question_answer

A light ray strikes centre of one end of a cylinder rod of refractive index n, at an angle \[\alpha \] with its axis.

Least value of n such that light ray does not emerge from curved surface of cylinder is

A) \[\sqrt{3}\]

B) \[\sqrt{2}\]

C) \[2\]

D) \[\sqrt{5}\]

Correct Answer: B

Solution :

From figure, we have \[{{r}_{2}}=90-{{r}_{1}}\]

\[{{({{r}_{2}})}_{\min }}=90-{{({{r}_{1}})}_{\max }}\]

Also, for TIR,

\[{{({{r}_{2}})}_{\min }}\ge {{\theta }_{c}}(\text{critical}\,\,\text{angle})\]

\[\sin \,{{({{r}_{2}})}_{\min }}\ge \sin {{\theta }_{c}}\]

\[\Rightarrow \]\[\sin \,{{(90-{{r}_{1}})}_{\max }})\ge \sin {{\theta }_{c}}\]

\[\Rightarrow \]\[\cos {{({{r}_{1}})}_{\max }}\ge \frac{1}{n}\]                  ? (i)

Also, \[\frac{\sin \alpha }{\sin {{r}_{1}}}=n\]\[\Rightarrow \]\[\frac{\sin \,{{(\alpha )}_{\max }}}{\sin \,{{({{r}_{1}})}_{\max }}}\le n\]

But \[{{\alpha }_{\max }}=90{}^\circ \]

\[\therefore \]\[\frac{\sin 90}{\sin {{({{r}_{1}})}_{\max }}}\le n\]\[\Rightarrow \]\[\frac{1}{\sin {{({{r}_{1}})}_{\max }}}\le n\]

\[\Rightarrow \]\[\sin {{({{r}_{1}})}_{\max }}\ge \frac{1}{n}\]                              ?(ii)

From Eqs. (i) and (ii), we have \[1-{{\cos }^{2}}{{({{r}_{1}})}_{\max }}={{\sin }^{2}}{{({{r}_{1}})}_{\max }}\]

\[1-\frac{1}{{{n}^{2}}}=\frac{1}{{{n}^{2}}}\]\[\Rightarrow \]\[\frac{2}{{{n}^{2}}}=1\]\[\Rightarrow \]\[n=\sqrt{2}\]

\[\therefore \]\[{{n}_{\min }}=\sqrt{2}\]