Question

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-

• A. \(\frac{d\,n_1n_2}{2 (n_1+n_2)}\)
• B. \(\frac{d\,n_1n_2}{ (n_1+n_2)}\)
• C. \(\frac{2d(n_1+n_2)}{n_1n_2}\)
• D. \(\frac{d(n_1+n_2)}{2n_1n_2}\)

Hint:

When calculating the apparent depth of a layered medium, consider the contribution of each layer separately and then add them together. Use the formula \( \text{Apparent Depth} = \frac{\text{Real Depth}}{\text{Refractive Index}} \) for each layer.

Answer 1

The Correct Option is D

To find the apparent depth of a vessel when viewed from above, we need to consider the influence of the refractive indices of the two different media (oil and water) involved. Each of these media affects the apparent depth differently. 

Given:

• The vessel is half filled with oil and the other half with water.
• Refractive index of oil, \(n_1\).
• Refractive index of water, \(n_2\).
• Total depth of the vessel, \(d\).
• Depth of oil = Depth of water = \(\frac{d}{2}\).

 

The apparent depth can be calculated using the formula for apparent depth due to each medium and then adding them up:

• Apparent depth due to oil: \(\text{Apparent depth by Oil} = \frac{\text{Actual depth of Oil}}{n_1} = \frac{d/2}{n_1}\)
• Apparent depth due to water: \(\text{Apparent depth by Water} = \frac{\text{Actual depth of Water}}{n_2} = \frac{d/2}{n_2}\)

Thus, the total apparent depth \((D_a)\) of the vessel when viewed from above is given by:

• \(D_a = \frac{d/2}{n_1} + \frac{d/2}{n_2}\)

On simplifying, we get:

• \(D_a = \frac{d}{2n_1} + \frac{d}{2n_2} = \frac{d(n_2 + n_1)}{2n_1n_2}\)

Therefore, the apparent depth of the vessel when viewed from above is:

• \(\frac{d(n_1+n_2)}{2n_1n_2}\)

This matches the given correct answer. Hence, the correct option is:

• \(\frac{d(n_1+n_2)}{2n_1n_2}\)