Speed of Light in Different Mediums: Exploring Maxwell Equations

[Cosmic Lollipop]

TL;DR Summary

Does medium permittivity change the speed of light?

Based on Maxwell equations, we can determine the speed of light as:

$$c=\frac{1}{\sqrt{\epsilon_{0}\mu_{0}}}$$

Where $\epsilon_{0}$ is the electric permittivity of vacuum and $\mu_{0}$ is the magnetic permittivity of vacuum. This notation makes me believe that in other mediums those parameters can acquire other values, implying that the speed of an electromagnetic wave changes depending on the medium and that goes against Einstein's second postulate of special relativity. What am I missing?

 

[PeterDonis]

implying that the speed of an electromagnetic wave changes depending on the medium

Yes. Light can travel slower in a medium than it does in vacuum.

that goes against Einstein's second postulate of special relativity.

No. The speed of light in vacuum is the only thing that needs to be invariant in SR.

 

[Cosmic Lollipop]

Yes. Light can travel slower in a medium than it does in vacuum.
No. The speed of light in vacuum is the only thing that needs to be invariant in SR.

So hypothetically we can find a medium where a mechanical wave is faster than an electromagnetic one. Is that correct?

 

[mitochan]

Yes, actually. For an example charged particles moving faster than light speed in a medium e.g. water, cause Cherenkov radiation. It is like a shock wave by supersonic jet plane.

 

[vanhees71]

Indeed, the phase velocity in an isotropic medium is $c_n=c/n$. This can be $<c$ (normal dispersion) or $>c$ (anomalous dispersion). There's no speed-limit for $c_n$. It doesn't contradict relativity, because one can show that the signal velocity (i.e., the velocity of the wave front of a signal) is alwasy $\leq c$ (Sommerfeld 1907, Sommerfeld and Brillouin 1914).

If a charged particle moves in a medium with $c_n<c$ Cherenkov radiation occurs. This was predicted before relativity has been theoretically predicted by Heaviside (1888) and by Sommerfeld (1904) for a particle faster than light in the vacuum, which is of course impossible due to relativity, and the theory of Cherenkov radiation in matter is due to Frank and Tamm (1937, Nobel prize 1958). The first experimental observation is due to M. Curie (1910) in a solution of radium.

https://en.wikipedia.org/wiki/Cherenkov_radiation

 

[PeroK]

implying that the speed of an electromagnetic wave changes depending on the medium and that goes against Einstein's second postulate of special relativity. What am I missing?

Second postulate (invariance of c)

As measured in any inertial frame of reference, light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body. Or: the speed of light in free space has the same value c in all inertial frames of reference.

See:

https://en.wikipedia.org/wiki/Postulates_of_special_relativity#Postulates_of_special_relativity

Note the conditions that light is traveling through empty space and is measured in an inertial frame of reference.

 

[MikeeMiracle]

This one used to get me too. Individual photons always travel at C, but if there is a medium the light is passing through those photon's get absorbed and re-emitted, thus the "wave" of light passing through a medium like water is slower than C. In water I believe the "wave" of light travels at 0.75C.

 

[Vanadium 50]

but if there is a medium the light is passing through those photon's get absorbed and re-emitted

Old wives tale. This is not what happens. I suggest a forum search.