The critical angle for glass is θ1 and that for water is θ2. The critical . . .

Question : The critical angle for glass is θ₁ and that for water is θ₂. The critical angle for glass-water surface would be (given _aµ_g=1.51, _aµ_w=1.33)

(A) less than θ₂

(B) between θ₁ and θ₂

(C) greater than θ₂

(D) less than θ₁

CBSE Delhi 2024

Solution :

We know, 
µ₂₁=1/sini_c

where µ₂₁ is the refractive index of 2nd (Denser Medium) medium w.r.t. 1st medium (Rarer Medium). 

aµg=1/sinθ₁
sinθ₁=1/_aµ_g
sinθ₁=1/1.51 
sinθ₁=1/(3/2)
sinθ₁=2/3 — (1)

_aµ_w=1/sinθ₂

sinθ₂=1/_aµ_w
sinθ₂=1/1.33 
sinθ₂=1/(4/3)
sinθ₂=3/4— (2)

 

Here the question is asking  critical angle for glass-water surface it means it is asking the refractive index of glass w.r.t. water ( and not water w.r.t. glass because TIR only happens when light travels from denser to rarer medium)

We know,
_wµ_g=1/sinθ
sinθ=1/_wµ_g
sinθ=1/[µ_g/µ_w]
sinθ=1/[(3/2)/(4/3)]
sinθ=1/(9/8)
sinθ=8/9 — (3)

From eq (1), (2) and (3)
8/9=0.88
3/4=0.75
2/3=0.66

8/9>3/4>2/3
sinθ>sinθ2>sinθ1
θ>θ2>θ1

Clearly θ is greater than θ₂ and θ₁

Hence, (C) greater than θ₂, is the correct option.