Do the electric and magnetic field change when we move from one frame to another?
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Yes it changes . You can infer that from the formula for Lorentz force , F = q(V x B) + qE ,which implies for inertial observers who have non-zero relative velocities will measure different velocity of the charge and therefore they will end up calculating different forces if there is no transformation of electric and magnetic fields for different frames . Therefore there it changes and the changes are given by Lorentz transformations.
