The Relativity Derivation
Reference[1] discusses this method carefully. In this web page we simply talk about the general idea to derive the Maxwell Equations from relativity. The first step is to use Lorentz transformation equation to get the Lorentz force from Coulomb’s force. Then the authors establish the concept of electric and magnetic fields, and prove that they satisfy the Maxwell Equations.
[1] Paul Lorrain and Dale Corson, Electromagnetic Fields and Waves, Second Edition