We know that a uniformly moving charged particle produces constant magnetic field and an accelerated charged particle produces electromagnetic waves. If we keep a charged particle inside a train which is moving uniformly with respect to ground, it is at rest for a person inside the train but moving for a person standing on the ground. If both of them measures magnetic field produced by it from their respective frames of, then the person outside the train should experience a non-zero magnetic field, but the person inside it should get it to be zero. How is this possible? Same question for the train while accelerating also.
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The Maxwell ecuations are invariant using the lorentz transformation of the Einstein's relativity theory.
So we have to talk about the electromagnetic field, wich is the same in the both systems. You can see this easy according to the Newton second law. Imagine two particles with the same charge A and B and the two systems S1, inside de train and S2 outside the train.
If A and B are inside the train the relative velocity between them is zero and there is not magnetic field. If A is outside and you are in the system S1 you will see that A is moving and B is quiet. So A is creating a magnetic field. Seen from S2 B is the particle with velocity and the one who creates the magnetic field. The second law of newton says that the force that A makes on B must be the same that B makes on A, so the magnetic fields in S1 and S2 must be the same.
As I say general relativity understand magnetic and electric field like the same phenomena and you can´t have the one without the other.
