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

Solution:
The apparent depth of a medium is given by: \[ \text{Apparent Depth} = \frac{\text{Real Depth}}{\text{Refractive Index}}. \]

Step 1: Calculate the contribution of oil.
For the oil layer of depth \( \frac{d}{2} \) and refractive index \( n_1 \): \[ \text{Apparent Depth of Oil} = \frac{\frac{d}{2}}{n_1} = \frac{d}{2n_1}. \]

Step 2: Calculate the contribution of water.
For the water layer of depth \( \frac{d}{2} \) and refractive index \( n_2 \): \[ \text{Apparent Depth of Water} = \frac{\frac{d}{2}}{n_2} = \frac{d}{2n_2}. \]

Step 3: Total apparent depth.
The total apparent depth is the sum of the apparent depths of the two layers: \[ \text{Total Apparent Depth} = \frac{d}{2n_1} + \frac{d}{2n_2}. \]

Step 4: Simplify the expression.
Taking a common denominator: \[ \text{Total Apparent Depth} = \frac{d n_2 + d n_1}{2n_1n_2} = \frac{d(n_1 + n_2)}{2n_1n_2}. \]