Question

Consider a tank made of glass $\text{(refractive index 1.5)}$ 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 $\mu$ is:

• A. $\sqrt{\frac{5}{3}}$
• B. $\frac{5}{\sqrt{3}}$
• C. $\frac{3}{\sqrt{5}}$
• D. $\frac{4}{3}$

Answer 1

The correct option is C $\frac{3}{\sqrt{5}}$


For any reflected rays not to completely get polarized,

$i_c<i_b---\left(1\right)$

According to Brewster's law,

$\tan i_b=\frac{1.5}{\mu }$

$\Rightarrow \sin i_b=\frac{1.5}{\sqrt{{\mu }^2+(1.5{)}^2}}$

The critical angle between air and the liquid is given by,

$\sin i_c=\frac{1}{\mu }$

From $\left(1\right)$, $\sin i_c<\sin i_b$

$\Rightarrow \frac{1}{\mu }<\frac{1.5}{\sqrt{{\mu }^2+(1.5{)}^2}}$

$\Rightarrow \sqrt{{\mu }^2+(1.5{)}^2}<1.5\times \mu$

$\Rightarrow {\mu }^2+(1.5{)}^2<(\mu \times 1.5{)}^2$

$\Rightarrow \mu >\frac{3}{\sqrt{5}}$

Therefore, the minimum value of $\mu$ should be $\frac{3}{\sqrt{5}}$

Hence, option $\left(B\right)$ is correct.