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Electric Motors
A an electric motor converts mechanical energy into rotational mechanical movement.
Brushed DC Motors
Conventional brushed motors used a coil wound around an iron core rotor with leads going to a commutator. External permanent magnets are magnetized in opposite directions and provide a fixed magnetic field.
Maxon makes fancy core-less brushed DC motors that have a thin cylindrical winding that sleeves over a permanent magnet.
The basics of electric motor operation relies on the Lorentz force which states that a current conductor in an external magnetic field experiences a force.
Formally, a particle of charge $q$ with a velocity $v$ in an electric field $\textbf{E}$ and a magnetic field $\textbf{B}$ experiences a force $\textbf{F}$:
$\textbf{F} = q\textbf{E}+q\textbf{v}\multiply\textbf{B}$