Question

For a prism kept in the air, it is found that for an angle of incidence ${60^0}$, the angle of refraction $A$, angle of deviation $\delta $ , and angle of emergence $e$ become equal. Then, the refractive index of the prism is:
$\left( A \right)1.73$
$\left( B \right)1.15$
$\left( C \right)1.5$
$\left( D \right)1.33$

Answer 1

Hint: When a light ray undergoes refraction, there will be some relation between the incident angle and emergent angle. When considering a prism, the minimum angle of deviation depends on the material of the prism. Apply the formula to find the minimum deviation. Then apply the snell's law at the incident surface. From applying the snell's law we can determine the refractive index of the prism.

Formula used:
$\delta = i + e - A$
$\delta $is the angle of deviation, $i$ is the incident angle, $e$ is the emergent ray, $A$ is the angle of prism.

Complete step by step solution:
The light ray undergoes refraction when the light ray is incident on the surface of the prism. The ray of the light bends towards the normal since the glass slab is the denser medium and it has a higher refractive index. That refracted light will act as the incident ray on the inner surface of the prism. The light ray will again undergo refraction and comes out of the glass prism as an emergent ray. Light bent through the smallest angle by an optical device is called an angle of deviation. If the angle of incidence is equal to the emergent angle then an angle of deviation is minimum. When considering a prism, the minimum angle of deviation depends on the material of the prism.
$\delta = i + e - A$
$\delta$ is the angle of deviation, $i$ is the incident angle, $e$ is the emergent ray, $A$ is the angle of prism.
Here the angle of incidence is equal to the angle of emergence.
$i = e$
The angle of the prism $A$ is ${60^0}$.
Then the angle of minimum deviation is ${30^0}$.
Now apply snell's law at the incident surface,
$\Rightarrow$ ${\mu _1}\sin {\alpha _1} = {\mu _2}\sin {\alpha _2}$
$\Rightarrow$ $1 \times \sin 60 = {\mu _2}\sin 30$
$\therefore$ $\mu = 1.732$
Hence option $A$ is the right option.

Note: We should always remember that in the case of a glass slab, the angle of emergence is always equal to the incident angle. But when we consider prisms the angle of emergence is always equal to the incident angle only at minimum deviation. Light bent through a smallest angle by an optical device is called the angle of deviation. When considering a prism, the minimum angle of deviation depends on the material of the prism.