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.”

1.
M.
Mansuripur
, “
Trouble with the Lorentz Law of Force: Incompatibility with special relativity and momentum conservation
,”
Phys. Rev. Lett.
108
,
193901
1
(
2012
).
2.
A.
Cho
, “
Textbook electrodynamics may contradict relativity
,”
Science
336
,
404
(
2012
).
3.
K. T.
McDonald
, “
Mansuripur's Paradox
,” ⟨www.physics.princeton.edu/mcdonald/examples/mansuripur.pdf⟩ (
14
pp);
D. A. T.
Vanzella
, “
Comment on ‘Trouble with the Lorentz law of force
,’” e-print arXiv:1205.1502 (
2
pp);
D. J.
Cross
, “
Resolution of the Mansuripur Paradox
,” e-print arXiv:1205:5451 (
3
pp);
P. L.
Saldanha
, “
Comment on ‘Trouble with the Lorentz law of force
,’” e-print arXiv:1205:6858 (
2
pp);
D. J.
Griffiths
and
V.
Hnizdo
, “
Comment on ‘Trouble with the Lorentz law of force
,’” e-print arXiv:1205.4646 (
3
pp);
Later critiques include
K. A.
Milton
and
G.
Meille
, “
Electromagnetic Angular Momentum and Relativity
,” e-print arXiv:1208:4826 (
4
pp);
F.
De Zela
, “
Comment on ‘Trouble with the Lorentz law of force
,’” e-print arXiv:1210.7344 (
2
pp);
T. M.
Boyer
, “
Examples and comments related to relativity controversies
,”
Am. J. Phys.
80
,
962
971
(
2012
);
A. L.
Kholmetskii
,
O. V.
Missevitch
, and
T.
Yarman
, “
Torque on a moving electric/magnetic dipole
,”
Prog. Elecromagn. Res. B
45
,
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
);
[PubMed]
and, under the same title,
S. M.
Barnett
,
Phys. Rev. Lett.
110
,
089402
1
(
2013
);
[PubMed]
P. L.
Saldanha
,
Phys. Rev. Lett.
110
,
089403
1
(
2013
);
[PubMed]
M.
Khorrami
,
Phys. Rev. Lett.
110
,
089404
1
(
2013
).
[PubMed]
6.
Paradox Lost
,” Science
339
,
496
(
2013
);
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
).
[PubMed]
7.
V.
Namias
, “
Electrodynamics of moving dipoles: The case of the missing torque
,”
Am. J. Phys.
57
,
171
177
(
1989
);
D.
Bedford
and
P.
Krumm
, “
On the origin of magnetic dynamics
,”
Am. J. Phys.
54
,
1036
1039
(
1986
).
8.
See, for instance,
V.
Hnizdo
, “
Magnetic dipole moment of a moving electric dipole
,”
Am. J. Phys.
80
,
645
647
(
2012
).
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
);
W. H.
Furry
, “
Examples of momentum distributions in the electromagnetic field and in matter
,”
Am. J. Phys.
37
,
621
636
(
1969
);
L.
Vaidman
, “
Torque and force on a magnetic dipole
,”
Am. J. Phys.
58
,
978
983
(
1990
);
V.
Hnizdo
, “
Conservation of linear and angular momentum and the interaction of a moving charge with a magnetic dipole
,”
Am. J. Phys.
60
,
242
246
(
1992
);
V.
Hnizdo
, “
Hidden mechanical momentum and the field momentum in stationary electromagnetic and gravitational systems
,”
Am. J. Phys.
65
,
515
518
(
1997
).
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
);
M. G.
Calkin
, “
Linear momentum of the source of a static electromagnetic field
,”
Am. J. Phys.
39
,
513
516
(
1971
).
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 B=(μ0/4π)(1/r3)[3(m·r̂)r̂m]+αμ0mδ3(r), where α=2/3 for Ampère dipoles and 1/3 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).
18.

See, for instance, Ref. 9, Eq. (12.118).

19.

The minus sign in Jb* 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 μ0H is our B [Eq. (3)].
22.

The ith component of the second term is (B·)Mi=·(MiB)Mi·B=·(MiB).

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 M=mδ3(rr0). 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).
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