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I understand that this question is asked many times on PSE, but my questions which I ask below are slightly different. I have read almost all the discussions on PSE on the topic but a few questions are still unanswered.

In the following post: Is there any potential associated with magnetism

Luboš Motl says the following:

The force qv×B acting on a charged particle is clearly not conservative because it depends on the velocity. Conservative forces are those that integrate to a fixed work – energy difference that is independent of the path – between two points so they may only depend on the location. We can't associate a potential with a force that is velocity-dependent.

My questions are the following.

  1. How Velocity-Dependence of a Force makes it Non-Conservative?

"We can't associate a potential with a force that is velocity-dependent."

  1. Magnetic force doesn't do any work, so it's work in a closed-loop will also be zero, so this in a way seems to make magnetic force a conservative force. Why not, it is then a conservative force?

  2. What about Magnetic Field? If there are no currents and there are no changing electric fields then also we can have magnetic fields due to Bar Magnets. Are the magnetic fields due to Bar Magnets Conservative?

  3. This idea of Conservative or Non-Consverative has become quite confusing, given several examples, which do not seem to cohere with each other. Is there any single rule with the help of which we can distinguish between Conservative and Non-Conservative?

Kindly help.

Devansh Mittal
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2 Answers2

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(1) a velocity dependent force cannot be conservative because when integrated along a closed contour if it was zero for one velocity distribution along the contour then increasing the speed (but not its direction) at one segment of the contour will make the integral imbalanced, so to speak, and thus nonzero.

(2) B does not do work for an electric charge moving in the field under its own force, that does not mean that a magnetic field does no work, for example over a current that is moved against that field.

(3) A permanent magnet (also paramagnets) has two kinds of macroscopic magnetic fields inside (outside the two fields are essentially the same): one is lamellar $\mathrm{curl}\mathbf{H}=0$ and the other is solenoidal $\mathrm{div}\mathbf{B}=0$. The field $\mathbf{H}$ is conservative in its interaction with macroscopic magnetic poles but $\mathbf{B}$ is not because $\mathrm{curl}\mathbf{B}=\mu_0\mathrm{curl}\mathbf{M} \ne 0$ inside.

(4) I think Lubos Motl has answered in [1].

[1] Is there any potential associated with magnetism

hyportnex
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I've always understood calling a magnetic field non-conservative to mean that the line integral of the magnetic flux density, $\mathbf B$, around a closed loop is in general non-zero. The magnetic fields due to bar magnets, like those due to current loops, are, according to this definition, non-conservative. For closed loops that pass through the magnet the line integral is non-zero.

Your question #3 is especially interesting. I think your argument is correct, provided that you're talking about the magnetic Lorentz force on a charged particle. But I think confusion arises because magnetic field strength used to be defined in terms of the force on a hypothetical unit pole, and this would be a non-conservative force in the same direction as the field.

Philip Wood
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