Question

A glass rod of rectangular cross-section is bent into the shape shown in figure. A parallel beam of light falls perpendicularly on the flat surface A.What should be the minimum value of R/d for which all the light entering at surface A will emerge at surface B. Index of refraction of the glass is 1.5?

Answer 1

A ray entering through surface A and travelling along the inner side of the rod will be reflected by the outer side with the smallest angle $\theta$ , at which the reflected ray is tangent to the inner side.
$\theta \ge {\theta }_{critical}=si{n}^{-1}\left[\frac{1}{n}\right]$
sin $\theta \ge \frac{1}{n}$
But sin $\theta =\frac{R}{R+d}$
$⟹\frac{R}{R+d}\ge \frac{1}{3/2}$

$\frac{R+d}{R}\le \frac{3}{2}$

$\frac{d}{R}\le \frac{1}{2}⟹\frac{R}{d}\ge 2$

$[\frac{R}{d}{]}_{min}=2$