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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?

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From: http://www.pmaweb.caltech.edu/Courses/ph136/yr2012/1202.1.K.pdf

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.

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  • $\begingroup$ Thank you very much! So basically if I trust the principle of relativity, waves created by the acceleration of the particle in both expirements will have the same speed! Very clever $\endgroup$ – Peppe Fabiano Sep 15 '17 at 17:35

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