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If a conductor carrying current is placed inside a magnetic field, we know that there is the Lorentz force pushing the wire. But what about the attraction force between the wire's field and the magnet/electromagnet's field? So, isn't there really two forces involved? Attraction due to two magnetic field, and Lorentz force?

Even if the pole of the "electromagnet" can attract the conductor.

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    $\begingroup$ The Lorentz force is the force between the magnetic field of a current-carrying wire and an external magnetic field. It's described in terms of the current because it's simpler that way. $\endgroup$ – rob Aug 1 '14 at 22:01
  • $\begingroup$ @rob, I don't think that's right. The force between two magnets depends on the inhomogeneity of the magnetic fields. I don't see how that translates to Lorentz force. Also, once you replace the electromagnet by a bar magnet, the magnetic attraction/repulsion persists while there can be no Lorentz force (no current!). So I'd say there are two forces, one of which is usually neglected. $\endgroup$ – Jonas Greitemann Aug 2 '14 at 8:50
  • $\begingroup$ @Jonas Note that a (straight) current-carrying conductor produces an inhomogeneous field which can't be reproduced by an arrangement of permanent magnetic dipoles. You're correct of course that a dipole in a uniform field experiences a torque, but not a force. $\endgroup$ – rob Aug 2 '14 at 13:49
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The experiment is simplified because it assumes the field generated by the wire to be small compared to that of the external field. That means there is a given external field and you can integrate Newtons law with the Lorentz force seperately. If that simplification doesn't hold any more, one would (in principal) have to consistently solve the coupled system of both the Maxwell's field equation and equations of motion, which might be a bit tricky in this case.

I wouldn't say that there are two forces involved, the field of the wire might weaken the external field which in turn weakens the Lorentz-force exerted on the wire.

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