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-
Show 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.
- \(\frac{d\,n_1n_2}{2 (n_1+n_2)}\)
- \(\frac{d\,n_1n_2}{ (n_1+n_2)}\)
- \(\frac{2d(n_1+n_2)}{n_1n_2}\)
- \(\frac{d(n_1+n_2)}{2n_1n_2}\)
The Correct Option is D
Solution and Explanation
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:
\[
\text{Total Apparent Depth} = \frac{d n_2 + d n_1}{2n_1n_2} = \frac{d(n_1 + n_2)}{2n_1n_2}.
\]
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