Question

The refractive index of the material of prism is 1.414 and its refracting angle is 30 degree. One of the refracting surfaces of the Prism is made a mirror inwards. A beam of monochromatic light entering the prism from the other face will retrace its path after reflection from the mirrored surface if its angle of incidence on the prism is

1. 60°
2. 0°
3. 30°
4. 45°

Answer 1

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Answer 2

Answer:

4. $45^\circ$

Explanation:

Given:

• A = angle of the prism = $30^\circ$
• $\mu$ = refractive index of the material of the prism = 1.414

Assume:

• a = angle of incidence at the first surface
• b = angle of refraction at the first surface
• c = angle of incidence at the second surface
• $\mu_a$ = refractive index of air = 1

In order to retrace the path after reflection at the second surface (mirrored surface), the incident ray at the second surface must be perpendicular to the reflecting surface. This means the angle of incidence is zero at the second surface.

i.e., $c=0^\circ$

So, according to the formula of a prism, we have

$A=b+c\\\Rightarrow b = A-c\\\Rightarrow b = 30^\circ-0^\circ\\\Rightarrow b = 30^\circ$

Now, using Snell's law of refraction at the first surface, we have

$\mu_a\sin a = \mu \sin b\\\Rightarrow 1\times \sin a= 1.414\sin 30^\circ\\\Rightarrow  \sin a= \sqrt{2}\times \dfrac{1}{2}\\\Rightarrow  \sin a=\dfrac{1}{\sqrt{2}}\\\Rightarrow  a=\sin^{-1}(\dfrac{1}{\sqrt{2}})\\\Rightarrow  a=45^\circ$

Hence, the angle of incidence should be $45^\circ$.