# Exactly how is the constant measured velocity of light deduced from Maxwell's equation?

For electromagnetic radiation the velocity of propagation is $c = 1/\sqrt{\mu_0 \epsilon_0}$. Since both $\mu_0$ and $\epsilon_0$ do not vary in any inertial frame, then $c$ must be constant in any inertial frame.

Now apply the argument to sound. The velocity of sound in a rod clearly doesn't change regardless of the velocity of the rod, but it's obvious that if the rod moves past me at the velocity of a sound wave in that rod, the sound wave will be standing still as I view it.

In other words, there appears to be something wrong with the argument in the first paragraph since the same argument fails for sound, which is also a wave with a propagation velocity.

So how exactly does light differ from sound here? I'm looking for an explanation that would be understandable to a budding scientist who is a non-physicist. (I'm trying to prepare to explain special relativity to my daughter (a geologist who's taking a course in cosmology), and want to be able to explain special relativity from first principles, in such a way that special relativity becomes obvious and simple).

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$\mu_0$ and $\epsilon_0$ are well defined in the vacuum, far from any reference frame-binding matter. material properties of the rod are undefined outside of the rod –  lurscher Sep 9 '11 at 13:45
Speaking quite literally (which is how one should speak in physics!) the velocity of a sound wave in a moving rod is different than the velocity of a sound wave in a rod at rest. That is, the crests and the troughs move (with respect to the observer) at a different velocity. The key difference with light is that there is nothing that plays the role of the rod. All observers will measure light's speed as c=1/sqrt(epsilon*mu), and you can use that as a starting point to derive all the other surprising basic facts about special relativity. –  Greg P Sep 9 '11 at 14:32
So the key observation is that electromagnetic waves propagate without a medium -- all other waves require a medium. OK, that's very helpful and literally is exactly what I wanted. (Given that, I can explain the rest of special relativity easily.) –  Walt Donovan Sep 9 '11 at 20:24
I don't like the title of this question. Maxwell's equations is a theoretical model of electromagnetism. The measured speed of light is an empirical observation. One cannot deduce the latter from the former. Rather, one can say that the model given by Maxwell's equations predicts that the speed of light is the same across reference frames. Whether the actual measurements agree can then be used as a test for the theory. –  Willie Wong Sep 11 '11 at 16:16

The key is, I think, that when you establish the equations for the rod, you implicitly assume a frame of reference, namely at rest w.r.t. the rod. However, Maxwell's equations are more fundamental and based on laws that shouldn't depend on the reference frame, at least they didn't seem to.

So, when you get an equation pointing to a fixed speed, you naturally ask the question about w.r.t. what reference frame. But there was no implicit assumption of a reference frame in the case of the Maxwell equations. So that was puzzling. So the first reaction was to postulate there must be one. An aether or something like that. But this turned out to be a failure.

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Are you really asking why the speed of light is invariant regardless of the observer's relative motion? I will try to answer that, although I'm really rusty on Maxwell.

It was commonly thought that there was an "aether" through which light propogated, just like sound waves in matter. If so, since the earth is moving through space at pretty high velocity, if you measure the speed of light along tubes oriented at right angles, you should see a difference. That was the Michaelson-Morley experiment, and oddly enough, there was no difference. It appears as long as you're moving steadily in a straight line, no matter where or how fast, if you measure the speed of light it comes out the same.

In order to explain this puzzle, a new theory was born, Special Relativity. It says our clocks and our measuring sticks (the things we measure speed with) start acting funny when we go fast with respect to someone else. Here's how it works.

Suppose you build a clock by having two parallel mirrors in a vacuum one half meter apart, and you bounce a small pulse of light vertically between the mirrors. You count the bounces. 299792458 round trips takes one second, because that's how fast light goes.

Now you mount this clock on a railroad car and look at it as it travels past you. Since the light is now traveling in a slanted direction, because the whole clock is moving, the light pulse has to travel further between bounces, which will take longer from your perspective. So from your perspective the clock on the train is running slower, but from the perspective of the person on the train, it appears to be going the same speed because that's the only clock he's got.

In fact, if you the supposedly stationary person also had a clock, the person on the train would say that your's appears to be running slower.

So, if the Michaelson-Morley experiment is correct, this is what you and the person on the train should observe. Well, experiment bears it out. That's what actually is seen. When very precise clocks are compared, and one of them is moving in a straight line at a constant high speed relative to the other, it does run slow. The faster it's moving, the slower it runs. In theory, if it could go at the speed of light, it would stop.

Also, measuring rods aligned in the direction of motion are shortened, for similar reasons.

So if you're trying to explain why the speed of light is constant regardless of the observer's motion, the answer is, we don't really know, but it appears to be, and the implications are that clocks and measuring rods should do funny things. And in fact, they do. This has deep implications, for better and worse, including nuclear energy and warfare.

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