Attraction & Lorentz force, at the same time?

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.

• 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.
– rob
Aug 1, 2014 at 22:01
• @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. Aug 2, 2014 at 8:50
• @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.
– rob
Aug 2, 2014 at 13:49