Question 181 of 191

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 $\mu$ is:

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

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Answer: (A) $\dfrac{3}{\sqrt{5}}$

For reflected light not be completely polarized,

$\tan ⁡r\neq \dfrac{{\mu }_{g}}{{\mu }_{w}}=\dfrac{1.5}{\mu }$

For minimum $\mu r$ is maximum. $r$ is maximum when $i$ is maximum i.e. $i={90}^{o}$

$\Rightarrow \sin ⁡r=\dfrac{1}{\mu }$

$\Rightarrow \tan ⁡r=\dfrac{1}{\sqrt{{\mu }^{2}-1}}$

$∴\dfrac{1}{\sqrt{{\mu }^{2}-1}}=\dfrac{1.5}{\mu }$

$\Rightarrow {\mu }^{2}=2.25\left({\mu }^{2}-1\right)$

$\Rightarrow 1.25{\mu }^{2}=2.25$

$\Rightarrow \mu =\sqrt{\dfrac{2.25}{1.25}}$

$=\sqrt{\dfrac{9}{5}}=\dfrac{3}{\sqrt{5}}$