A light ray is incident on the surface of a sphere of refractive index n at an angle of incidence θ 0 . The ray partially refracts into the sphere with angle of refraction ϕ 0 and then partly refl

Question

A light ray is incident on the surface of a sphere of refractive index $n$ at an angle of incidence $θ_{0}$ . The ray partially refracts into the sphere with angle of refraction $ϕ_{0}$ and then partly reflects from the back surface. The reflected ray then emerges out of the sphere after a partial refraction. The total angle of deviation of the emergent ray with respect to the incident ray is $α$ . Match the quantities mentioned in List-I with their values in List-II and choose the correct option. Screenshot 2026-02-08 at 9.42.56 PM.png

TestHub · Accepted Answer

​ α = ( θ 0 ​ − ϕ 0 ​ ) + ( 180 − 2 ϕ 0 ​ ) + ( θ 0 ​ − 2 ϕ 0 ​ ) α = 180 + 2 θ 0 ​ − 4 ϕ 0 ​ ​ (P) α = 180 + 2 θ 0 ​ − 4 ϕ 0 ​ 180 = 180 + 2 θ 0 ​ − 4 ϕ 0 ​ ⇒ θ 0 ​ = 2 ϕ 0 ​ ...(i) sin θ 0 ​ = 2 sin ϕ 0 ​ ...(ii) From (i) & (ii) ​ sin θ 0 ​ = 2 sin ( θ 0 ​ /2 ) ⇒ cos ( 2 θ 0 ​ ​ ) = 1 2 θ 0 ​ ​ = 0 ⇒ θ 0 ​ = 0 ​ (Q) θ 0 ​ = 2 ϕ 0 ​ ...(i) sin θ 0 ​ = 3 ​ sin ϕ 0 ​ ...(ii) From (i) & (ii) ​ sin θ 0 ​ = 3 ​ sin ( 2 θ 0 ​ ​ ) ⇒ cos ( 2 θ 0 ​ ​ ) = 2 3 ​ ​ 2 θ 0 ​ ​ = 30 , 150 θ 0 ​ = 60 , 300 (Rejected) θ 0 ​ = 60 , 0 ​ (R) θ 0 ​ = 2 ϕ 0 ​ ​ sin θ 0 ​ = 3 ​ sin ϕ 0 ​ sin 2 θ 0 ​ = 3 ​ sin ϕ 0 ​ cos ϕ 0 ​ = 2 3 ​ ​ ϕ 0 ​ = 30 , 150 (Rejected) ​ ϕ 0 ​ = 30 , 0 ...(iii) ​ (S) sin 45 = 2 ​ cos ϕ 0 ​ cos ϕ 0 ​ = 1/2 ϕ 0 ​ = 60 α = 180 + 2 θ 0 ​ − 4 ϕ 0 ​ ​ α = 180 + 90 − 120 ...(iv) = 180 − 30 ; α = 15 0 ∘

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