In the given figure, the face AC of the equilateral prism is...

In the given figure, the face $AC$ of the equilateral prism is immersed in a liquid of refractive index $n$ . For incident angle $60 °$ 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 $= 1 . 5$ )

$In the given figure, the face AC of the equilateral prism is immersed in a liquid of refractive index n . For incident$

Correct answer

Step-by-step solution

$In the given figure, the face AC of the equilateral prism is immersed in a liquid of refractive index n . For incident$
When the refracted light just grazes the boundary the incident angle is called a critical angle.
Using snell's law at face $AC$ .
$1 . 5\sin 60 ° = n \times \sin 90 °$
$⇒ 1 . 5 \times \dfrac{\sqrt{3}}{2} = n = \dfrac{\sqrt{x}}{4}$
$⇒ 3\sqrt{3} = \sqrt{x}$
$⇒ x = 27$

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