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 \[1-{{\cos }^{2}}{{({{r}_{1}})}_{\max }}={{\sin }^{2}}{{({{r}_{1}})}_{\max }}\] |
\[1-\frac{1}{{{n}^{2}}}=\frac{1}{{{n}^{2}}}\]\[\Rightarrow \]\[\frac{2}{{{n}^{2}}}=1\]\[\Rightarrow \]\[n=\sqrt{2}\] |
\[\therefore \]\[{{n}_{\min }}=\sqrt{2}\] |
You need to login to perform this action.
You will be redirected in
3 sec