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QJEEM22Ph9a7b: JEE Main 2022 Physics - Class 12 JEEM22Ph9a7b Solution: In the given figure, the face of the equilateral prism is…

In the given figure, the face ACA C of the equilateral prism is immersed in a liquid of refractive index ' nn '. For incident angle 60∘60^{\circ} at the side ACA C , the refractive light beam just grazes along face ACA C . The refractive index of the liquid n=x4n=\frac{\sqrt{x}}{4} . The value of xx is ____________.

(Given refractive index of glass =1.5=1.5 )

JEE Main 2022 (Online) 26th July Evening Shift Physics - Geometrical Optics Question 86 English

Step-by-Step Verified Solution

Given prism is equilateral so, angel of prism, A =60∘=60^{\circ}

On first surface light is passing straight without any deviation, hence angle of refraction r1=0∘\mathrm{r}_1=0^{\circ} .

amp;r1+i2=Aamp;0+i2=60∘amp;⇒i2=60∘\begin{aligned}& r_1+i_2=\mathrm{A} \\\\& 0+i_2=60^{\circ} \\\\& \Rightarrow i_2=60^{\circ}\end{aligned}

On 2nd 2^{\text {nd }} surface ray, refracting ray is becoming parallel to the junction of surfaces, so i2i_2 will act as critical angle.

From snell's law μ1sin⁡θ1=μ2sin⁡θ2\mu_1 \sin \theta_1=\mu_2 \sin \theta_2

amp;⇒32sin⁡i2=μsin⁡θ2amp;⇒μ=32×sin⁡60∘sin⁡90∘=32×32=334 or 274\begin{aligned}& \Rightarrow \frac{3}{2} \sin i_2=\mu \sin \theta_2 \\\\& \Rightarrow \mu=\frac{3}{2} \times \frac{\sin 60^{\circ}}{\sin 90^{\circ}}=\frac{3}{2} \times \frac{\sqrt{3}}{2}=\frac{3 \sqrt{3}}{4} \text { or } \frac{\sqrt{27}}{4}\end{aligned}

⇒\Rightarrow Comparing above equation with

given n=x4n=\frac{\sqrt{x}}{4} , we get x=27x=27

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