Question

A parallel beam of light is incident on a glass prism in the shape of a quarter cylinder of radius $8\, cm$ and refractive index $5 / 3$ placed on a horizontal table as shown in the figure. Beyond the cylinder, nearest distance from the cylinder where emergent light can struck the table is _____ m.

Answer 1

The distance at which the beam of light will fall on the table is $x$. This distance will be minimum when the incident beam strikes the cylindrical surface of prism at critical angle as shown in the figure.

$\sin i _{ c }=\frac{1}{\mu}=\frac{3}{5}$
From figure,
$ \cos i _{ c }=\frac{ R }{ R + x } $
$\therefore \sqrt{1-\sin ^{2} i _{ c }}=\frac{ R }{ R + x }$
$\therefore \sqrt{1-\frac{9}{25}}=\frac{ R }{ R + x } $
$\therefore \frac{ R }{ R + x }=\sqrt{\frac{16}{25}}=\frac{4}{5} $
$\therefore 4( R + x )=5 R $
$\therefore 4 x = R$
$\therefore x =\frac{ R }{4}=\frac{8}{4}=2 \,cm =0.02\, m$