How do you prove from Maxwell's equations that the speed of light is independent of the motions of the sources that produce the wave?
Measuring the Speed of Light Without Light:
In some inertial reference frame, we perform two experiments using two particles, one with a large charge $Q$; the other, a test particle, with a much smaller charge $q$ and mass $\mu$. In the first experiment we place the two particles at rest, separated by a distance |∆x| ≡ r and measure the electrical repulsive acceleration $a_e$ of $q$. In Gaussian cgs units (where the speed of light shows up explicitly instead of via $\epsilon_0\,\mu_0 = 1/c^2$, the acceleration is $a_e = qQ/r2μ$. In the second experiment, we connect $Q$ to ground by a long wire, and we place q at the distance |∆x| = r from the wire and set it moving at speed $v$ parallel to the wire. The charge $Q$ flows down the wire with an $e$-folding time τ so the current is I = dQ/dτ = (Q/τ)e−t/τ. At early times 0 < t ≪ τ, this current I = Q/τ produces a solenoidal magnetic field at q with field strength B = (2/cr)(Q/τ), and this field exerts a magnetic force on q, giving it an acceleration $a_m = q(v/c)B/μ = 2vqQ/c2τr/μ$. The ratio of the electric acceleration in the first experiment to the magnetic acceleration in the second experiment is $a_e/a_m = c2τ/2rv$. Therefore, we can measure the speed of light c in our chosen inertial frame by performing this pair of experiments, carefully measuring the separation r, speed v, current Q/τ, and accelerations, and then simply computing c = (2rv/τ)(ae/am). The Principle of Relativity insists that the result of this pair of experiments should be independent of the inertial frame in which they are performed. Therefore, the speed of light c which appears in Maxwell’s equations must be frame-independent. In this sense, the constancy of the speed of light follows from the Principle of Relativity as applied to Maxwell’s equations.