Suppose there is a test isolated charge and I am moving around it so in my frame of reference the charge is moving. So in my frame of reference does the charge produce a magnetic field? If yes, how can we calculate it? (which formula to use?)
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Yes, the charge will produce a magnetic field in your frame of reference. The calculation of it can be difficult depending on the way you move around it. Most generally, the magnetic field in your frame can be calculated by taking the curl of the magnetic vector potential, which can be extracted from the Liénard–Wiechert potential which describes the effect of an arbitrarily moving electric point charge.
If your motion is uniform, however, it would be easier to simply calculate the electric field in the rest frame of the charge (the magnetic field being zero in this frame) and Lorentz transform into your rest frame to find the electric and magnetic fields in your frame.
Radu Moga
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"So in my frame of reference does the charge produce a magnetic field?"
No.
Charge doesn't not produce a magnetic field. The magnetic field is sourced from currents and changing currents, as seen on the past light cone (see https://en.wikipedia.org/wiki/Jefimenko%27s_equations).
However: one frame's charge is another frame's current:
$$\vec j = q\vec v$$
so that any frame that sees a moving charge will see a magnetic field.
The calculation of the fields is tractable, but tedious. A complete description is given in the venerable Feynman Lectures: https://www.feynmanlectures.caltech.edu/II_26.html
JEB
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