Question 99 of 295

In the given figure, the face $AC$ of the equilateral prism is immersed in a liquid of refractive index $n$. For incident angle $60^{\circ}$ at the side $AC$, the refracted light beam just grazes along face $AC$. The refractive index of the liquid $n=\dfrac{\sqrt{x}}{4}$. The value of $x$ is _____ .
(Given refractive index of glass $=1.5$)

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Answer: 27

When the refracted light just grazes the boundary the incident angle is called a critical angle.
Using snell's law at face $AC$.
$1.5\sin 60^{\circ}=n\times \sin 90^{\circ}$
$\Rightarrow 1.5\times \dfrac{\sqrt{3}}{2}=n=\dfrac{\sqrt{x}}{4}$
$\Rightarrow 3\sqrt{3}=\sqrt{x}$
$\Rightarrow x=27$