A hemispherical vessel is completely filled with a liquid of refractive index $$\mu$$. 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

For grazing emergence along the surface BE to reach point E:

$$\text{Angle of refraction at surface near B}, \quad r = 90^\circ$$

From the geometry of the hemispherical vessel of radius $$R$$:

$$OC = R, \quad CB = R \implies \angle OBC = 45^\circ$$

$$\text{Angle of incidence at point B}, \quad i = \angle OBC = 45^\circ$$

Applying Snell's law at interface at B:

$$\mu \sin i = 1 \cdot \sin r$$

$$\mu \sin(45^\circ) = 1 \cdot \sin(90^\circ)$$

$$\mu \left(\frac{1}{\sqrt{2}}\right) = 1 \implies \mu = \sqrt{2}$$