B Magnetic field in frame of moving charge

A charge moving relative to some observer produces magnetic field in space around it. Now I want to ask that how does the magnetic field of the charge at any fixed point varies as the charge passes by. Or is the B field only dependent on current and not a single moving charge?

If I introduce one more observer to be moving with the charge then if a time comes at which both the observer coincide, then at that instant will magnetic field be experienced by only stationary observer or both?
 

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Or is the B field only dependent on current and not a single moving charge?
A single moving charge does produce a current. The B field of the current in a conductor is the sum of the B fields of all of the charges moving in the conductor.

Now I want to ask that how does the magnetic field of the charge at any fixed point varies as the charge passes by.
The electromagnetic field satisfies Maxwell's equations. The solution for a single moving charge can be found, for example, here (slide 8): https://www.physics.rutgers.edu/ugrad/227/L15 Magnetic Field of Currents Biot-Savart.pdf

If I introduce one more observer to be moving with the charge then if a time comes at which both the observer coincide, then at that instant will magnetic field be experienced by only stationary observer or both?
This question does not make any sense. If an observer is not moving with respect to the charge, then they will either coincide at all times or at no time.

For an observer at rest with respect to the charge, there would be no magnetic field. The electric and magnetic field are not invariant under boosts (changes in inertial frames with a relative velocity), i.e., if you change to a moving frame, the electric and magnetic fields will change.
 
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Now I want to ask that how does the magnetic field of the charge at any fixed point varies as the charge passes by. Or is the B field only dependent on current and not a single moving charge?
The fields for a classical point charge are given by the Lienard Wiechert potentials: https://en.wikipedia.org/wiki/Liénard–Wiechert_potential

In the appropriate limits these fields reduce to Coulomb's law and the Biot Savart law but you can use them for any arbitrary motion.
 
I think the question is about when the moving observer coincides with the stationary observer. The stationary observer would measure a magnetic field with her compass (and there would be some electric field as well) but the observer moving with the charge would measure only an electric field.
 

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