JEE Main 2023 · 13 April · morning · Single correct · +4 / −1

A vessel of depth 'd' is half filled with oil of refractive index n₁ and the other half is filled with water of refractive index n₂. The apparent depth of this vessel when viewed from above will be-

  1. (a) (2 d(n₁+n₂))/(n₁ n₂)
  2. (b) d(n₁+n₂)/(2 n₁ n₂)
  3. (c) (d n₁ n₂)/2(n₁+n₂)
  4. (d) (d n₁ n₂)/(n₁+n₂)
Answer and solution

Answer (b) d(n₁+n₂)/(2 n₁ n₂)

Solution

Physics sir · 4 steps

Each layer shrinks on its own: divide each half-depth by its own refractive index, then add.

  1. One layer. Seen from above, near normal viewing, a layer of real thickness t and refractive index n looks only t/n thick.
  2. Two layers. Each liquid is d/2 thick, so the oil looks d/2n₁ thick and the water looks d/2n₂ thick. The total apparent depth is the sum.

    hₐₚₚ = d/2n₁ + d/2n₂ = d(n₁ + n₂)/(2n₁ n₂)

  3. Quick check. Put n₁ = n₂ = 1, which is the same as an empty vessel. The apparent depth must be d. Option (b) gives 2d/2 = d. Options (a), (c) and (d) give 4d, d/4 and d/2.
  4. So. The apparent depth is d(n₁+n₂)/(2n₁ n₂), option (b).

The idea to keep. For stacked layers, apparent depth adds up as Σ tᵢ/nᵢ. Before you choose an option, check it with n = 1: the formula must give back the real depth.

Answer from the NTA final answer key. Solution by PrepWiser.

Study Ray Optics

Asked about 200 times in JEE Main since 2020. This question is from 2023.