Science 27 April 2012:
Vol. 336 no. 6080 pp. 404-404
DOI:10.1126/science.336.6080.404
Vol. 336 no. 6080 pp. 404-404
DOI:10.1126/science.336.6080.404
News & Analysis
Textbook Electrodynamics May Contradict Relativity
A scientist claims that the classic formula for the force exerted by electric and magnetic fields—the so-called Lorentz force—clashes with Einstein's special theory of relativity.
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Zhulin Zhang
Comment on "Textbook Electrodynamics May Contradict Relativity"
Lai Zhang，Zhu Lin Zhang Anhui University of Science and Technology, Huainan, Anhui, 232001, P. R. China
Abstract We present an exact solution for Mansuripur’s Paradox.
Recently, Prof. M. Mansuripur considered a point charge q and a magnetic point dipole at rest in the lab frame, there is no force between them. But, in a moving frame, q creates electric field which exerts the force (Torque) on the magnetic dipole. Then, he claimed: “Lorentz force clashes with Einstein's special theory of relativity” [1]. In the following, we are going to prove that such a "paradox" is a mere consequence of putting the boot on the wrong leg. We use a square Ampérian loop ABCD to represent the magnetic dipole, the loop is made of electrically neutral copper wire with a current flowing. It is well known that this simple model is good enough for any scale relatively small（point）magnetic dipole. Let AB remain static in an inertial frame S, extend along the x axis from A(x=0) to B(x>0) and a point charge q be fixed on the x axis(x<0), the physics picture core in [2] is established. In AB, the copper ions keep still, the free electrons move with the velocity -v along the x axis; the positive and negative linear charge densities in AB are ±w respectively, so the net linear charge density is λ=0. Let Q stand for any one point charge of the copper ions or free electrons in AB, the resultant Coulomb force exerted on all of Q by q is zero. Let S’ be another inertial frame moving along the x axis with the velocity -u relative to S and a test point charge p be at rest in S’, in Textbook Relativity, the net linear charge density in AB has been derived for p: λ'(p) =γuvwc^(-2) where γ≡[1-(u⁄c)^2 ]^(-0.5). If uv≠0, λ'(p)≠0 can be understood in terms of the relativity of simultaneity. In S’, q and AB are moving with the velocity u. Textbook Electrodynamics showed that the magnetic field created by q vanishes on AB, and the Lorentz force exerted on a single Q in AB by q is F' =kqQ(γd)^(-2) where, d and γd are distances from q to Q in S’ and S respectively. Immediately, we realize that F’ is nothing, but the Coulomb force exerted on Q by q in S. That is to say, in S’, the force exerted on AB by q is zero too, or λ'(q) =λ=0. Mistaking λ'(p) for λ'(q) is the main reason leading to the “Paradox”.
References [1] A. Cho, Science 336, 404 (2012). [2] Masud Mansuripur, Phys. Rev. Lett. 108, 193901 (2012).
Zhulin Zhang
Mansuripur’s Paradox is nothing, but pranks and jokes. We suggest stop the time consuming discussion on this issue.
Recently, we have read two paper [1][2], which said that “Lorentz force clashes with Einstein's special theory of relativity”. Prof. Masud Mansuripur considered a point charge and a magnetic dipole at rest in our lab reference system, there no Force between them. However, in a moving reference system, the point charge creates electric field which exerts the force (Torque) on the magnetic dipole. He believes that “Lorentz force clashes with Einstein's special theory of relativity”.
Several authors have submitted very complex comments [3~5] on this issue, unfortunately, the clumsy mathematics in their comment papers swamped the simple physics core.
A magnet (ideal magnetic dipole) moving in our Lab system appears to be electrically polarized is a pure effect of the special relativity [6]. In the magnet frame, no such electric dipole exists, therefore, “No force or torque is generated in cases involving a charge and a magnet with their relative velocity zero” [3]. Our earth is a magnet moving in the Solar system; it could generate the electric dipole near the equator of the earth too. If you can sit on Jupiter, you would measure the electric dipole due to the relative motion between them. But, it’s not possible to find the electric dipole if sitting anywhere on the earth. A similar phenomenon was proofed by Röntgen experimentally [6] in 1888. An ideal electric dipole moving in our Lab system appears to be magnetically polarized due to the pure effect of the special relativity [6].
