Question

Calculate the critical angle for diamond having a refractive index of \[2.42\] and crown glass having refractive index of \[1.52\] ?

Answer 1

Hint: First of all, we will find the relation of the refractive index of the medium and the critical angle. Then use the values of the refractive index in the formula and manipulate accordingly.

Complete step by step answer:
We have from Snell’s law that:
The equation which relates refractive index, incident angle and refracted angle.
\[{n_1}\sin {\theta _1} = {n_2}\sin {\theta _2}\] …… (1)
Where,
\[{n_1}\] indicates the refractive index of a medium from which light is coming.
\[{\theta _1}\] indicates the angle of incidence.
\[{n_2}\] indicates the refractive index of a medium into which light is entering into.
\[{\theta _2}\] indicates the angle of refraction.

We know, at the critical angle, the refracted ray emerges at right angle.
In this case, it can be written as:
\[{\theta _2} = 90^\circ \]
And \[{\theta _1} = c\]
Where, \[c\] is the critical angle.

Now, equation (1) can be re-written as:
\[{n_1}\sin {\theta _1} = {n_2}\sin {\theta _2}\]
\[
  {n_1}\sin c = {n_2}\sin 90^\circ \\
  {n_1}\sin c = {n_2} \times 1 \\
  {n_1}\sin c = {n_2} \\
  \sin c = \dfrac{{{n_2}}}{{{n_1}}} \\
 \]
When the second medium is air, then the refractive index of the second medium is \[1\] .
Then the expression for the critical angle becomes:
\[
  \sin c = \dfrac{{{n_2}}}{{{n_1}}} \\
  \sin c = \dfrac{1}{{{n_1}}} \\
\]
\[c = {\sin ^{ - 1}}\left( {\dfrac{1}{{{n_1}}}} \right)\] …… (2)
Calculating critical angle for diamond:
Given, the refractive index for diamond is \[2.42\] .
Putting the value in equation (2), we get:
\[
  {c_{{\text{diamond}}}} = {\sin ^{ - 1}}\left( {\dfrac{1}{{{n_1}}}} \right) \\
  {c_{{\text{diamond}}}} = {\sin ^{ - 1}}\left( {\dfrac{1}{{2.42}}} \right) \\
  {c_{{\text{diamond}}}} = {\sin ^{ - 1}}\left( {0.41} \right) \\
  {c_{{\text{diamond}}}} = 24.2^\circ \\
\]
Hence, the critical angle of the diamond is \[24.2^\circ \] .

Calculating critical angle for crown glass:
Given, the refractive index for diamond is \[1.52\] .
Putting the value in equation (2), we get:
\[
  {c_{{\text{crown}} - {\text{glass}}}} = {\sin ^{ - 1}}\left( {\dfrac{1}{{{n_1}}}} \right) \\
  {c_{{\text{crown}} - {\text{glass}}}} = {\sin ^{ - 1}}\left( {\dfrac{1}{{1.52}}} \right) \\
  {c_{{\text{crown}} - {\text{glass}}}} = {\sin ^{ - 1}}\left( {0.65} \right) \\
  {c_{{\text{crown}} - {\text{glass}}}} = 40.5^\circ \\
\]
Hence, the critical angle of crown glass is \[40.5^\circ \] .

Additional Information:
Refraction in physics is the shift in the direction of a wave moving from one medium to another, or from a gradual shift in the medium. The critical angle is defined as the incidence angle that provides a 90-degree angle of refraction.

Note:
While deriving the relation, keep in mind that in case of critical angle, the refracted angle is always the right angle. It is important to note that, higher the refractive index is, lower is the magnitude of critical angle.