Why does the magnitude of magnetic force which one moving charge exerts on another charge appears inconsistent in different frames of reference?

Question

I have a fundamental misunderstanding about the definition of magnetic force as arises from Lorenz force: $F = q\vec{v} \times \vec{B}$. According to a basic fact of electromagnetism, electric currents are the source of magnetic fields, and a single moving charge can be viewed as a current. Therefore, in a situation of one moving charge and another stationary charge, the moving charge creates a magnetic field, but by the Lorenz definition of magnetic force, the force experienced by the stationary charge iz zero (since it's velocity is zero). But if we shift to a frame of reference moving at half the speed ($\frac{{v}}{{2}}$) of the first charge and in the same direction, we get one charge moving in a speed $+\frac{{v}}{{2}}$ and a second charge moving at $-\frac{{v}}{{2}}$. In this scenario, the magnetic force experienced by the same charge is not zero.

So what exactly am i missing? i view it as a serious bug in my fundamental understanding of electromagnetism, and i will be extremely thankful if someone will complete for me the picture of charge-charge electromagnetic interaction.

Answer 1

Actually what you said is all correct. Magnetism and electricity are not independent things, they should always be taken together as one complete electromagnetic field. In one frame you have only an electric field. Whether or not you see an magnetic field depends on your reference frame, but if you would apply both the effects of the magnetic and electric field in that frame to another particle, it will result in the same motion as in the first frame.

I suggest you read The Feynman Lectures Vol. II 13-6, he gives a more comprehensible explanation than I ever could.