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In a brushless DC motor, the permanent magnet of the rotor creates a magnetic flux which moves through one stator pole shoe, through the back iron and returns to the opposite pole. The magnetic field lines pass through two poleshoes/coil cores. If the flux is contained to this iron path how does it generate force? As the Lorentz force is given by $I \times B$ and $B$ is zero outside the iron (where the current is), shouldn't $F$ be zero as well?

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I believe that magnetic Lorentz forces on the current-carrying conductors are not very significant in the operation of the motor.

Instead, consider the way you might explain the operation of an electric motor to someone who just wants a simple explanation. You'd probably talk about the poles of the electromagnets in the stator attracting those of the magnets in the rotor, the need to reverse the current in the electromagnet at appropriate intervals to keep the force attractive between poles approaching each other, and so on. There's actually not much wrong with this explanation, nor with the implication that the iron plays a much bigger role than the wires (which serve mainly to magnetise the iron).

At a deeper level, we have to go back to magnetic Lorentz forces, or their microscopic counterparts – applied to the electrons in the atoms of the iron!

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