Q. A rod of glass $(\mu = 1.5)$ and of square cross section is bent into the shape shown in the figure. A parallel beam of light falls on the plane flat surface $A$ as shown in the figure. If $d$ is the width of a side and $R$ is the radius of circular arc then for what maximum value of $\frac{d}{R}$ light entering the glass slab through surface $A$ emerges from the glass through $B$
A) 1.5
B) 0.5
C) 1.3
D) None of these
Explanation
(b) Consider the figure if smallest angle of incidence $\theta$ is greater than critical angle then all light will emerge out of $B$
$\Rightarrow \theta \geq \sin^{- 1}\left( \frac{1}{\mu} \right) \Rightarrow \sin\theta \geq \frac{1}{\mu}$
From figure $\sin\theta = \frac{R}{R + d}$
$\Rightarrow \frac{R}{R + d} \geq \frac{1}{\mu} \Rightarrow \left( 1 + \frac{d}{R} \right) \leq \mu$
$\Rightarrow \frac{d}{R} \leq \mu - 1 \Rightarrow \left( \frac{d}{R} \right)_{\max} = 0.5$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.