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Refractive index of medium $m$, $n_m$, is given by $$n_m = \sqrt{\mu_r\epsilon_r}$$where $\mu_r$ and $\epsilon_r$ are permittivity and permeability of the medium respectively.

And, velocity of light in medium $v_m$, is given by, $$v_m = {c \over n_m}$$so if we prove $n_m <1$, can light move faster than $c$?

Also, we know that $\mu_r$ can even reach $0$, in case of superconductor.

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The index of refraction can indeed be less than $1$. This means that the phase velocity of light can exceed $c$ (the vacuum speed of light), and in fact it does exceed $c$ over a large range of frequencies in the ionosphere. The article [1] is a relatively concise and easy-to-read review. It says:

As the refractive index [at radio frequencies in the ionosphere] is less than unity, phase velocity exceeds the velocity of light in free space. It also depends on the frequency. When the information is modulated with the radio wave, it travels with the group velocity which is always less than the velocity of light.

The group velocity is a better indicator of the speed of propagation of any modulation applied to the light wave, and as the excerpt indicates, the modulation will not propagate faster than $c$. Information cannot propagate faster than $c$ in any medium, vacuum or otherwise.

(Keep in mind, though, that this statement refers to the local speed, as explained in another post.)


Reference:

[1] Bora (2017), "Ionosphere and Radio Communication," https://www.ias.ac.in/article/fulltext/reso/022/02/0123-0133

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