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I am just reading David Morins "Introduction to Classical Mechanics". He writes about Newtons third law the following:

It holds for forces of the “pushing” and “pulling” type, but it fails for the magnetic force, for example. In that case, momentum is carried off in the electromagnetic field (so the total momentum of the particles and the field is conserved).

Some questions about this:

  • I can imagine that somehow (but it's not clear to me) the colinearity (opposide direction of the action and reaction) fails in the case of two currents because of the Biot-Savart law. However is it also possible that the action and reaction force doesn't have equal magnitudes? So which part of the law fails for magnetic forces?

  • Does the law already fails in magnetostatics? I guess not, but how to prove it?

  • This is my main question: Is there any experiment, ideally something which can be accomplished with high school lab equipment, which shows in a convincing way that Newton's third law doesn't hold for magnetic forces in general?

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  • $\begingroup$ I think this is baloney. For example, two parallel wires carrying current in the same direction are pulled toward one another, each with the same force. $\endgroup$
    – DanielSank
    Sep 23, 2014 at 9:52
  • $\begingroup$ Have a look at this suggestion youtube.com/watch?v=PhfX51FpjwI. One could do the experiment and measure the velocities; simpler for same weight magnets measure distance traveled. This is a suggestion that will show it works for simple magnetic forces. $\endgroup$
    – anna v
    Sep 24, 2014 at 5:39
  • $\begingroup$ have a look also at this which clears up the situations of the "naive" violation of third law physics.stackexchange.com/q/43269 $\endgroup$
    – anna v
    Sep 24, 2014 at 5:41
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    $\begingroup$ From the link above "Feynman gives a simple example, two charge particles, one moving directly towards the other and the other one moving in some other random direction ." from this I would guess that if one set up two wires one in the x direction and the other in the y, with two independent dc circuits and started a dc current at the same time on both (with the correct direction) , only one of them would move. It would be interesting to try if you have the lab equipment. see relevant paragraph digitalcommons.unl.edu/cgi/… $\endgroup$
    – anna v
    Sep 24, 2014 at 5:59

3 Answers 3

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Let charge A be at the origin, moving to the right (along the positive x axis). Let charge B be at coordinates (1,0), moving in the positive y direction.

A's magnetic force on B vanishes, since by symmetry the magnetic field due to A is zero at B's position.

B's magnetic force on A doesn't vanish.

Does the law already fails in magnetostatics? I guess not, but how to prove it?

In magnetostatics there can't be any radiation. If there's no radiation, then mechanical momentum is the only form of momentum we have. If Newton's third law fails, then mechanical momentum isn't conserved. This would lead to a violation of conservation of momentum, which is impossible. So no, the third law can't fail in magnetostatics.

This is my main question: Is there any experiment, ideally something which can be accomplished with high school lab equipment, which shows in a convincing way that Newton's third law doesn't hold for magnetic forces in general?

It would have to be an experiment in which a large amount of momentum was carried away by radiation. Seems tough to me. Even if you build a very powerful and directional radio transmitter, the amount of momentum carried away is tiny in mechanical terms.

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  • $\begingroup$ Thanks, do you have any estimations about the rough magnitudes of the relevant quantities (to get a clearer picture of what means "tough" and "large amount" in this case)? $\endgroup$
    – Julia
    Sep 24, 2014 at 15:06
  • $\begingroup$ Why is magnetic field due to A zero at B's position? I'm sorry for asking after a year... $\endgroup$
    – A Googler
    Apr 23, 2015 at 18:09
  • $\begingroup$ "If there's no radiation, then mechanical momentum is the only form of momentum we have." This is not true. Even if you just have static currents flowing in wires, there is a momentum density associated with the moving charges. The momentum just isn't flowing anywhere in the static case, where as in the time dependent case it can move around, and in the special case of radiation it can escape to infinity. $\endgroup$
    – ApproximatelyTrue
    Jun 4, 2015 at 7:34
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It is an old conundrum how and why Newton's 3rd law fails for the differential form of Biot-Savart. To quote Bleaney & Bleaney:"Page and Adams {1945) have shown that there is no real violation, since the electromagnetic field of the current elements possesses momentum which is changing at a rate just equal to the difference of the two forces." Leigh Page and Norman I. Adams Jr.: Action and Reaction Between Moving Charges American Journal of Physics 13, 141 (1945);

