Question

Consider a tank made of glass (refractive index $1.5$ ) with a thick bottom. It is filled with a liquid of refractive index $\mu$ . A student finds that, irrespective of what the incident angle $i$ (see figure) is for a beam of light entering the liquid, the light reflected from the liquid glass interface is never completely polarized. For this to happen, the minimum value of μ is

(A) $\dfrac{3}{{\sqrt 5 }}$
(B) $\dfrac{5}{{\sqrt 3 }}$
(C) $\sqrt {\dfrac{5}{3}}$
(D) $\dfrac{4}{3}$

Answer 1

For finding out the minimum value of the refractive index, we need to use the relation between Brewster angle and the critical angle which is given as $sin{\text{ }}c < sin{\text{ }}{i_{b\;}}$ . Brewster angle is an angle of incidence at which light with a particular polarization is perfectly transmitted through a transparent dielectric surface, with no reflection.

Formula used:
$sin{\text{ c}} < sin{\text{ }}{i_{b\;}}$
$\ sin90 = \mu \sin c \\\ \Rightarrow sinc = \dfrac{1}{\mu } \\\$
Where, the critical angle be ${\text{ c}}$ , the Brewster angle be ${\text{ }}{i_{b\;}}$ , ${\text{ }}\mu$ is the refractive index.

Complete step by step solution:
Let us consider the critical angle be c, the Brewster angle be ib.
The relation comes between the critical angle and the Brewster angle is,
$sin{\text{ }}c < sin{\text{ }}{i_{b\;}}$
For the ray travelling from air to liquid,
$sin90 = \mu \sin c$
Now the value of $sin90$ is 1 so we get
$sinc = \dfrac{1}{\mu }$
Since, we know that,
$\tan {i_b} = {\mu _{{0_{rel}}}}$
And,
$sin{\text{ }}c < sin{\text{ }}{i_{b\;}}$
Then substituting the values in the equation we get,
$\Rightarrow \dfrac{1}{\mu } < \dfrac{{1.5}}{{\sqrt {{\mu ^2} + {{(1.5)}^2}} }}$
Thus, after simplification, we get,
$\Rightarrow \mu < \dfrac{3}{{\sqrt 5 }}$
Hence, the correct answer is option A.

Note:
There are numerous applications of Brewster angle in real life. It includes,
-Polarized sunglasses use the principle of Brewster's angle to reduce glare from the sun reflecting off horizontal surfaces such as water or road.
-Photographers use the same principle to remove reflections from water so that they can photograph objects beneath the surface.
-Brewster angle prisms are used in laser physics. The polarized laser light enters the prism at Brewster's angle without any reflective losses.
-In surface science, Brewster angle microscopes are used in imaging layers of particles or molecules at air-liquid interfaces.