Question

A light ray is an incident at one face of an equilateral prism such that the angle of incidence is ${45}^0$. if the light ray emerges from the other face of the prism by making the same angle with the normal, determine the refractive index of the material of the prism.

• A. $\sqrt{\frac{3}{2}}$
• B. $\frac{{}^{\sqrt{3}}}{2}$
• C. $\sqrt{3}$
• D. $\sqrt{2}$

Answer 1

The correct option is D $\sqrt{2}$

Given: A light ray is an incident at one face of an equilateral prism such that the angle of incidence is ${45}^{\circ }$. if the light ray emerges from the other face of the prism by making the same angle with the normal

To find the refractive index of the prism.

Solution:

As it is an equilateral triangle all the angles are equal to ${60}^{\circ }$

Hence, refractive angle, $A={60}^{\circ }$

Incident angle is ${45}^{\circ }⟹$ angle of incidence, $i=90-45={45}^{\circ }$ refer the above fig.

And, angle of emergence, $e={45}^{\circ }$

Let $\delta$ be angle of deviation, as $i=e$, it is angle of minimum deviation

And $\delta =i+e-A⟹i=45+45-60={30}^{\circ }$

And for prism,

Refractive index, $\mu =\frac{\sin \left(\frac{\delta +A}{2}\right)}{\sin \left(\frac{A}{2}\right)}$

Substituting the corresponding values, we get

$⟹\mu =\frac{\sin \left(\frac{30+60}{2}\right)}{\sin \left(\frac{60}{2}\right)}⟹\mu =\frac{\sin \left(45\right)}{\sin \left(30\right)}⟹\mu =\frac{\frac{1}{\sqrt{2}}}{\frac{1}{2}}⟹\mu =\frac{2}{\sqrt{2}}$

Multiply numerator and denominator with $\sqrt{2}$, we get

$\mu =\frac{2\times \sqrt{2}}{\sqrt{2}\times \sqrt{2}}=\sqrt{2}$

Hence the refractive index of the prism is $\sqrt{2}$