Suppose in a reference system K' a charge and a body are moving with the same velocities v'. The charge should not produce a magnetic field. But for the reference frame K( at rest relative to K') the charge produces a magnetic field.
Is there an absolute answer to this?(whether a magnetic field is produced or not) and how do we compute it? Or are there some basic conceptual errors in the assumptions?
The simple (relatively speaking) answer is that magnetism is fundamentally a relativistic phenomenon, and relativity is what unites electricity and magnetism.
What appears to be a purely electric field in the reference frame in which the charge and body are at rest turns out to be a combination of electric and magnetic fields in a reference frame in which they are moving. By the first postulate of special relativity, neither interpretation is more correct than the other.
Wikipedia has a good explanation of electromagnetism under special relativity, however as the Maxwell Equations are compatible with special relativity they (specifically the Ampere-Maxwell law) will generally be sufficient to calculate the magnetic field produced.
A common example is to consider the magnetic field produced by a current-carrying wire. In the laboratory frame, the wire is neutrally charged, however in a moving frame, the moving electrons and the protons will contract at a different rate and hence the wire will appear to be charged. The apparent electric field is equivalent to the magnetic field in the laboratory frame.
