Question

A ray of light is incident at an angle of incidence $45^\circ $ on an equilateral prism and emerge at an angle $45^\circ $ then the angle of deviation of the ray will be
(1) $\dfrac{\pi }{6}$
(2) $\dfrac{\pi }{3}$
(3) $\dfrac{\pi }{4}$
(4) $\dfrac{\pi }{8}$

Answer 1

Hint the above problem can be solved by using the concept of the angle of deviation. The angle of deviation is the deflection of the incident ray when it passes through the prism. The angle of the emergent ray depends on the prism angle. The prism for the above problem is equilateral prism, so the angle of the prism will be $60^\circ $.

Complete step by step answer
Given: The angle of the incident ray is $i = 45^\circ = 45^\circ \times \dfrac{\pi }{{180^\circ }} = \dfrac{\pi }{4}$.
The angle of the emergence of the ray is $r = 45^\circ = 45^\circ \times \dfrac{\pi }{{180^\circ }} = \dfrac{\pi }{4}$ .
The angle of the equilateral prism is $A = 60^\circ = 60^\circ \times \dfrac{\pi }{{180^\circ }} = \dfrac{\pi }{3}$ .
The equation to calculate the angle of deviation of the ray is,
$\delta = i + r - A$
Substitute $\dfrac{\pi }{4}$for $i$ , $\dfrac{\pi }{4}$ for $r$ and $\dfrac{\pi }{3}$ for A in the above equation to find the angle of deviation of the ray.
$\delta = \dfrac{\pi }{4} + \dfrac{\pi }{4} - \dfrac{\pi }{3}$
$\delta = \dfrac{\pi }{6}$

Thus, the angle of deviation of the ray is $\dfrac{\pi }{6}$ and the option (1) is the correct answer.

Additional Information The angle of deviation becomes minimum if the angle of incidence of ray becomes equal to the angle of emergence of the light from the prism. The condition for the no emergence from the prism is that the angle of the prism is greater than the half of the critical angle. The splitting of the beam of the light into a spectrum of various colors of light is called the angular dispersion.

Note The angle of deviation in the emergence ray is minimum because the angle of incidence of the ray is equal to the angle of emergence of the ray. If the angle of incidence of the ray and angle of prism is very small then the angle of deviation can be found by using the formula$\delta = \left( {\mu - 1} \right)A$. Here, $\mu $ is the refractive index of the medium.