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Consider a current carrying wire in a uniform magnetic field. Obviously the wire moves due to the force exerted onto the wire. Ok, but why? I have heard 2 explanations. The first one talks about the electrons moving in the wire have Lorentz forces induced onto them according to $F=qv\times B$. This force then sort of "bumps" into the wire and moves the whole thing. There is an obvious fault with this explanation in that it assumes the collision is perfectly conservative.

The second explanation is that the current carrying wire induced a magnetic field, this field interacts with the external field to form repelling or attracting forces, which we call the motor effect. I am more so confused by this explanation. How does magnetic field interaction work? Don't the field vectors just add up instead of actually exerting forces on each other? Is it because the magnetic field from the wire exerts force on the external magnets, and according to Newton's third law, an opposite force is induced on the wire?

Are either of these explanations correct? If so, how? If not, what is the correct way of thinking about it? Thank you.

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They are the same explanation. There is only one electromagnetic field. You can think of charges and the forces they experience due to the field, or you can think about the evolution of the field itself.

The first one talks about the electrons moving in the wire have Lorentz forces induced onto them according to $F=qv×B$. This force then sort of "bumps" into the wire and moves the whole thing. There is an obvious fault with this explanation in that it assumes the collision is perfectly conservative.

The moving electrons would prefer to move along helical paths in the magnetic field. However, for the conduction electrons to leave the wire entirely (leaving behind a lattice of positive ions) requires more energy than is available in the circuit. So the moving electrons concentrate on one side of the wire. The positive lattice is attracted to the concentration of negative charge, and the whole wire moves.

(The Hall effect is a consequence of this separation of the moving and stationary charges.)

I don't follow your "obvious fault"; it costs energy to push a current through a resistive wire, so there's no need to assume any collisions are conservative.

The second explanation is that the current carrying wire induced a magnetic field, this field interacts with the external field to form repelling or attracting forces.

This is the "field-only" approach. The electromagnetic field stores energy with density

$$ U = \frac12\left( \epsilon_0 E^2 + \frac1{\mu_0}B^2 \right). $$

In general this means that the total energy of an electromagnetic field can be reduced by minimizing the total volume of "strong" fields —— even if the fields become locally stronger to do so. Two magnetic dipoles oriented north-to-north will release energy by moving further apart ("repel each other"): the antiparallel north fields cancel out near the interaction region, but the fringing fields add together over a large volume. Two magnetic dipoles oriented north-to-south will have a strong field where they interact, but their fringing fields mostly cancel out. Those parallel dipoles can make the strong field volume smaller if they get closer together: an attraction.

If you are thinking about the fields produced by wires and the forces between the wires and the fields, you can think either about the charges on the wires or about minimizing the volume of strong field. Either approach should lead you to the rules that

  1. parallel currents attract each other
  2. antiparallel currents repel each other
  3. skew currents (neither parallel nor antiparallel nor intersecting) should feel a torque that makes them want to become parallel.
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  • $\begingroup$ Ah, superb explanation. Just one thing to clarify though, on an intuitive level, when the magnetic field from the wire interacts with the external field, does it interact directly with the magnet, or with the magnet's field? I am a bit confused on how the interaction of fields lead to the production of a force if the latter case is true. $\endgroup$ Nov 11, 2021 at 4:41
  • $\begingroup$ The only reason you call it “the magnet” is because of its external field. But you have the same duality there. You can think about the entire field of the permanent magnet. Or you can think about the field produced by the wire producing aligning torques and attraction/repulsion on all of the microscopic electron-orbit currents that are associated with the permanent magnet. $\endgroup$
    – rob
    Nov 11, 2021 at 5:22
  • $\begingroup$ Can you elaborate? When do we consider the entire field and when do we consider the effect on the actual magnet? $\endgroup$ Nov 11, 2021 at 5:25
  • $\begingroup$ We choose the approach where the integral is easiest. If we are teaching, we do both the easy and the hard integral to prove to the students that they are the same. $\endgroup$
    – rob
    Nov 11, 2021 at 5:30
  • $\begingroup$ by integral do you mean the potential energy equation you provided above? $\endgroup$ Nov 11, 2021 at 8:37

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