  The Coulomb force, established in the rest frame of a source-charge $Q$, when
transformed to a new frame moving with a velocity $\vec{V}$ has a form
$\vec{F}= q\vec{{E}} + q\vec{v} \times \vec{{B}}$, where
$\vec{{E}}=\vec{E}'_\parallel + \gamma\vec{E}'_\perp $ and $\vec{{B}}=
\vec{\frac{V}{c^2}} \times \vec{{E}}$ and $\vec{E}'$ is the electric field in
the rest frame of the source. The quantities $\vec{E}$ and $\vec{B}$ are then
manifestly interdependent. We prove that they are determined by Maxwell's
equations, so they represent the electric and magnetic fields in the new frame
and the force $\vec{F}$ is the well known from experiments Lorentz force. In
this way Maxwell's equations may be discovered theoretically for this
particular situation of uniformly moving sources. The general solutions of the
discovered Maxwell's equations lead us to fields produced by accelerating
sources.
