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Isn't a magnetic field basically an electric field originating from a line charge, but as seen in a boosted frame? I mean, that's where the relativistic explanation comes from right?

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You're going the right way. See here:… – jinawee May 8 '13 at 22:22
The EM field is a field of oriented planes through spacetime. These planes are spanned by (1) the direction of the current (whether through space or time) and (2) the direction from the observer to the current (in space). Electric field planes come from currents that only go through time (such currents represent static charges), but when those currents have spatial components, they generate magnetic planes also. – Muphrid May 9 '13 at 3:31

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