Question

Is there a way to derive the field of arbitrarily moving charge from Maxwell's equations?

Answer 1

yes there is a way to derive the field of arbitrary moving charge from Maxwell equations

Answer 2

The Liénard–Wiechert potentials describe the classical electromagnetic effect of a moving electric point charge in terms of a vector potential and a scalar potential in the Lorenz gauge. Built directly from Maxwell's equations, these potentials describe the complete, relativistically correct, time-varying electromagnetic field for a point charge in arbitrary motion, but are not corrected for quantum-mechanical effects. Electromagnetic radiation in the form of waves can be obtained from these potentials. These expressions were developed in part by Alfred-Marie Liénard in 1898[1] and independently by Emil Wiechert in 1900.[2][3]