# Can light in a medium travel faster than speed of light $c$ in vacuum?

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.

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:
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.