Reciprocity of magnetic forces

Consider the following interaction between a proton and an electron:

The electric forces between the proton and electron exhibit “reciprocity.” That is, the electric force that the proton exerts on the electron is equal in magnitude and opposite in direction to the electric force that the electron exerts on the proton. The reason of course is that the electric force is proportional to q1q2 (which equals q2q1) and is also proportional to the vector that points from one charge toward the other. (Similarly, gravitational forces also exhibit reciprocity, because the gravitational force is proportional to m1m2 (which equals m2m1) and is also proportional to the vector that points from one mass toward the other.)

The magnetic force is quite different. The moving proton contributes zero magnetic field at the location of the electron and so no magnetic force acts on the electron, whereas the electron does contribute a nonzero magnetic field at the location of the proton, into the page, so there is a magnetic force on the proton, in the +y direction. The magnetic force does not have the property of reciprocity. To put it another way, the forces do not obey Newton’s third law, which means that this “law” is not fundamental but rather a relationship that applies to electric and gravitational interactions but not to all types of interaction.

This raises an interesting issue. Choose the proton plus electron as the system of interest. The net force on this seemingly isolated system is nonzero, which means that the total momentum of the system will change, which implies that momentum is not a conserved quantity in the presence of magnetic forces. In a 1942 paper by J. M. Keller, “Newton’s Third Law and Electrodynamics” (American Journal of Physics 10, 302-307, https://doi.org/10.1119/1.1990405), the author discusses this situation and shows that if one adds the field momentum to the particle momentum, this total momentum indeed does not change.

In magnetic interactions between closed current loops, reciprocity of the forces (Newton’s third law) is valid. It is interesting to see in detail how this comes about. In the GlowScript VPython program shown below, the two current-carrying loops of wire are divided into many short segments. The Biot-Savart law is used to calculate the magnetic field that a loop contributes at the location of each segment of the other loop, and the magnetic force exerted on that segment. Also calculated is the torque of this force around the center of the scene (at the left edge of the right loop, which is the origin of the coordinate system, where x is to the right, y is up, and z is out of the page). The cyan arrows represent the magnetic field at a location around a loop, due to the other loop, and red arrows represent the magnetic force on that segment, due to the magnetic field of the other loop. Ignoring tiny components due to numerical roundoff, the calculation shows that the net forces on the two loops have the same magnitude and opposite directions. The result is not obvious from looking at the very different distribution of forces over the two loops.

Right button drag or Ctrl-drag to rotate “camera” to view scene.
To zoom, drag with middle button or Alt/Option depressed, or use scroll wheel.
On a two-button mouse, middle is left + right.
Shift-drag to pan left/right and up/down.
Touch screen: pinch/extend to zoom, swipe or two-finger rotate.

Since the net force on the left loop is in the –z direction and the net force on the right loop is in the +z direction, one might expect the torques about the center of the scene would both be in the –y direction. The forces on the segments of the left loop are all in the –z direction, so the torque must necessarily be in the –y direction. However, on the right loop the force is in the +z direction only on about a third of the loop, and the distance to the center of the screen is small (recall that the center of the screen is at the left edge of the right loop). There are smaller forces on the rest of the loop, but they are all in the –z directions and the distances from the center of the screen are large. The net effect is that the net torque on the right loop is in fact in the +y direction, with the same magnitude as the torque on the left loop. The net torque on the combined system is zero.

If the loops are free to move and start from rest, in the first short time interval the left loop will acquire some momentum in the –z direction and the right loop will acquire the same magnitude of momentum, in the +z direction. The left loop will acquire some angular momentum in the –y direction and the right loop will acquire the same magnitude of angular momentum, in the +y direction.

Bruce Sherwood

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11 Responses to Reciprocity of magnetic forces

  1. Gonzalo Fuster says:

    Excelente!
    Saludos, Bruce.
    Gonzalo Fuster

  2. Dusan says:

    1. If the loops are free to move and start from rest, wouldn’t they eventually become parallel to each other and attract, since current in these two loops would be in the same direction?

    2. Do you have any qualitative explanation as to why reciprocity exists when the loops are closed? It’s a very cool discovery.

    Hope you can find the time to answer, all the best professor Bruce!

    • BruceSherwood says:

      1. It’s not entirely obvious to me what the final state would be, but note that if you place two bar magnets side by side, with their magnetic moments in the same direction, they will flip so that their magnetic moments are opposite to each other; that’s the stable position.

      2. I don’t actually know the nature of the proof of reciprocity for closed current loops. I’ve been told that it can be proved, and I merely cite the result but then show the details of the fields and forces, which I hadn’t seen before.

      • Dusan says:

        “It’s not entirely obvious to me what the final state would be, but note that if you place two bar magnets side by side, with their magnetic moments in the same direction, they will flip so that their magnetic moments are opposite to each other; that’s the stable position.”

        I’ve read that explanation in your book, regarding magnetic materials (beautiful explanation btw, really).

        I think that that’s correct, when they can *only* rotate. But if they can move spatially, I think it would be like when we place two magnets next to each other, they would just pinch.

        Right now I’m holding 2 magnets, disk-shaped, so I can find a stable position when they are side by side (like you mentioned), because of the disk-shape. But they generally want to be parallel to each other. It’s obviously shape-dependent also.

  3. I thought I’d mention, for the record, that the same interaction is presented by Feynman as a puzzle for students “to worry about,” in The Feynman Lectures on Physics, Volume II, Section 26-2; it is illustrated in figure 26-6: http://www.feynmanlectures.caltech.edu/II_26.html#Ch26-F6. The puzzle is resolved at the end of the following chapter, on the subject of field energy and field momentum.

  4. BruceSherwood says:

    Thanks much for that reference. I thought I remembered Feynman having posed that puzzle but when I looked for it recently I failed to find it. Another puzzle is whether it was reasonable to imagine even a very good Caltech student being able to find the answer.

  5. BruceSherwood says:

    Here is what Feynman says at the end of Volume II, Chapter 27:

    “We will mention two further examples of momentum in the electromagnetic field. We pointed out in Section 26–2 the failure of the law of action and reaction when two charged particles were moving on orthogonal trajectories. The forces on the two particles don’t balance out, so the action and reaction are not equal; therefore the net momentum of the matter must be changing. It is not conserved. But the momentum in the field is also changing in such a situation. If you work out the amount of momentum given by the Poynting vector, it is not constant. However, the change of the particle momenta is just made up by the field momentum, so the total momentum of particles plus field is conserved.”

    • Dusan says:

      Could you please explain, in a little simpler language, what exactly does “field momentum” means? It’s a little vague term in my head, can’t really figure out what it’s meant by that. And, ensuing question, how can momentum in the field change?

      • BruceSherwood says:

        I can’t provide a tutorial here on electromagnetic fields, which carry both energy and momentum. Here is a Wikipedia article on “radiation pressure” associated with field momentum:

        https://en.wikipedia.org/wiki/Radiation_pressure

        In the final (radiation) chapter of our textbook is a brief analysis of why light can change the momentum of an object. The electric field in the light makes the charge move, and the magnetic field in the light exerts a force on the moving charge which turns out to be in the direction of the propagating light. The field momentum decreases and the particle momentum increases. If the fields in the light change (say by changing the acceleration of the emitting charges) then of course the field momentum changes.

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