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The magnetic part of the Lorentz force is $$\vec F_L=q\left(\vec v\times\vec B\right)$$ As this force is always perpendicular to the direction of the movement, we learned that no work is done by it.

However, it's easily observed that permanent magnets attract each other (at least with south pointing to north). This attraction accelerates the magnets in the direction of the movement, so a work is being done. How is this possible?

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Easier (for me, at least) than two magnets is two parallel current-carrying wires attracting each other. The force of attraction is equal to the component at right angles to the wire of the magnetic Lorentz force on the charge carriers – the so-called Laplace force.

As soon as the wires start to move together (and work is done on them) the charge carriers' velocity acquires a component at right angles to the wire, and the magnetic Lorentz force acquires a component along the wire. So if the charge carriers are to keep moving at the same speed, work has to be done on them by the electric field set up by the battery; in other words a back-emf has to be overcome.

So it's the battery that ultimately supplies the work done on the wires as they move together. The magnetic Lorentz force acts a bit like a pulley, changing the direction of the force that does work.

I imagine that the case of two magnets attracting can be analysed in a similar sort of way. But then I do have a vivid imagination...

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  • $\begingroup$ That helps a lot. The only question remaining is, where does the energy come from in permanent magnets? What is the "battery" equivalent there? $\endgroup$
    – MetaColon
    Commented Oct 6, 2019 at 8:45
  • $\begingroup$ The Laplace force should be part of the Lorentz force. Also, you need to explain the force felt when the magnets (the sources of the magnetic field) are at rest. $\endgroup$
    – my2cts
    Commented Oct 6, 2019 at 9:10
  • $\begingroup$ @MetaColon: Frankly I don't know where the energy comes from in permanent magnets, though I note that the electrons, whose angular momenta are related to their magnetic moments, are not isolated; they interact with the rest of the magnet.I think this is probably relevant, $\endgroup$
    – Philip Wood
    Commented Oct 6, 2019 at 21:52
  • $\begingroup$ @my2acts: (a) I wrote only about attracting wires not about magnets. (b) I agree that the Laplace force is equal to a component of the Lorentz force. I held back from saying that it IS a component of the Lorentz force because strictly it is the force on the wire itself, which is the Newton's third law partner to the electrostatic force that the wire exerts on the moving charge carriers, preventing them from being pushed sideways out of the wire by the Lorentz force. $\endgroup$
    – Philip Wood
    Commented Oct 6, 2019 at 22:02
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A straightforward, but uncommon, explanation is the following. The Lorentz force expression can be rewritten as $f_k =\frac{d(p+qA)_k} {dt} = qv_i \frac{dA_i} {dx_k} $. When two magnets are near and held at rest, dA/dt=0. An acceleration results in the direction of the gradient of A, proportional to the component of v parallel to A. This force performs work if the separation of the magnets is changed.

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  • $\begingroup$ The problem is that the concepts of E and B stem from electro- and magnetistatics. They are not really suitable otherwise. The potential conceived by L. V. Lorenz is seen as just an auxiliary, in spite of its dominant role in for example quantum mechanics. So we keep confusing ourselves and we keep believing in gauge invariance. $\endgroup$
    – my2cts
    Commented Oct 6, 2019 at 12:22
  • $\begingroup$ My2acts: Straightforward, you say... Any help would be appreciated. $\endgroup$
    – Philip Wood
    Commented Oct 6, 2019 at 22:51
  • $\begingroup$ If you write @my2cts next time I am notified of your comment. As E=0 in magnetostatics one has $f_i = \epsilon_{ijk} v_j B_k = \epsilon_{ijk} v_j \epsilon_{klm} \partial_l A_m$. Work out the sum over k involving the Levi-Civita symbols and you find the expression in the answer. $\endgroup$
    – my2cts
    Commented Oct 8, 2019 at 19:36
  • $\begingroup$ Forgot the @ and inserted a spurious a. Many thanks for elaborating. $\endgroup$
    – Philip Wood
    Commented Oct 9, 2019 at 19:14
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The Lorentz force is defined as the total electromagnetic force exerted by an electromagnetic field on an electric point charge. Coming to the magnetic part of lorentz force, it can never do work.

However, it's easily observed that permanent magnets attract each other (at least with south pointing to north). This attraction accelerates the magnets in the direction of the movement, so a work is being done.

As the definition itself conveys that it is a magnetic force on an electric point charge but magnet is not an electric charge. It's well understood that we can't use lorentz force to define the forces between two magnets.

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    $\begingroup$ How can we define the forces between two magnets then? $\endgroup$
    – MetaColon
    Commented Oct 6, 2019 at 6:45
  • $\begingroup$ @MetaColon google.com/url?sa=t&source=web&rct=j&url=https://… $\endgroup$
    – Shreyansh Pathak
    Commented Oct 6, 2019 at 6:47
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    $\begingroup$ But don't the dipole interactions originate in the Lorentz force? Does the Lorentz force not describe all forces that appear in electromagnetism? And if not, which other fundamental law exists? $\endgroup$
    – MetaColon
    Commented Oct 6, 2019 at 8:17
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    $\begingroup$ @Metacolon Your comment sums up all that is wrong with this answer. Also the Wikipedia article is of low quality. In defense of the answer, the theory is very counterintuitive on this topic. $\endgroup$
    – my2cts
    Commented Oct 6, 2019 at 8:34

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