another inertial frame S'-which moves relative to S with in different inertial frames. the same velocity -;, so that in this frame the charge q is at History of the Magnetic Force rest (-;' = O)-the force would be zero if there were no transfonnation of the fields. But the force must exist in all According to Whittaker (pp. 306-310)10 the first to arrive inertial frames, which means that somehow there must exist at an expression like this (except for the factor 1/2) was J.J. Thomson in 1881, two years after Maxwell's death. Thomson an electric field If'in the frame S' that will exert a force on ~ ~ q. We present here the general Lorentz transfonnations of the arrived at the expression qv x B 12 when studying theoret­ fields: If in an inertial frame S we have the electric and ically the force of a magnet on a charge q moving through a magnetic fields given by E and B, then in an inertial frame medium characterized by a dielectric constant € and magnetic S', which moves relative to S with a velocity -;, the fields will permeability /-I. According to him the velocity -; in this --> --> 18 expression (which he called the "actual velocity" of the be E ' and B " and these are related to the previous fields by : charge) was not relative to the magnet or to the observer, but relative to the medium. III his own words: "It must be re­ E (2) E'II marked that what we have for convenience called the actual velocity of the particle is, in fact, the velocity of the particle B'II ~ B (3) relative to the medium through which it is moving ... [a] medium whose magnetic pemleability is /-1:.19 Eight years ~ (E + v x -B ) (4) later Heaviside, in another theoretical paper, corrected the E' = 1- h-lf;c2 .1 factor 1/2 of lllOmson's work. He did not comment on Thomson's meaning of the velocity -;, so we can assume that ->--> ..... 2 (B -v xE/c) (5) he accepted Thomson's interpretation. This is even more B' = .; 22.1 evident by the title of his work: On the electromagnetic 1- 1 - v Ie effects due to the motion of electrification through a dielec­ tric.2o A detailed and careful analysis of the works of Thom­ In these equations the subscripts I I and..l mean the compo­ son and Heaviside can be found in BUchwald's book.21 nents of the fields parallel and perpendicular to the direction 111eoretical phYSicist Lorentz presented his force law, Eq. 22 of the velocity -;, respectively. (1), for the first time in 1895. Contrary to the interpretations As a matter of fact, if-; were, for instance, the velocity of ofl1lOmson and Heaviside, and also contrary to om present- ~ q relative to the macroscopic source of B (let us suppose a day interpretation, Lorentz maintained that the velocity v in magnet) then the magnetic force would not change from an Eg. (1) was the velocity of the charge q relative to the ether, inertial frame S to another inertial frame S' moving with which according to him was in a state of absolute rest relative velocity -; relative to S. If this were the case the magnetic to the frame of the fixed stars. I This can also be seen in his -> --> -> --> component of Eg. (1) would read {; = q (Vq - v,) x B ",' most famous book, The Theory ofElectrolls,23 and also ill his Lectures 011 Theoretical Physics. 24 111e present-day interpre­ where are the velocities of the charge and magnet, ~ and~! tation that the velocity -; in Eq. (1) is to be understood as the respectively, relative to an inertial frame of reference. More­ velocity of the charge q relarive to an i/tenial frame of over 8", would be the magnetic field at the position of the reference, the same being true for the electric fields E and 25 charge q, relative to the magnet (and therefore having the if, appeared for the first time in Einstein's paper of 1905. same value to all observers since it would depend only on the There he presents the difference between the old paradigm position of q relative to the magnet). If in frame S we had of electromagnetism and the new one based on his theory of ~n = 0 (which means that the magnet is at rest in this frame), relativity. It seems that later on Lorentz accepted tllis inter­ then according to our last expression the magnetic force pretation by Einstein.

