Question

A ray of light is incident on a \[60^\circ \] prism at the minimum deviation position. The angle of refraction at the first face (i.e. incident face) of the prism is:
(A) Zero
(B) \[30^\circ \]
(C) \[45^\circ \]
(D) \[60^\circ \]

Answer 1

Hint: Generally, the angle of refraction from the first face plus the incident angle on the second face (which is due to the refracted light from the first face) is equal to the angle of prism. For minimum deviation, the first refraction angle (refraction from first face), and the second incident angle (incidence on second face) are equal.
Formula used: In this solution we will be using the following formulae;
\[A = {r_1} + {i_2}\] where \[A\] is the angle of prism, \[{r_1}\] is the angle of refraction from the first face, and \[{i_2}\] is the angle of incidence on the second face.

Complete Step-by-Step Solution:
Generally, when light enters a prism from one side, there is refraction towards the normal. Now this refracted light travels in the prism, and becomes an incident ray on the second face of the prism. Then the light gets refracted as it exits the prism, and the refraction is away from the normal. This second refracted light is called an emergent ray.
Generally, in a prism of a particular angle, the following equation is obeyed
\[A = {r_1} + {i_2}\] where \[A\] is the angle of prism, \[{r_1}\] is the angle of refraction from the first face, and \[{i_2}\] is the angle of incidence on the second face.
Now, for minimum deviation, the refraction angle \[{r_1}\] is equal to the incident angle \[{i_2}\] i.e. \[{r_1} = {i_2} = r\]
Then
\[A = r + r = 2r\]
Then the refraction angle is
\[r = \dfrac{A}{2}\]
By substitution of known values, we get
\[r = \dfrac{{60^\circ }}{2} = 30^\circ \]

Hence, the correct option is B

Note: For clarity, the angle of prism formula can be proven from the diagram as shown below

The angle between the normal and the prism is 90 degrees, hence, the angle between the first refracted ray
and the first face of prism would be
\[z = 90 - {r_1}\]
Similarly for the second face of prism,
\[y = 90 - {i_2}\]
But the total angle in a triangle is 180 degrees, hence
\[A + z + y = 180\]
\[ \Rightarrow A = 180 - (90 - {r_1}) - 90 - {i_2}\]
By simplifying, we obtain
\[A = {r_1} + {i_2}\]