Question

A prism of refracting angle $30^\circ$ is coated with a thin film of transparent material of refractive index 2.2 on face AC of the prism. A light of wavelength 6600 $\mathring{A}$ is incident on face AB such that angle of incidence is $60^\circ$. Find
(a) the angle of emergence and
(b) the minimum value of thickness of the coated film on the face AC for which the light emerging from the face has maximum intensity. (Given refractive index of the material of the prism is $\sqrt{3}$)

Answer 1

(a) sin$i_1=\mu sin\, r_1$
$or sin\, 60^\circ sin\, r_1$
$\therefore sin r_1=\frac{1}{2}\, or\, \, \, \, \, \, \, \, r_1=30^\circ$
$Now, \, \, \, \, \, \, \, \, \, \, r_1+r_1=A$
$\therefore r_2=A-r_1=30^\circ-30^\circ=0^\circ$
Therefore, ray of light falls normally on the face A C and
angle of emergence$i_2=0^\circ$
(b) Multiple reflections occur between surface of film.
Intensity will be maximum if constructive interference
takes place in the transmitted wave.
For maximum thickness
$\Delta x=2\mu t=\lambda $ (t = thickness)
$\therefore \, \, \, \, \, \, \, \, \, t=\frac{\lambda}{2\mu}=\frac{6600}{2\times 22}=1500 \mathring{A}$