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Accordingly, we can rephrase the question: imagine that in the lab from an
electron is moving through a magnetic field. At any given point, it feels
a force transverse to its motion. So the question is, what happens in some
other reference frame, such as one comoving with the electron?
Answer: The electron sees an electric field. You can easily calculate the
details of this field by Lorentz transforming the electromagnetic field
tensor from the one frame to the other.