A recent article by Mansuripur claims that the Lorentz force law is incompatible with special relativity. We discuss the “paradox” on which this claim is based. The resolution depends on whether one assumes a “Gilbert” model for the magnetic dipole (separated monopoles) or the standard “Ampère” model (a current loop). The former case was treated in these pages many years ago; the latter, as several authors have noted, constitutes an interesting manifestation of “hidden momentum.”
The authors thank Kirk McDonald, Daniel Vanzella, and Daniel Cross for useful correspondence. V.H. coauthored this paper in his private capacity; no official support or endorsement by the Centers for Disease Control and Prevention is intended or should be inferred.
Article outline: I. INTRODUCTION II. GILBERT DIPOLES: NAMIAS'S RESOLUTION III. AMPÈRE DIPOLES: HIDDEN MOMENTUM IV. MAGNETIZED MATERIALS V. THE EINSTEIN-LAUB FORCE LAW VI. CONCLUSION
3.A. L. Kholmetskii, O. V. Missevitch, and T. Yarman, “Torque on a moving electric/magnetic dipole,” Prog. Elecromagn. Res. B45, 83–99 (2012). See also M. Mansuripur, “Trouble with the Lorentz Law of Force: Response to critics,” Proc. SPIE, 8455, 845512 (2012).
4.The details are worked out in Cross and Vanzella (Ref. 3).
5.D. A. T. Vanzella, “Comment on ‘Trouble with the Lorentz Law of Force: Incompatibility with Special Relativity and Momentum Conservation,’” Phys. Rev. Lett.110, 089401–1 (2013);
6.A. Cho, “Purported relativity paradox resolved,” ⟨http://scim.ag/Lorpara⟩ (2013). Mansuripur himself does not accept this verdict, though he does appear to have softened his assertions somewhat: M. Mansuripur, “Mansuripur Replies,” Phys. Rev. Lett.110, 089405–1 (2013).
7.V. Namias, “Electrodynamics of moving dipoles: The case of the missing torque,” Am. J. Phys.57, 171–177 (1989);
9.See, for example, D. J. Griffiths, Introduction to Electrodynamics, 4th ed. (Pearson, Boston, 2013), Eq. (7.69).
10.We calculate all torques (in the lab frame) with respect to the origin. But because the net force on the dipole is zero in all cases, it does not matter—we could as well use any fixed point, including the (instantaneous) position of the dipole.
11.W. Shockley and R. P. James, “‘Try simplest cases' discovery of ‘hidden momentum’ forces on ‘magnetic currents,’” Phys. Rev. Lett.18, 876–879 (1967);
12.This is of course an unrealistic model for an actual current-carrying wire. Vaidman (Ref. 11) explores more plausible models, but the result is unchanged.
13.If the center of energy of a closed system is at rest, the total momentum of the system must be zero. See, for example, S. Coleman and J. H. Van Vleck, “Origin of ‘Hidden Momentum Forces' on Magnets,” Phys. Rev.171, 1370–1375 (1968);
14.Mansuripur variously calls hidden momentum an “absurdity” ( M. Mansuripur, “Resolution of the Abraham-Minkowski controversy,” Opt. Commun.283, 1997–2005 (2010), p. 1999), a “problem” to be “solved” (Ref. 1), and “as applied to magnetic materials… an unnecessary burden” (Ref. 6).
15.D. J. Griffiths, “Dipoles at rest,” Am. J. Phys.60, 979–987 (1992). The magnetic field of a point dipole is , where for Ampère dipoles and for Gilbert dipoles. The delta function term leads, for example, to hyperfine splitting in the ground state of hydrogen, and provides experimental confirmation that the proton is an Ampére dipole.
16.We do not know a simple way to prove this directly, but we will confirm it implicitly in Sec. IV.
17.This is certainly not the first time such issues have arisen. How can there be a torque in the lab frame, when there is none in the proper frame? See J. D. Jackson, “Torque or no torque? Simple charged particle motion observed in different inertial frames,” Am. J. Phys.72, 1484–1487 (2004). How can there be a torque, with no accompanying rotation? See D. G. Jensen, “The paradox of the L-shaped object,” Am. J. Phys. 57, 553–555 (1989).
19.The minus sign in is due to the switched sign in “Ampère's law” for magnetic monopoles [see Ref. 9, Eq. (7.44)]. Note that in Eq. (31) [and hence also Eq. (32)], M and P are the densities of the dipole moments of magnetic monopoles and magnetic-monopole currents, respectively.
20.There is some dispute as to the correct form of the Lorentz force law for magnetic monopoles in the presence of polarizable and magnetizable materials, but not when (as here) the polarization/magnetization is itself due to monopoles. See K. T. McDonald, “Poynting's Theorem with Magnetic Monopoles,” (11 pp), ⟨http://puhep1.princeton.edu/ mcdonald/examples/poynting.pdf⟩ (2013).
21.A. Einstein and J. Laub, “Über die im elektromagnetischen Felde auf ruhende Körper ausgeübten ponderomotorischen Kräfte,” Ann. Phys. (Leipzig)26, 541–550 (1908); English translation in The Collected Papers of Albert Einstein, Vol. 2 (Princeton U.P., Princeton, NJ, 1989). In evaluating the force, Mansuripur uses the field H due to q (see Eq. (12b) in Ref. 1); his is our B [Eq. (3)].
22.The ith component of the second term is .
23.B. D. H. Tellegen, “Magnetic-dipole models,” Am. J. Phys.30, 650–652 (1962). Tellegen's force (6), which assumes a Gilbert magnetic dipole, can be obtained by integrating the magnetization-dependent terms in the Einstein–Laub force density in which it is assumed that . Another derivation of the force on a Gilbert magnetic dipole is by A. D. Yaghjian, “Electromagnetic forces on point dipoles,” IEEE Antenna. Prop. Soc. Symp. 4, 2868–2871 (1999).