JEE Advanced · Physics · 24. Ray Optics

A monochromatic light is incident from air on a refracting surface of a prism of angle $75^{o}$ and refractive index $n_{0} = \sqrt{3} .$ The other refracting surface of a prism is coated by a thin film of material of refractive index $n$ as shown in figure. The light suffers total internal reflection at the coated prism surface for an incidence angle of $θ \leq 60^{o} .$ The value of $n^{2}$ is_______.

A 1.5
B 1.2
C 1.1
D 2.5

Verified Solution

Answer & Solution

Correct Answer

(A) 1.5

Step-by-step Solution

Detailed explanation

For T.I.R at coating

sinc = \frac{n}{\sqrt{3}}

…(i)

[\sin C = \frac{n_{2}}{n_{1}}]

Appling Snell's law at first surface

\sin θ = \sqrt{3} \sin (75 - c)

for limiting conduction, at

θ = 60^{o}

\sin 60 = \sqrt{3} \sin (75 - c)

\frac{\sqrt{3}}{2} = \sqrt{3} \sin (75 - c)

\begin{matrix} \frac{1}{2} = \sin (75 - c) & \Rightarrow & \sin 30 = \sin (75 - c) \\ \Rightarrow 30 = 75 - c & \Rightarrow & c = 45^{o} \\ U s i n g (i) \frac{n}{\sqrt{3}} = \frac{1}{\sqrt{2}} & \Rightarrow & n^{2} = \frac{3}{2} = 1.50 \end{matrix}

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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