Question

How electron produce both magnetic and electric field?

Answer 1

Magnetic fields come from changing electric fields, which come from the electron's motion.

Explanation:

Simple Answer
The equations which tell us how magnetic fields and electric fields interact show that a moving charge produces a moving electric field which in turn generates a moving magnetic field. All of these things are perpendicular.

Less Simple Answer
It was observed that electromagnetism worked identically no matter what frame of reference you were in. This has deep implications with regards to what are called the Maxwell equations, which dictate how electric and magnetic fields relate to one another.

The real problem comes from the fact that moving magnets cause currents. This is a problem because we could imagine being in a reference frame that isn't moving relative to the magnet and therefore, there should not be any electric field. This initially seems broken: in one frame, there's an electric and magnetic field while in another there is no electric field.

However, one can actually transform the field and find that the magnetic field is actually a byproduct of this shift and therefore magnets and electrical flows are basically two sides of the same coin. The two concepts are actually one, with their relationship being special relativity.

You can look up the actual math for this if you so wish, but it's one of the most important and interesting E&M proofs one can find.