Does the existence of magnetic field follow from special relativity and invariant of electric charges? as the title suggest, Can I show the existence of magnetic field using only special relativity and  invariant of electric charges?
 A: The magnetic field has some origin from special relativity and you can show How the concept of magnetic field appears, starting from special relativity. This is very well done on Electricity and Magnetism By Purcell. You can look at it if interested.
But the book sometimes leads to the false statement that special relativity is sufficient for the magnetic field which is not true.
The fundamental sources of the magnetic field are electric currents and magnetic dipole moments (changing electric field also can also create magnetic field). Intrinsic magnetic dipole moments exist at the fundamental level and they directly create magnetic fields. Most subatomic particles have magnetic dipole moments and therefore act like little bar magnets. For instance, an electron has an electric charge and an intrinsic magnetic dipole moment. Therefore, an electron directly creates an electric field from its electric charge, and a magnetic field from its magnetic dipole moment, even when at rest. When moving, an electron also creates a magnetic field from its motion (because a moving electric charge constitutes an electric current).

More on this can be found here

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*Does special relativity make magnetic fields irrelevant?

*How is a magnetic field just an electric field with relativity applied?
