Join the Most Relevant JEE Main 2025 Test Series & get 99+ percentile! Join Now
Search any question & find its solution
Question: Answered & Verified by Expert
A glass prism has a right-triangular cross section $\mathrm{ABC}$, with $\angle \mathrm{A}=90^{\circ}$. A ray of light parallel to the hypotenuse BC and incident on the side AB emerges grazing the side AC. Another ray, again parallel to the hypotenuse BC, incident on the side AC suffers total internal reflection at the side $\mathrm{AB}$. Which one of the following must be true about the refractive index $\mu$ of the material of the prism?
PhysicsRay OpticsKVPYKVPY 2016 (SB/SX)
Options:
  • A $\sqrt{\frac{3}{2}} < \mu < \sqrt{2}$
  • B $\mu>\sqrt{3}$
  • C $\mu < \sqrt{\frac{3}{2}}$
  • D $\sqrt{2} < \mu < \sqrt{3}$
Solution:
2755 Upvotes Verified Answer
The correct answer is: $\sqrt{\frac{3}{2}} < \mu < \sqrt{2}$



$\mathrm{r}+\theta_{\mathrm{C}}=90^{\circ}......\mathrm{(1)}$
$1 \times \sin \left(90^{\circ}-\alpha\right)=\mu \sin \mathrm{r}$
$\cos \alpha=\mu \sin \mathrm{r}......\mathrm{(2)}$
$90^{\circ}-\mathrm{e}>\theta_{\mathrm{C}}......\mathrm{(3)}$
$\mu \sin \mathrm{e}=1 \times \sin \alpha......\mathrm{(4)}$
$(3) \&(4)$
$90^{\circ}-\theta_{C}>\mathrm{e}$
$\cos \theta_{\mathrm{C}}>\sin \mathrm{e}$
$\cos \theta_{\mathrm{C}}>\frac{\sin \alpha}{\mu}$
$1-\sin ^{2} \theta_{\mathrm{C}}>\frac{1}{\mu^{2}}\left[1-\mu^{2} \sin ^{2} \mathrm{r}\right]$
$1-\frac{1}{\mu^{2}}>\frac{1}{\mu^{2}}\left[1-\mu^{2} \sin ^{2}\left(90^{\circ}-\theta_{\mathrm{C}}\right)\right]$
$1-\frac{1}{\mu^{2}}>\frac{1}{\mu^{2}}-\cos ^{2} \theta_{\mathrm{C}}$
$1-\frac{2}{\mu^{2}}>-\left[1-\frac{1}{\mu^{2}}\right]$
$2>\frac{3}{\mu^{2}}$
$\mu>\sqrt{\frac{3}{2}}$
$(1) \&(2)$
$\cos \alpha=\mu \sin \left(90^{\circ}-\theta_{C}\right)$
$\cos \alpha=\mu \cos \theta_{C}$
$\cos \alpha < 1$
$\mu \cos \theta_{C} < 1$
$\sqrt{1-\frac{1}{\mu^{2}}} < \frac{1}{\mu}$
$1-\frac{1}{\mu^{2}} < \frac{1}{\mu^{2}}$
$1 < \frac{2}{\mu^{2}}$
$\mu < \sqrt{2}$
$\therefore \sqrt{\frac{3}{2}} < \mu < \sqrt{2}$

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.