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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?

Qmechanic
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Peppe Fabiano
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  • Possible duplicates: https://physics.stackexchange.com/q/1574/2451 , https://physics.stackexchange.com/q/14482/2451 , https://physics.stackexchange.com/q/77634/2451 and links therein. – Qmechanic Sep 15 '17 at 14:13
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    Possible duplicate of Exactly how is the constant measured velocity of light deduced from Maxwell's equation? – Javier Sep 15 '17 at 14:17
  • I'm sorry but I think that none of these questions answer mine – Peppe Fabiano Sep 15 '17 at 14:26
  • What i wanr to do is, take the maxwell equations and prove that they give the same speed c when solving them in any inertial reference frame – Peppe Fabiano Sep 15 '17 at 14:27
  • How are the inertial frames connected? By a Lorentz transformation? – lalala Sep 15 '17 at 14:31
  • Let's assume that I do not know anything about relativity and Lorentz transformations – Peppe Fabiano Sep 15 '17 at 14:33
  • In classical terms, having just developed maxwell equations, how do I see from them that the speed of Light is constant in any inertial reference frame (independent of the motion of the sources)? – Peppe Fabiano Sep 15 '17 at 14:34
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    Take for example the oscillating charge. How do I prove wit Maxwell equations that the speed of the wave produced by the charge is the same if the charge is just oscillating or if the charge is oscillating and translatin? – Peppe Fabiano Sep 15 '17 at 14:35
  • I know how to derive the wave equation, I just want to know what changes when I consider moving sources – Peppe Fabiano Sep 15 '17 at 14:57
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    @PeppeFabiano That's precisely the point - there are no changes when you consider moving sources. It may change the form of the source terms, but that does not affect the speed of propagation of the resulting waves. – J. Murray Sep 15 '17 at 16:18
  • @J.Murray That is exactly what I'm asking. I'm asking how does one deduce that in fact there are no changes when I consider moving sources – Peppe Fabiano Sep 15 '17 at 16:36
  • @PeppeFabiano Maxwell's equations lead to inhomogeneous wave equations for the electric and magnetic fields. The speed with which those waves propagate in vacuum is $1/\sqrt{\epsilon_0 \mu_0}$, which is a constant and therefore obviously independent of the motion of the sources which generated the waves. Beyond that, I'm not sure what you're asking. – J. Murray Sep 15 '17 at 16:45

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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.

JEB
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  • 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 – Peppe Fabiano Sep 15 '17 at 17:35