In a medium the speed of light wave decreases to 0 . 2 times to its...

Speed of light in medium is related to speed of light in vacuum $\left(c\right)$ and refractive index $\left(\mu\right)$ by $v = \dfrac{c}{\mu}$.

Then, we have $\mu = \dfrac{c}{v} = \dfrac{c}{0 . 2c}$

$⇒ \mu = 5$

Refractive index in terms of relative permeability $\left({\mu}_{r}\right)$and relative permittivity$\left({ε}_{r}\right)$ of the medium is stated as $\mu = \sqrt{{ϵ}_{r}{\mu}_{r}}$

$⇒ {ϵ}_{r} = \dfrac{{\mu}^{2}}{{\mu}_{r}}$

$∴ \dfrac{{ϵ}_{r}}{\mu} = \dfrac{\mu}{{\mu}_{r}} = 5$

Hence, the value of $x = 5$.