# Two explanations of how electric motor works: are they correct?

Let's consider the simplest model for a DC electric motor:

For simplicity, we assume that over a short period of time, the current through the motor is constant. All the discussion below refers to this time period.

In high school, apparently, two explanations of the mechanism of this motor are acceptable:

1. "The moving electrons in the wire experience a force in the magnetic field created by the permanent magnets." This is clearly talking about the Lorentz force.
2. "The coil in the armature is essentially an electromagnet. So the two magnets (permanent and electric) attract and repel each other, creating a torque."

After learning some degree level electromagnetism, I am pretty certain that explanation 1 is valid. However, explanation 2 is quite strange, in that there is no simple formula in electromagnetics that describes "the force between two magnets". We have the electrical forces between charges and Lorentz force on charges due to the magnetic field, but no analogy for forces that only involve magnets and magnetic fields only; after all, all magnetic fields are supposed to be generated by electric currents.

Let's have a closer look: the strength of the magnetic field generated by a wire should be approximately inversely proportional to the distance from the wire. The magnetic field on the wire is, therefore, undefined; thus we cannot really analyze how the magnetic field generated by the coil affects the motion of the coil, given that the current is constant. So, the second explanation above isn't really supported by any rigorous formulae and mathematics.

Can I say the second explanation is unacceptable or wrong based on that?