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Physics Question on Refraction of Light

For the given incident ray as shown in figure, the condition of total internal refraction of this ray the required refractive index of prism will be :-

A

3+12\frac{\sqrt 3+1}{ 2}

B

2+12\frac{\sqrt 2+1}{ 2}

C

32\sqrt{\frac{3}{2}}

D

76\sqrt{\frac{7}{6}}

Answer

32\sqrt{\frac{3}{2}}

Explanation

Solution

For given condition snell's law give
1 sin⁡ 45∘=μ ⋅sin⁡(90−θc)1 \,\sin \,45^{\circ}=\mu \,\cdot \sin \left(90-\theta_{ c }\right)
12=μ cos⁡θc=μ2−1\frac{1}{\sqrt{2}}=\mu \,\cos \theta_{ c }=\sqrt{\mu^{2}-1}
⇒μ2=1+12\Rightarrow \mu^{2}=1+\frac{1}{2}
⇒μ2=32\Rightarrow \mu^{2}=\frac{3}{2}
⇒μ=32\Rightarrow \mu=\sqrt{\frac{3}{2}}
sin⁡θc=1μ\sin \theta_{c}=\frac{1}{\mu}
cos⁡θc=μ2−1μ\cos \theta_{ c }=\sqrt{\frac{\mu^{2}-1}{\mu}}
Alternate solution (objective method)
for given condition μ=1+sin⁡2θ\mu=\sqrt{1+\sin ^{2} \theta}
⇒μ=1+sin⁡245∘\Rightarrow \mu=\sqrt{1+\sin ^{2} 45^{\circ}}
=μ=1+12=\mu=\sqrt{1+\frac{1}{2}}
=32=\sqrt{\frac{3}{2}}