This editorial discusses "Trouble with the Lorentz Law of Force: Incompatibility with Special Relativity and Momentum Conservation"

Barnett, Stephen M.

English

Barnett, S.M.

Enlighten

        Barnett, S.M. (2013) Comment on “Trouble with the Lorentz Law of Force: Incompatibility with Special Relativity and Momentum Conservation”. Physical Review Letters, 110 (8). 089402. ISSN 0031-9007   Copyright © 2013 American Physical Society.   A copy can be downloaded for personal non-commercial research or study, without prior permission or charge   The content must not be changed in any way or reproduced in any format or medium without the formal permission of the copyright holder(s)   When referring to this work, full bibliographic details must be given      http://eprints.gla.ac.uk/89039/        Deposited on:  10 Jan 2014          Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk Comment on “Trouble with the Lorentz law offorce: Incompatibility with special relativity andmomentum conservation.”The recent claim that the Lorentz law of force is in-compatible with the requirements of special relativity [1],essentially that the Lorentz force law is incompatible withthe Lorentz transformations, was based on treating onlythe spatial part of the Lorentz force. Including the ratesof change for both the energy and the momentum re-stores compatibility. We work throughout in the naturalsystem of units: ε0, µ0 and c are all set equal to unity.We first review, briefly, the argument given in [1]. TheLorentz force law describes the force exerted by the elec-tric and magnetic fields on charges and currents. Ex-pressed as a force density it becomesF = ρE+ J×B . (1)The objection under consideration deals with the actionof a charge on a magnetic dipole. We were asked to re-strict our attention to bound charges so as to rewrite theforce density in terms of the polarization and magnetisa-tion:F = −(∇ ·P)E+ (P˙+∇×M)×B . (2)Consider a charge q at the origin of a primed coordinatesystem and a magnetic dipole oriented in the x′ directionplaced a distance d from the charge along the z′ axis.Naturally, the charge exerts neither a force nor a torqueon the dipole in this frame, as may readily be verifiedusing the magnetizationM = m0xˆ′δ(x′)δ(y′)δ(z′ − d) (3)and the force density (1). Consider now the situation asviewed from an unprimed frame in which the charge andthe dipole are moving at speed V in the positive z direc-tion. In this frame the charge is moving and so producesa magnetic field. The moving magnetic dipole, more-over, acquires some electric dipole character by virtueof its motion. It follows, therefore, that the force givenin Eq. (2) will include electric and magnetic fields andalso both a magnetization and a polarization centeredon the magnetic dipole. The net force exerted by thecharge on the dipole remains zero but there is a torque,T = (V qm0/4pid2)xˆ, suggesting that the charge coercesthe dipole to spin about its axis and, moreover, that thetorque inducing this is proportional to the velocity atwhich the charge and the dipole are seen to be moving.Were this to induce a rotation of the dipole then therewould indeed be a conflict between the Lorentz force lawand special relativity.The problem with this analysis is that the Lorentz forcelaw is not a three-vector but a four-vector [2]. The time-component of this is J ·E, which gives the rate of changeof the dipole’s energy. The integral of J ·E over space iszero and there is no net transfer of energy just as there isno net force. If we consider the torque, however, then theJ · E term produces a change to the moment of inertiaof the dipole, just as the force density (1) contributes tothe torque. When combined, these two effects result inan angular acceleration:aangular =1I∫d3r r× [ρE+ J×B−V(J · E)] , (4)where I is the moment of inertia of the dipole. For thesituation just described, this angular acceleration is zeroas expected.The −V(J · E) term in (4) is reminiscent of thatfound when calculating the acceleration of a particle withcharge Q and rest-mass m, moving with velocity v [3]:a =Qmγ[E+ v ×B− v(v · E)] , (5)where γ = (1 − v2)−1/2 is the usual Lorentz factor. Wecan associate the additional, v ·E, term with the changein energy of the particle caused by its change in speed. InMansuripur’s magnetic-dipole thought experiment thereis no change in the velocity of the dipole because thereis no net force acting on it, but there is a change in themoment of inertia and this balances exactly the torquederived from (1). This effect is simply the angular analogof Mansuripur’s own argument for resolving the apparentconflict with relativity for the force on a current carryingwire [1].Finally, we note that there are related treatments inthe literature [4].I am grateful to Colin Baxter and Alison Yao for help-ful comments and to Masud Mansuripur for helpful dis-cussions. This work was supported by the UK EPSRC,the Royal Society and the Wolfson Foundation.Stephen M. BarnettDepartment of Physics, University of Strathclyde,Glasgow G4 0NG, UK.[1] M. Mansuripur, Phys. Rev. Lett. 108, 193901 (2012).[2] W. Rindler, Essential Relativity 2nd ed. (Springer-Verlag,New York, 1977).[3] L. D. Landau and E. M. Lifshitz, The Classical Theory ofFields 4th ed. (Butterworth Heinemann, Oxford, 1973).[4] D. Bedford and P. Krumm, Am. J. Phys. 54, 1036 (1986);V. Namias, Am. J. Phys. 57, 171 (1989); L. Vaidman,Am. J. Phys. 58, 978 (1990); G. Spavieri, Found. Phys.Lett. 3, 291 (1990); D. J. Griffiths and V. Hnzido, arXiv:1205.4646; K. A. Milton and G. Meille, arXiv: 1208.4826.I am grateful to all those who kindly brought these to myattention.

