Calculate the velocity of the light ray in the medium, if the critical angle for TIR from medium to the vacuum is 30°.

The correct option is 1.5 x 10⁸ m/s.

To find the velocity of the light ray in the medium, we can use the relationship between the critical angle for total internal reflection (TIR) and the refractive index of the medium.

The formula for the critical angle (${\theta }_c$) when light travels from a denser medium (with refractive index $n$) to a rarer medium (vacuum or air, with refractive index $n_{\text{vacuum}}=1$) is given by:

$sin\left({\theta }_c\right)=\frac{1}{n}$

Given that the critical angle ${\theta }_c={30}^{\circ }$, we can substitute this value into the equation:

$sin\left({30}^{\circ }\right)=\frac{1}{n}$

Since $sin\left({30}^{\circ }\right)=0.5=\frac{1}{2}$, we have:

$\frac{1}{2}=\frac{1}{n}$

Therefore, the refractive index of the medium is:
$n=2$

The refractive index ($n$) of a medium is also defined as the ratio of the speed of light in a vacuum ($c$) to the velocity of light in that medium ($v$):

$n=\frac{c}{v}$

We know that the speed of light in a vacuum ($c$) is approximately $3\times {10}^8\text{m/s}$. Rearranging the formula to solve for $v$:

$v=\frac{c}{n}$

Substituting the known values:

$v=\frac{3\times {10}^8\text{m/s}}{2}$

$v=1.5\times {10}^8\text{m/s}$

Thus, the velocity of the light ray in the medium is 1.5 x 10⁸ m/s.