482 THE PHYSICS TEACHER VOL. 30, NOV. 1992 "On the Velocity in the Lorentz Force Law" We think that it is very instructive to ,,------.,---.... see this conceptual change in the mean­ ing of one of the most utilized expres­ sions of physics. Perhaps this is one of the reasons for the lack of clarity in the major 1328 W, Palo Alto Ave, • Fresno, CA 93711 part of textbooks when presenting the Phone: (209) 435·5273 • Fax (209) 449·9760 Lorentz force law. &--d~ Acknowledgments Bring your documents alive with a click of the mouse! Author A.K.T.A. wishestothankFAPESP and CNPq (Brazil) for their fmancial support. The Art or Physics & Hewitt Drewlt • comprehensive library of lecture demo References 1. A. Pais, Subtle Is the Lord (Oxford Uni­ and laboratory equipment versity Press, Oxford, 1982), p. III. • conveniently organized in HyperCard 2. J.e. Maxwell, A Treatise on Electricity and Magnetism (Dover, New York, The Art or Biology 1954), Vol. 2, Chap. 23. 3. A. O'Rahilly, Electromagnetic Theory­ • comprehensive library of biology graphics A Critical Examination a/Fundamentals (Dover, New York, 1965), Vol. 2, Chap. • 195 graphics organized in HyperCard & II. as 300 dpi PICTfiles • Send for free Demo Disk 4. J.P. Wesley, "Weber's electrodynamics, Available in either Macintosh or IBM compatible formats Parts I to III," Found. Phys. Lett. 3, 443, '-____...... __ ..... ______.....1 471,and586. 5. A.K.T. Assis, "Weber's law and mass variation," Phys. Lett. A, 136, 277 (1989). - 6. A.K.T. Assis, "Deriving Ampere's law from Weber's law," HadronicJ. 13,441 (1990). 17. W,K.H. Panofsky and M. Phillips, Classical Electricity and 7. R.A. Clemente and A.K.T. Assis, "Two-body problem for Magnetism (Addison-Wesley, Reading, MA, 1964), p. 182. Weber-like interactions," 1m. J. Thea!: Phys. 30, 537 (I991), 18. Ref. 12, Chap. 26,26.9. 8. A.K.T. Assis and ].1, Caluzi, "A limitation of Weber's law," 19. 1.1. Thomson, "On the electric and magnetic effects produced Phys. Lett. A. 160,25 (1991). by the motion of electrified bodies," Philos. Mag. 11, 229 9. T.E. Phipps, Jr., 'Toward modemizationofWeber's force law," (1881). Phys. Essays 3, 414 (1990). 20. O. Heaviside, "On the electromagnetic effects due to the motion 10. E.T. Whittaker, A History of the Theories ofAether and Elec­ of electrification through a dielectric," Phi/os. Mag. 27,324 tricity (Humanities Press, New York, 1973), Vol. 1: The Clas­ (1889). sical Theories, pp. 234-235; Also Ref. 3, Vol. I, p. 222. 21. J.Z. Buchwald, From Maxwell to Microphysics-Aspects of 11. K.R. Symon, Mechanics, 3rd ed. (Addison-Wesley, Reading, Electromagnetic Theory in the Last Quarter ofthe Nineteenth MA, 1971), p, 141. Century (University of Chicago Press, Chicago, 1985), Appen­ 12. R.P. Feynman, R.B. Leighton, and M, Sands, The Feynman dix I. Lectures on Physics (Addison-Wesley, Reading, MA, 1977), 22. H.A. Lorentz, Versuch Einer Theorie der Electrischetl und Vol. 2, pp. 1.2 and 13.t. Optischen Erscheimmgen in Bewegten Korpern (Brill, Leiden, 13. 1.0, Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New 1895), Section 12; H.A, Lorentz, Collecled Papers (Ny hoff, York, 1975), pp. 2 and 238. the Hague, 1936), Vol. 5, p. L 14. 1.R. Reitz and F.1. Milford, FOllndations 0/ Electromagnetic 23. H.A. Lorentz, The Theory of Electrons (Teubner, Leipzig, Theory (Addison-Wesley, Reading, MA, 1967), p. 148. 1915), pp. 14-15. 15. F.W. Sears, Electriciry and Magnetism (Addison-Wesley, Read­ 24. H.A, Lorentz, Lectures Oil Theoretical Physics (MacMillan, ing, MA, 1958), pp. 227-229. London, 1931), Vol. 3, p, 306; Also, Ref. 3, Vol. 2, p, 563. 16. E.M. Purcell, Ell'ctriciry and Magnetislll, Berkeley Physics 25. A. Einstein, "On electrodynamics of moving bodies," The Course, Vol. 2 (McGraw-Hili, New York, 1965), p. 150. Principle ofRelativity (Dover, New York, 1952); sec especially p.54.

··On the Velocity in the Lort'ntz Force Law" VOL. 30, NOV. 1992 THE PHYSICS TEACHER 483