Question

A ray of light travelling in impure water is incident on a glass plate immersed in it. When the angle of incidence is ${{51}^{\circ }}$the reflected ray is completely plane polarized. Given that the refractive index of impure water is 1.4. The refractive index of glass should be, ($(\tan {{51}^{\circ }}=1.235)$)

Answer 1

It is given to us that the light travels from impure water to the glass which gets polarized. Hence we will use the condition for polarization and obtain the refractive index of glass. This refractive index of glass will be with respect to impure water. Refractive index of a material with respect to medium 1 when the ray of light enters from medium 1 to that material is the ratio of the refractive index of the material with respect to air to that of medium 1 with respect to air.
Formula used:
If a ray of light is incident form medium 1 to medium 2 such that the reflected light from the surface of medium 2 gets polarized, then refractive index $(\eta )$ of medium 2 with respect to 1 is $\eta =\tan {{i}_{p}}$ where ${{i}_{p}}$ is the angle of incidence with respect to the normal of medium 2.

Complete answer:
It is given in the question that the glass is immersed in water. Hence when we use the condition for refractive index by polarization, the refractive index will be with respect to the impure water. Let us say the refractive index of glass with respect to air is ${{\eta }_{G}}$ and refractive index of water with respect to air be ${{\eta }_{W}}$ hence the refractive index of glass with respect to water such that the reflected light gets polarized is given by,
$\begin{aligned} & \eta =\dfrac{{{\eta }_{G}}}{{{\eta }_{W}}}=\tan {{i}_{p}} \\\ & \dfrac{{{\eta }_{G}}}{1.4}=\tan {{51}^{\circ }},since\tan {{51}^{\circ }}=1.235 \\\ & \Rightarrow {{\eta }_{G}}=1.235\times 1.4=1.729 \\\ \end{aligned}$

Hence the refractive index of glass is 1.729

Note:
The angle of incidence for which the reflected light will get polarized depends on the medium on which the light is incident on. To be more precise, it also depends on both the mediums, the rarer as well as the denser. The effect of a polarized light is that the intensity of the light decreases. It is also to be noted that the refractive index is dimensionless.