Question

If the critical angle for total internal reflection from a medium to vacuum is \[30^\circ \]. Then velocity of light in the medium is:
A. \[3 \times {10^8}{{\rm{m}} {\left/ {\vphantom {{\rm{m}} {\rm{s}}}} \right.} {\rm{s}}}\]
B. \[4.5 \times {10^8}{{\rm{m}} {\left/ {\vphantom {{\rm{m}} {\rm{s}}}} \right.} {\rm{s}}}\]
C. \[1.5 \times {10^8}{{\rm{m}} {\left/ {\vphantom {{\rm{m}} {\rm{s}}}} \right.} {\rm{s}}}\]
D. \[2.5 \times {10^8}{{\rm{m}} {\left/ {\vphantom {{\rm{m}} {\rm{s}}}} \right.} {\rm{s}}}\]

Answer 1

Hint: We will utilize the relationship between the critical angle for total internal reflection and the refractive index of the medium. Also, we will use the relationship between speed of light in vacuum, speed of light in the medium and refractive index of the medium.

Complete step by step answer:
We are given the critical angle for total internal reflection from a medium to vacuum is \[{\theta _c} = 30^\circ \]. We are required to calculate the velocity of light in the medium.

We know that the refractive index of air is unity.
\[{n_1} = 1\]

We also know the relationship between the refractive index of the medium and critical angle for total internal reflection when light travels from a medium to vacuum.
\[{n_2} = \dfrac{1}{{\sin {\theta _c}}}\]

Substitute \[30^\circ \] for \[{\theta _c}\] in the above expression.
\[\begin{array}{c}
{n_2} = \dfrac{1}{{\sin 30^\circ }}\\
 = \dfrac{1}{{{1 {\left/ {\vphantom {1 2}} \right.} 2}}}\\
 = 2
\end{array}\]

Speed of light in the refractive medium can be written as the ratio of the speed of light in a vacuum to the refractive index of that medium.
\[v = \dfrac{c}{n}\]……(1)

We know that the speed of light in a vacuum is given as \[c = 3 \times {10^8}{{\rm{m}} {\left/ {\vphantom {{\rm{m}} {\rm{s}}}} \right.} {\rm{s}}}\].
Substitute \[3 \times {10^8}{{\rm{m}} {\left/ {\vphantom {{\rm{m}} {\rm{s}}}} \right.} {\rm{s}}}\] for c and \[2\] for n in equation (1).
\[\begin{array}{c}
v = \dfrac{{3 \times {{10}^8}{{\rm{m}} {\left/ {\vphantom {{\rm{m}} {\rm{s}}}} \right.} {\rm{s}}}}}{2}\\
 = 1.5 \times {10^8}{{\rm{m}} {\left/ {\vphantom {{\rm{m}} {\rm{s}}}} \right.} {\rm{s}}}
\end{array}\]
Therefore, we can say that if the critical angle for total internal reflection from a medium to vacuum is \[30^\circ \], the velocity of light in the medium is equal to \[1.5 \times {10^8}{{\rm{m}} {\left/ {\vphantom {{\rm{m}} {\rm{s}}}} \right.} {\rm{s}}}\]

So, the correct answer is “Option C”.

Note:
It will be an added advantage to remember the value of the speed of light in vacuum. Also, the critical angle for total internal reflection is given, do not confuse it with the angle of reflection when light is travelling from a medium to vacuum.