Question

why does a moving charge experience a force in a magnetic field?

Answer 1

A charged particle moving without acceleration produces an electric as well as a magnetic field. It produces an electric field because it's a charge particle.
But when it is at rest, it doesn't produce a magnetic field. All of a sudden when it starts moving, it starts producing a magnetic field.

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Answer 2

Any moving charge in space constitute convection density which is proportional to its velocity hence due to this static magnetic field is produced. This field is produced by moving charge and magnetic field interact hence charge is experiences centripetal force nd follow circular path