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    $\begingroup$ Well, yes. But exhibiting a case where it is clear that the mechanical momentum of the system is not conserved would be nice. $\endgroup$
    – dmckee --- ex-moderator kitten
    Sep 23, 2014 at 23:05
  • $\begingroup$ That would be a very difficult experiment because it would involve non-closed circuit currents. In the article quoted it is also shown that the total forces on closed loops do cancel. For accelerating charges, "open circuit" current, as @Ben Crowell is alluding to it, the missing momentum is then carried away by the radiation of the accelerating charges. One may plausibly say that the charges are also accelerated in a closed circuit necessarily, and this is the source of the apparent discrepancy but then everything cancel when integrated in the loop, as they should for the 3rd law. $\endgroup$
    – hyportnex
    Sep 24, 2014 at 11:24
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This might be a long answer but hope it helps in resolving the curiosity about this controversial topic. A better understanding of the nature and origin of magnetic force is needed to understand how the Newton’s third law does not fail in magnetostatic. The conservation of momentum explanation is a solid proof that the third law is satisfied in magnetostatics but this explanation does not say HOW, some may make general statements such as “there are corresponding and opposite net forces and torques on the electromagnetic field” and this is not satisfying for me.

It is known that electric currents generate magnetic field/force but since these currents are electrically neutral the magnetic force and the electric force were treated as different aspects of the same subject. Therefore, the actual nature of magnetic field/force was a mystery.

A new work published recently provided a successful explanation to the origin of the magnetic force that is able to explain how the Newton’s third law is satisfied in magnetism, how opposite forces exist, and why they do not appear in some cases, and it was satisfying to me. Shadid in his work “Two new theories for the current charge relativity and the electric origin of the magnetic force” analyzed the electric field pattern spreading in the space due to the movement of electric current charges. Biot-Savart law and Magnetic force law depend on two properties of the current only: Amount and direction. So regardless how the current is generated, e.g., (moving negative charges, moving positive charges, or both moving positive and moving negative charges) as long as it has the same amount and direction it will produce the same magnetic field/force effect in the space. This effect is studied using the minimum possible equal amount of positive and negative charges moving in opposite directions that is needed to produce the current. This minimum amount is obtained when these charges are moving at the maximum possible speed, which is at the speed of light.

In this situation, when the positive and negative charges switch positions in a current element due to their movement the spreading electric field changes from inward to outward and from outward to inward indicating the change of charge positions. This alternation in electric field produces discontinuity points in the spreading electric field. By applying Gauss law at these discontinuity points, the Gauss law indicates the existence of charges, referred to as discontinuity charges. These discontinuity charges surround current elements and are produced when charges move to carry on electric field changes in the space. These charges are partially explained by the photons that travel to indicate the changes in the electric field, these photons are assumed to be charged as assumed in Altschul’s work "Bound on the photon charge from the phase coherence of extragalactic radiation".

These discontinuity charges interact with the moving positive and negative charges in a current element to produce the magnetic force. By applying the electric force law on these interactions the exact magnetic force law and Biot-Savart law are derived as specified in the electromagnetic theory. So, moving electric charges of current elements interact with each other through discontinuity charges, since current elements are electrically neutral. The electric force between a current charge and a discontinuity charge obeys Newton's third law as in Coulomb’s law. The forces exerted on current charges allow the charges to produce either a non-zero or zero net force on the containing infinitesimal current element. This net force on the current element is the observed magnetic force. The produced net force is non-zero on the current element when the positive and negative charges push the current element in the same direction. The push occurs when the exerted forces are perpendicular to the motion direction of the charges and they are not allowed to move outside the containing filamentary current element, while they are free to move along that element. Notice that the push interaction between the current charges and the containing current element obeys Newton’s third law as in particles interaction. However, the net force is zero when these charges push the current element in opposite directions thereby canceling each other or when the exerted forces on the current charges are completely along the direction of movement for the charges so no push force is produced on the containing current element.

The full details of the proof and calculation are a bit long; I gave a brief overview of it. Details can be found at http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=7546893

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