question_answer

A monochromatic light is incident from air on a refracting surface of a prism of angle \[75{}^\circ \]  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 \[\theta \,\le \,{{60}^{0}}.\]The value of \[{{n}^{2}}\] is _______ .

A) 1.5

B) 2.5

C) 3.5

D) 4.5

Correct Answer: A

Solution :

For TIR at coating sin \[\text{C}\,=\,\frac{n}{\sqrt{3}}\]

Applying snells? law

\[\text{sin}\theta =\sqrt{3}\text{sin}(75{}^\circ -C)\]

For limiting condition \[\theta =\,60{}^\circ \]

\[\text{sin}60{}^\circ \,=\,\sqrt{3}\text{sin}(75{}^\circ -\text{C})\]

\[\therefore \]      \[\text{C}=\,45{}^\circ \]

and       \[{{n}^{2}}=\,\frac{3}{2}=1.5.\]