Search any question & find its solution
Question:
Answered & Verified by Expert
A ray of light is incident on one face of an equilateral glass prism having refractive index $\sqrt{2}$. It produces the emergent ray which just grazes along the adjacent face. The value of angle of incidence is
Options:
Solution:
2663 Upvotes
Verified Answer
The correct answer is:
$\sin ^{-1}\left(\sqrt{2} \sin 15^{\circ}\right)$
The emergent ray just grazes the second face.
Hence angle of emergence, $\mathrm{e}=90^{\circ}$
$\begin{aligned} & \mu=\frac{\sin \mathrm{e}}{\sin r_2}=\frac{\sin 90^{\circ}}{\sin r_2}=\frac{1}{\sin r_2} \\ & \therefore \frac{1}{\sin r_2}=\sqrt{2} \text { or } \sin r_2=\frac{1}{\sqrt{2}}\end{aligned}$
$\begin{aligned} & \therefore \mathrm{r}_2=45^{\circ} ; \mathrm{A}=\mathrm{r}_1+\mathrm{r}_2 \\ & \therefore \mathrm{r}_1=\mathrm{A}-\mathrm{r}_2=60-45=15^{\circ}\end{aligned}$
Also, $\frac{\sin i}{\sin r_1}=\mu$
$\begin{aligned} & \therefore \sin \mathrm{i}=\mu \sin \mathrm{r}_1=\sqrt{2} \sin 15^{\circ} \\ & \therefore \mathrm{i}=\sin ^{-1}\left(\sqrt{2} \sin 15^{\circ}\right)\end{aligned}$
Hence angle of emergence, $\mathrm{e}=90^{\circ}$
$\begin{aligned} & \mu=\frac{\sin \mathrm{e}}{\sin r_2}=\frac{\sin 90^{\circ}}{\sin r_2}=\frac{1}{\sin r_2} \\ & \therefore \frac{1}{\sin r_2}=\sqrt{2} \text { or } \sin r_2=\frac{1}{\sqrt{2}}\end{aligned}$
$\begin{aligned} & \therefore \mathrm{r}_2=45^{\circ} ; \mathrm{A}=\mathrm{r}_1+\mathrm{r}_2 \\ & \therefore \mathrm{r}_1=\mathrm{A}-\mathrm{r}_2=60-45=15^{\circ}\end{aligned}$
Also, $\frac{\sin i}{\sin r_1}=\mu$
$\begin{aligned} & \therefore \sin \mathrm{i}=\mu \sin \mathrm{r}_1=\sqrt{2} \sin 15^{\circ} \\ & \therefore \mathrm{i}=\sin ^{-1}\left(\sqrt{2} \sin 15^{\circ}\right)\end{aligned}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.