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Question: A parallel beam of light is incident from air at an angle \[\alpha \] on the side PQ of a right angl...

A parallel beam of light is incident from air at an angle α\alpha on the side PQ of a right angled triangular prism of a refractive index η=2\eta = \sqrt 2 . Light undergoes total internal reflection in the prism at the face PR when α\alpha has a minimum value of 45o{45^o}. The angle θ\theta of the prism is:

A. 15o{15^o}
B. 22.5o{22.5^o}
C. 30o{30^o}
D. 45o{45^o}

Explanation

Solution

Hint – In questions we should remember the laws of refraction and specially the Snell’s law i.e.
n1sin⁡α=n2sin⁡β{n_1}\sin \alpha = {n_2}\sin \beta , where α\alpha is angle of incidence and β\beta is angle of refraction.

Formula used-

  1. n1sin⁡α=n2sin⁡β{n_1}\sin \alpha = {n_2}\sin \beta
  2. Angle sum of a triangle is 180o{180^o}

Complete step-by-step answer:
The ray will bend toward the normal since it goes from the rarer to denser medium and then strikes with the face PR and then TIR happens.

Given, η=2\eta = \sqrt 2

αmin⁡=45o{\alpha _{\min }} = {45^o}

We know that, n1sin⁡α=n2sin⁡β{n_1}\sin \alpha = {n_2}\sin \beta

1×sin⁡α=n×sin⁡β1 \times \sin \alpha = n \times \sin \beta

When the value of αmin⁡=45o{\alpha _{\min }} = {45^o}and η=2\eta = \sqrt 2 .

Then the value of sin⁡β=12\sin \beta = \dfrac{1}{2}
Therefore the value ofβ=30o\beta = {30^o}

Here the new term δ\delta is called the angle of deviation. It is defined as the difference of the angle between refraction and the incidence angle and this angle is made when there is a minimum two mediums of different refractive index as here it is air and glass.

δ=90−(180−(90+θ+β))\delta = 90 - (180 - (90 + \theta + \beta ))

∴ Applying Snell's law at surface PR we get the equation as:

∴2sin⁡δ=1 ∴sin⁡(θ+β)=12 θ+β=45o θ+30o=45o Therefore,θ=15o  \therefore \sqrt 2 \sin \delta = 1 \\\ \therefore \sin (\theta + \beta ) = \dfrac{1}{{\sqrt 2 }} \\\ \theta + \beta = {45^o} \\\ \theta + {30^o} = {45^o} \\\ Therefore,\theta = {15^o} \\\

Hence, the correct option is A.

Note: In this question we should use Snell's law of refraction and basic geometry of triangles to find the angle as the sum of angles of a triangle is 180o{180^o}. Doing this will solve your problem. Generally these types of problems arise from this chapter. So, learning the important formulas will be a better option for students.