Comment on “Trouble with the Lorentz Law of Force: Incompatibility with Special Relativity and Momentum Conservation”

Enlighten: Publications

  
 
 
 
 
 
 Barnett, S.M. (2013) Comment on “Trouble with the Lorentz Law of Force: 
Incompatibility with Special Relativity and Momentum Conservation”. 
Physical Review Letters, 110 (8). 089402. ISSN 0031-9007 
 
 
Copyright © 2013 American Physical Society. 
 
 
A copy can be downloaded for personal non-commercial research or 
study, without prior permission or charge  
 
The content must not be changed in any way or reproduced in any format 
or medium without the formal permission of the copyright holder(s)  
 
When referring to this work, full bibliographic details must be given 
 
 
 
 
 
http://eprints.gla.ac.uk/89039/ 
 
 
 
 
 
 
 
Deposited on:  10 Jan 2014 
 
 
 
 
 
 
 
 
 
Enlighten – Research publications by members of the University of Glasgow 
http://eprints.gla.ac.uk 
Comment on “Trouble with the Lorentz law of
force: Incompatibility with special relativity and
momentum conservation.”
The recent claim that the Lorentz law of force is in-
compatible with the requirements of special relativity [1],
essentially that the Lorentz force law is incompatible with
the Lorentz transformations, was based on treating only
the spatial part of the Lorentz force. Including the rates
of change for both the energy and the momentum re-
stores compatibility. We work throughout in the natural
system of units: ε0, µ0 and c are all set equal to unity.
We first review, briefly, the argument given in [1]. The
Lorentz force law describes the force exerted by the elec-
tric and magnetic fields on charges and currents. Ex-
pressed as a force density it becomes
F = ρE+ J×B . (1)
The objection under consideration deals with the action
of a charge on a magnetic dipole. We were asked to re-
strict our attention to bound charges so as to rewrite the
force density in terms of the polarization and magnetisa-
tion:
F = −(∇ ·P)E+ (P˙+∇×M)×B . (2)
Consider a charge q at the origin of a primed coordinate
system and a magnetic dipole oriented in the x′ direction
placed a distance d from the charge along the z′ axis.
Naturally, the charge exerts neither a force nor a torque
on the dipole in this frame, as may readily be verified
using the magnetization
M = m0xˆ
′δ(x′)δ(y′)δ(z′ − d) (3)
and the force density (1). Consider now the situation as
viewed from an unprimed frame in which the charge and
the dipole are moving at speed V in the positive z direc-
tion. In this frame the charge is moving and so produces
a magnetic field. The moving magnetic dipole, more-
over, acquires some electric dipole character by virtue
of its motion. It follows, therefore, that the force given
in Eq. (2) will include electric and magnetic fields and
also both a magnetization and a polarization centered
on the magnetic dipole. The net force exerted by the
