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

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For the rays from coin to reach the point $E$ , the refracted rays must grazing the surface, i.e. they must be incident at critical angle $θ_{c}$ inside the liquid. $μ = \frac{1}{s i n θ _{c}}$ $μ$ is minimum when $θ_{c}$ is maximum. Maximum value of $θ_{c} = 4 5^{\circ}$ $\Rightarrow μ$ has a minimum value of $2$ .

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