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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 the 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 $$\theta \le 60^o$$. The value of $$n^2$$ is ____ .

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For T.I.R at coating

$$\sin c = \dfrac{n}{\sqrt{3}}$$

Appling snell's law at first surface

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

for limiting condition, at $$\theta = 60^o$$

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

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

$$\dfrac{1}{2} = \sin (75 - c) \Rightarrow \sin 30 = \sin (75 - c)$$

$$30 = 75 - c \Rightarrow c = 45^o$$

$$\dfrac{n}{\sqrt{3}} = \dfrac{1}{\sqrt{2}} \Rightarrow n^2 = \dfrac{3}{2} = 1.50$$

$$\sin c = \dfrac{n}{\sqrt{3}}$$

Appling snell's law at first surface

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

for limiting condition, at $$\theta = 60^o$$

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

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

$$\dfrac{1}{2} = \sin (75 - c) \Rightarrow \sin 30 = \sin (75 - c)$$

$$30 = 75 - c \Rightarrow c = 45^o$$

$$\dfrac{n}{\sqrt{3}} = \dfrac{1}{\sqrt{2}} \Rightarrow n^2 = \dfrac{3}{2} = 1.50$$

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