charge on the dipole remains zero but there is a torque,
T = (V qm0/4pid
2)xˆ, suggesting that the charge coerces
the dipole to spin about its axis and, moreover, that the
torque inducing this is proportional to the velocity at
which the charge and the dipole are seen to be moving.
Were this to induce a rotation of the dipole then there
would indeed be a conflict between the Lorentz force law
and special relativity.
The problem with this analysis is that the Lorentz force
law is not a three-vector but a four-vector [2]. The time-
component of this is J ·E, which gives the rate of change
of the dipole’s energy. The integral of J ·E over space is
zero and there is no net transfer of energy just as there is
no net force. If we consider the torque, however, then the
J · E term produces a change to the moment of inertia
of the dipole, just as the force density (1) contributes to
the torque. When combined, these two effects result in
an angular acceleration:
aangular =
1
I
∫
d3r r× [ρE+ J×B−V(J · E)] , (4)
where I is the moment of inertia of the dipole. For the
situation just described, this angular acceleration is zero
as expected.
The −V(J · E) term in (4) is reminiscent of that
found when calculating the acceleration of a particle with
charge Q and rest-mass m, moving with velocity v [3]:
a =
Q
mγ
[E+ v ×B− v(v · E)] , (5)
where γ = (1 − v2)−1/2 is the usual Lorentz factor. We
can associate the additional, v ·E, term with the change
in energy of the particle caused by its change in speed. In
Mansuripur’s magnetic-dipole thought experiment there
is no change in the velocity of the dipole because there
is no net force acting on it, but there is a change in the
moment of inertia and this balances exactly the torque
derived from (1). This effect is simply the angular analog
of Mansuripur’s own argument for resolving the apparent
conflict with relativity for the force on a current carrying
wire [1].
Finally, we note that there are related treatments in
the literature [4].
I am grateful to Colin Baxter and Alison Yao for help-
ful comments and to Masud Mansuripur for helpful dis-
cussions. This work was supported by the UK EPSRC,
the Royal Society and the Wolfson Foundation.
Stephen M. Barnett
Department of Physics, University of Strathclyde,
Glasgow G4 0NG, UK.
[1] M. Mansuripur, Phys. Rev. Lett. 108, 193901 (2012).
[2] W. Rindler, Essential Relativity 2nd ed. (Springer-Verlag,
New York, 1977).
[3] L. D. Landau and E. M. Lifshitz, The Classical Theory of
Fields 4th ed. (Butterworth Heinemann, Oxford, 1973).
[4] D. Bedford and P. Krumm, Am. J. Phys. 54, 1036 (1986);
V. Namias, Am. J. Phys. 57, 171 (1989); L. Vaidman,
Am. J. Phys. 58, 978 (1990); G. Spavieri, Found. Phys.
Lett. 3, 291 (1990); D. J. Griffiths and V. Hnzido, arXiv:
1205.4646; K. A. Milton and G. Meille, arXiv: 1208.4826.
I am grateful to all those who kindly brought these to my
attention.


