The critical angle for a denser-rarer interface is 45 ° . The speed...

The critical angle and the refractive index of a media are related by the formula:

$sin{\theta}_{c} = \left(\dfrac{1}{\mu}\right) . . . \left(1\right)$

Substitute the value of the critical angle into equation (1) and solve to calculate the refractive index of the medium.

$\begin{cases}\sin 45 ° & = & \dfrac{1}{\mu} \\ ⇒ & \dfrac{1}{\sqrt{2}} = \dfrac{1}{\mu} \\ ⇒ & \mu = \sqrt{2}\end{cases}$

The refractive index of the medium can also be written as

$\mu = \dfrac{c}{v} . . . \left(2\right)$

Substitute the values of the known parameters into equation (2) and solve to calculate the required speed of light within the medium.

$\begin{cases}\sqrt{2} & = & \dfrac{3 \times {10}^{8}m{s}^{- 1}}{v} \\ ⇒ & v = \dfrac{3 \times {10}^{8}m{s}^{- 1}}{\sqrt{2}} \\ = & 2 . 12 \times {10}^{8}m{s}^{- 1}\end{cases}$