A hemispherical vessel is completely filled with a liquid of refractive index μ. A small coin is kept at the lowest point (O) of the vessel as shown in figure. The minimum value of the refractive index of the liquid so that a person can see the coin from point E (at the level of the vessel) is ____.
(A)3
(B)23
(C)2✓ Correct
(D)23
Answer: (C)
Solution
The coin at the lowest point O sends light in all directions through the liquid; for a ray to reach the observer at point E — level with the flat top surface, i.e. at the rim — it must travel along the straight line from O to the rim and refract there into air.
By the geometry of the hemisphere of radius R, the line from the lowest point O to a point on the rim has equal horizontal and vertical projections (each equal to R), so this ray strikes the flat liquid surface at 45° to the normal.
For the observer to just be able to see the coin (the limiting, minimum-μ case, where the ray emerges grazing the surface), this 45° angle of incidence must equal the critical angle:
sinC=μ1,C=45∘
μ=sin45∘1=2
JEE Main · 2025 · 28 Jan 2025 · Shift 1 · Single correct option
