Question

A parallel beam of light is incident on a glass prism in the shape of a quarter cylinder of radius $8cm$ 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

Answer: 0.02

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,
$\cos i_c=\frac{R}{R+x}$
$\therefore \sqrt{1-{\sin }^2{i}_c}=\frac{R}{R+x}$
$\therefore \sqrt{1-\frac{9}{25}}=\frac{R}{R+x}$
$\therefore \frac{R}{R+x}=\sqrt{\frac{16}{25}}=\frac{4}{5}$
$\therefore 4(R+x)=5R$
$\therefore 4x=R$
$\therefore x=\frac{R}{4}=\frac{8}{4}=2cm=0.02m$