Magnetic forces do no (net) work

Because the magnetic force on a moving charge is perpendicular to the velocity, the work done by a magnetic force is zero. However, in a multiparticle system it can happen that magnetic forces can rearrange the energy within the system, as long as these forces do as much negative work as they do positive work, so that the net work done is zero. An interesting example of this is the case of a rotating wheel carrying point charges in a magnetic field:

There is a fixed, current-carrying coil on the left with a magnetic dipole moment pointing to the right. When you click Run, the wheel is given an initial spin so that it too has a magnetic dipole moment pointing to the right, and it is attracted toward the coil, sliding with negligible friction on a long rod. Now that it is moving to the left, it is easy to show that there are now magnetic forces that act tangentially on the point charges, thereby exerting a torque that slows down the rotation and eventually reduces the angular speed to zero and even makes the wheel spin in the opposite direction to the original spin direction. Now that the wheel’s magnetic dipole moment is to the left, the current-carrying coil repels the wheel, bringing the wheel momentarily to a stop and then pushing the wheel to the right. The torque on the wheel again reverses and restores the original spin direction, and the wheel is again attracted to the coil so that the translational motion decreases. The effect is to lead to an oscillation of the wheel, back and forth. (Right button drag or Ctrl-drag to rotate the “camera”; to zoom, drag with middle button or Alt/Option depressed, or use a scroll wheel; on a two-button mouse, middle is left + right. After clicking Run, use Shift-drag to pan left/right, up/down. With a touch screen, pinch/extend to zoom, swipe or two-finger rotate.)

The graph that appears when you click Run shows the rotational kinetic energy of the wheel (red) which initially decreases as the translational kinetic energy (green) increases. The sum of the rotational and translational kinetic energies remains constant (the blue line), consistent with the fact that magnetic forces do zero net work on a system.

Another example of this is the case of a conducting bar sliding along conductive rails in the presence of a constant magnetic field perpendicular to the plane of the rails. With a resistor attached across the rails, it is necessary to exert a force on the bar to keep it moving at a constant speed, and a current runs in the resistor due to a “motional emf” of amount vBL, where v is the speed of the bar, B is the magnitude of the magnetic field, and L is the distance between the two rails. As the bar slides along, a mobile electron in the bar moves at an angle to a perpendicular to the rails, and when this path direction is taken into account one sees that the magnetic force on the moving electron has a component parallel to the rails which does negative work, opposing the force you apply, and a component perpendicular to the rails, which does positive work that heats the resistor. The total work done by the magnetic force can be shown to be zero.

Bruce Sherwood

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6 Responses to Magnetic forces do no (net) work

  1. Dan MacIsaac says:

    Nice, and thank you. Just subscribing directly to your blog.

  2. Chris Fuller says:

    Another excellent article! Thank you. What about the work performed by the force and resulting acceleration caused by two magnetic moments of the same magnitude and phase? The force is proportional to the divergence of the dot product of the two magnetic moments. This is similar to the force between two magnets of the same pole with the magnets parallel to each other. If the magnets are not constrained, then the result of the force is acceleration and kinetic energy. If one of the magnets was permanent and the other an electromagnet, why does all the energy only ever come from the electromagnet? Why don’t both contribute equally to the imparted force and kinetic energy if the field of the electromagnet exactly equals the field of the permanent magnet?

    • BruceSherwood says:

      Sorry, I don’t quite understand. I don’t see how there can be any significant difference in the behavior of a bar magnet and a current-carrying coil, if they have the same magnetic moment. There is of course the difference that energy has to be provided to the coil to maintain its current, but even that wouldn’t be necessary if the coil is superconducting.

  3. Gaurav says:

    Sir, as far as motional emf is concerned, let us consider a similar arrangement where a straight current wire is perpendicular to magnetic field. Now if we allow, this current wire to move under the influence of magnetic field, then displacement of wire is in direction of magnetic force, and hence can it be said that magnetic force here is doing some net work. Please put some light into this.

    • BruceSherwood says:

      Good question, and the explanation is subtle. A battery is driving mobile electrons through the wire. Supply a magnetic field, and the mobile electrons will be driven by the magnetic force to one side of the wire, leaving positive cores exposed near the other side of the wire. This means that the cross-sectional area of the electron current is now smaller than the cross-sectional area of the wire, so the battery has to work harder to force the electrons through the wire. The energy to move the wire comes from the battery.

      • Bruce Sherwood says:

        What I said is incorrect. Yes, the cross-sectional area containing the mobile electrons gets smaller, but the density of the mobile electrons increases. It seems like a better viewpoint is to say that the displaced electrons pull electrically on the positive charges.

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