Question

A vessel of depth d is filled upto a depth of d₁ by a liquid of refractive index n₁ upto a depth of d₂ by a liquid of refractive index n₂ and upto a depth of d₃ by a liquid of refractive index n₃. Show that the apparent depth of the vessel when viewed normally is given by:

`h_(app) = (d_1/n_1 + d_2/n_2 + d_3/n_3)`

[here d = d₁ + d₂ + d₃ and n₁ > n₂ > n₃]

Answer 1

When light travels from a denser medium to a rarer one, it bends away from the normal. Due to this bending (refraction), submerged objects appear to be at a shallower depth than they actually are. This is called apparent depth.

If a vessel is filled with three transparent liquids of refractive indices n₁ > n₂ > n₃ and respective depths d₁, d₂, and d₃​, each layer contributes an apparent thickness when viewed normally from above.

Using the relation for a single medium:

Apparent depth = `"Real depth"/"Refractive index"`

So for all three layers:

`h_(app) = (d_1/n_1 + d_2/n_2 + d_3/n_3)`