Question

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 = \((\sqrt x)/4\). The value of \(x\) is _______.(Given refractive index of glass = \(1.5\))

Answer 1

Correct Answer: 27

The prism is equilateral, so each angle is 60°. When the light refracts out into the liquid, it grazes along face AC. Thus, the angle of refraction \( r = 90^\circ \).
Using Snell's law at the AC interface: $n\; \backslash sin\; 90^\backslash circ\; =\; \backslash mu\; \backslash sin\; 60^\backslash circ$
Since \( n = \frac{\sqrt{x}}{4} \) and \( \mu = 1.5 \), we substitute: $\backslash frac\{\backslash sqrt\{x\}\}\{4\}\; \backslash cdot\; 1\; =\; 1.5\; \backslash cdot\; \backslash frac\{\backslash sqrt\{3\}\}\{2\}$
Simplify and multiply both sides by 4: $\backslash sqrt\{x\}\; =\; 3\; \backslash cdot\; \backslash frac\{\backslash sqrt\{3\}\}\{2\}\; =\; \backslash frac\{3\backslash sqrt\{3\}\}\{2\}$
Square both sides: $x\; =\; \backslash left(\backslash frac\{3\backslash sqrt\{3\}\}\{2\}\backslash right)^2\; =\; \backslash frac\{27\}\{4\}\; \backslash times\; 3\; =\; 27$
This verifies \( x=27 \), which is within the given range. Hence, the value of \( x \) is 27.