Zhu Lin Zhang and Lai Zhang Department of Physics, College of Mathematics and Physics, Anhui University of Science and Technology, Huainan, Anhui, 232001, China
REFERENCES [1] Masud Mansuripur ， Trouble with the Lorentz Law of Force: Incompatibility with Special Relativity and Momentum Conservation，Phys. Rev. Lett. 108, 193901 (2012) [4 pages]
[2] Adrian Cho, Textbook Electrodynamics May Contradict Relativity, Science 27 April 2012: Vol. 336 no. 6080 p. 404.
[3] C. S. Unnikrishnan, Does Lorentz Force Law Contradict the Principle and Theories of Relativity for Uniform Linear Motion? arXiv: 1205. 1080v2 [physics. class-ph] 8 May 2012.
[4] Daniel A. T. Vanzella, Comment on “Trouble with the Lorentz law of force: Incompatibility with special relativity and momentum conservation”, arXiv:1205.1
Francis Redfern
I would like to clarify a point i made in my previous comment. The reason, in my view, that you can consider a charge at rest with respect to a current-carrying wire to be at rest with the magnetic field, but you cannot consider charge moving with respect to a current-carrying wire to be at rest with the magnetic field in its rest frame is that the former does not violate charge conservation, but it is easy to imagine situations where the latter does. Again, i refer you to my website, http://prism.texarkanacollege.edu/physicsjournal/paradox.pdf.
Francis Redfern
If you replace the magnet in the figure with a current-carrying loop, i think the resolution to this problem may be fairly simple and involve the fact that an observer moving parallel to a wire carrying a current must see the wire in electrostatic equilibrium, just as an observer at rest with respect to the wire must see the wire in electrostatic equilibrium. This is required by the principle of relativity. The loop will then not exhibit charge polarization when moving in the frame of a stationary observer and no torque will act on the loop. Consider an observer beginning to speed up from rest parallel to a current-carrying wire in the direction of the positive current. The electrons will appear closer together due to Lorentz contraction, but the positive ions will also appear to move closer together - not because of Lorentz contraction but to maintain electrostatic equilibrium as seen by the moving observer. The wire as a whole will be shorter. If this were not the case you can invent situations where electric charge is not conserved. The Lorentz transformation of the magnetic field of the wire from its rest frame to a moving one in the positive current direction results in a radial electric field pointing toward the wire. There is no need for there to be an electric charge on the wire from the perspective of the moving observer any more than there needs to be one at the center of a cyclotron orbit. I have posted a more detailed discussion of this idea on my website, http://prism.texarkanacollege.edu/physicsjournal/paradox.pdf.
Lynn Wilson
There is a mistake in the assumptions associated with the bottom left figure in Cho's article. Unless the "magnetic" is a superconductor, a charged particle will have an effect on the magnetization/polarization of the magnet. Both the magnetic and electric fields need to be transformed and John David Jackson's book on electrodynamics deals with this in chapter 12.10 Section C. Jackson actually explains why one must be careful to use the Lorentz force density on 4-vector charge and current densities.
There is another problem with the different reference frames argument concerning the wire vs. particle frame. The particle frame is not an inertial frame and would require infinite Lorentz transformations to account for the accelerating reference frame.
Unless I am completely missing something, this is not an issue.
Peter Miles
The observer sees the circulating electrons more densely bunched together when they are moving away from the positive charge, however she sees them as more massive by an equal relativistic factor. Because of this, even in the absence of dissipative forces, they come to the end of a complete circuit of their path with the same velocity as when they started. They do not acquire angular momentum, despite the fact that they spend more time on the part of their path when they move away from the positive particle, so they cannot impart torque to the magnet.
oliver elbs
As usual: physicists will just have to reset and reformulate their maps. That is all (about this debate). After all, it is never some "truth" that is at stake, but only maps. And maps are malleable and can always be adapted... (for social and technical purposes). Life (and maps) will go on... Hence, there is less religiosity (and philosophy) here concerning this debate than it seems at first glance...
Ari Diacou
I would have liked a citation stating an experiment that shows that there is no torque on the magnet in the lab frame. I would think that it would be a pretty easy experiment. I suggest: Take a magnet with a dot painted on it, levitate it over a superconductor, fire an electron/or charged particle beam to the side of it, and see if the dot changes speed. This would be pretty hard to do in a vacuum, since that would boil away the refrigerant for the superconductor, so you would have to use a heavy particle particle beam.
Michael Walker
So what is wrong with polarization and magnetization being fundamental, given that point particles carry angular momentum and the quantum vacuum can be polarized?
Mingsheng Tan
This issue aroused mine curiosity and interest, although I'm not reseach this field, it's also significantly to know about it. I want to say truth can undergo the verification of time.