Stephen M. Barnett

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Physical Review Letters

University of Strathclyde Institutional Repository

Strathprints Institutional RepositoryBarnett, Stephen M. (2013) Comment on "Trouble with the Lorentz Law of Force: Incompatibilitywith Special Relativity and Momentum Conservation". Physical Review Letters, 110 (8). ISSN0031-9007Strathprints is designed to allow users to access the research output of the University of Strathclyde.Copyright c© and Moral Rights for the papers on this site are retained by the individual authorsand/or other copyright owners. You may not engage in further distribution of the material for anyprofitmaking activities or any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or study, educational, ornot-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to Strathprints administrator:mailto:strathprints@strath.ac.ukhttp://strathprints.strath.ac.uk/Comment on ‘‘Trouble with the Lorentz Law of Force:Incompatibility with Special Relativity and MomentumConservation’’The recent claim that the Lorentz law of force isincompatible with the requirements of special relativity[1], essentially that the Lorentz force law is incompatiblewith the Lorentz transformations, was based on treatingonly the spatial part of the Lorentz force. Including therates of change for both the energy and the momentumrestores compatibility. We work throughout in the naturalsystem of units: "0, 0, and c are all set equal to unity.We first review, briefly, the argument given in Ref. [1].The Lorentz force law describes the force exerted by theelectric and magnetic fields on charges and currents.Expressed as a force density, it becomesF ¼ Eþ J B: (1)The objection under consideration deals with the action ofa charge on a magnetic dipole. We were asked to restrictour attention to bound charges so as to rewrite the forcedensity in terms of the polarization and magnetization:F ¼ ðr  PÞEþ ð _PþrMÞ  B: (2)Consider a charge q at the origin of a primed coordinatesystem and a magnetic dipole oriented in the x0 directionplaced a distance d from the charge along the z0 axis.Naturally, the charge exerts neither a force nor a torqueon the dipole in this frame, as may readily be verified usingthe magnetizationM ¼ m0x^0ðx0Þðy0Þðz0  dÞ (3)and the force density (1). Consider now the situation asviewed from an unprimed frame in which the charge and thedipole are moving at speed V in the positive z direction. Inthis frame the charge is moving and so produces a magneticfield. The moving magnetic dipole, moreover, acquiressome electric dipole character by virtue of its motion. Itfollows, therefore, that the force given in Eq. (2) will includeelectric and magnetic fields and also both a magnetizationand a polarization centered on the magnetic dipole. The netforce exerted by the charge on the dipole remains zero, butthere is a torque, T ¼ ðVqm0=4d2Þx^, suggesting that thecharge coerces the dipole to spin about its axis and, more-over, that the torque inducing this is proportional to thevelocity at which the charge and the dipole are seen to bemoving. Were this to induce a rotation of the dipole, therewould indeed be a conflict between the Lorentz force lawand special relativity.The problem with this analysis is that the Lorentzforce law is not a three-vector but a four-vector [2]. Thetime component of this is J E, which gives the rate ofchange of the dipole’s energy. The integral of J E overspace is zero and there is no net transfer of energy just asthere is no net force. If we consider the torque, however,then the J  E term produces a change to the moment ofinertia of the dipole, just as the force density (1) contributesto the torque. When combined, these two effects result inan angular acceleration:aangular ¼ 1IZd3rr ½Eþ JB VðJ  EÞ; (4)where I is the moment of inertia of the dipole. For thesituation just described, this angular acceleration is zero asexpected.The VðJ EÞ term in (4) is reminiscent of that foundwhen calculating the acceleration of a particle with chargeQ and rest mass m, moving with velocity v [3]:a ¼ Qm½Eþ vB vðv  EÞ; (5)where  ¼ ð1 v2Þ1=2 is the usual Lorentz factor. Wecan associate the additional v E term with the change inenergy of the particle caused by its change in speed. InMansuripur’s magnetic-dipole thought experiment there isno change in the velocity of the dipole because there is nonet force acting on it, but there is a change in the momentof inertia and this balances exactly the torque derivedfrom (1). This effect is simply the angular analog ofMansuripur’s own argument for resolving the apparentconflict with relativity for the force on a current carryingwire [1].Finally, we note that there are related treatments in theliterature [4].I am grateful to Colin Baxter, Alison Yao, and MasudMansuripur for helpful comments and discussions and toall those who kindly brought to my attention the paperscited in Ref. [4]. This work was supported by the U.K.EPSRC, the Royal Society, and the Wolfson Foundation.Stephen M. BarnettDepartment of PhysicsUniversity of StrathclydeGlasgow G4 0NGUnited KingdomReceived 8 May 2012; published 20 February 2013DOI: 10.1103/PhysRevLett.110.089402PACS numbers: 41.20.q[1] M. Mansuripur, Phys. Rev. Lett. 108, 193901 (2012).[2] W. Rindler, Essential Relativity (Springer-Verlag, NewYork, 1977), 2nd ed.[3] L. D. Landau and E.M. Lifshitz, The Classical Theory ofFields (Butterworth-Heinemann, Oxford, 1973), 4th ed.[4] D. Bedford and P. Krumm, Am. J. Phys. 54, 1036 (1986); V.Namias, Am. J. Phys. 57, 171 (1989); L. Vaidman, Am. J.Phys. 58, 978 (1990); G. Spavieri, Found. Phys. Lett. 3, 291(1990); D. J. Griffiths and V. Hnzido, arXiv:1205.4646;K. A. Milton and G. Meille, arXiv:1208.4826.PRL 110, 089402 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending22 FEBRUARY 20130031-9007=13=110(8)=089402(1) 089402-1  2013 American Physical Society

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