Question:
A vessel of depth \(x\) is half filled with oil of refractive index \(\mu_1\) and the other half is filled with water of refractive index \(\mu_2\). The apparent depth of the vessel when viewed from above is
A vessel of depth \(x\) is half filled with oil of refractive index \(\mu_1\) and the other half is filled with water of refractive index \(\mu_2\). The apparent depth of the vessel when viewed from above is
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For multiple transparent layers viewed normally,
\[
d_{\text{apparent}}
=
\sum \frac{t_i}{\mu_i},
\]
where \(t_i\) is the thickness of the \(i^{\text{th}}\) layer and \(\mu_i\) is its refractive index.
Updated On: Jul 9, 2026
- \(\dfrac{x(\mu_1+\mu_2)}{2\mu_1\mu_2}\)
- \(\dfrac{x\mu_1\mu_2}{2(\mu_1+\mu_2)}\)
- \(\dfrac{2x\mu_1\mu_2}{\mu_1+\mu_2}\)
- \(\dfrac{2x(\mu_1+\mu_2)}{\mu_1\mu_2}\) \bigskip
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The Correct Option is A
Solution and Explanation
Concept:
For normal viewing through a medium of refractive index \(\mu\),
\[
\text{Apparent depth}
=
\frac{\text{Real depth}}{\mu}.
\]
When several transparent layers are stacked, the total apparent depth is the sum of the apparent depths of the individual layers.
Step 1: Find the apparent depth of the oil layer. Depth of oil layer, \[ \frac{x}{2}. \] Its apparent depth is \[ d_1 = \frac{\frac{x}{2}}{\mu_1} = \frac{x}{2\mu_1}. \]
Step 2: Find the apparent depth of the water layer. Depth of water layer, \[ \frac{x}{2}. \] Its apparent depth is \[ d_2 = \frac{\frac{x}{2}}{\mu_2} = \frac{x}{2\mu_2}. \]
Step 3: Calculate the total apparent depth. \[ d = d_1+d_2. \] \[ d = \frac{x}{2\mu_1} + \frac{x}{2\mu_2}. \] \[ d = \frac{x}{2} \left( \frac{1}{\mu_1} + \frac{1}{\mu_2} \right). \] \[ d = \frac{x}{2} \left( \frac{\mu_1+\mu_2}{\mu_1\mu_2} \right). \] \[ d = \frac{x(\mu_1+\mu_2)} {2\mu_1\mu_2}. \]
Step 4: Write the final answer. \[ \boxed{ d= \frac{x(\mu_1+\mu_2)} {2\mu_1\mu_2} } \] \[ \boxed{\text{Answer = (A)}} \]
Step 1: Find the apparent depth of the oil layer. Depth of oil layer, \[ \frac{x}{2}. \] Its apparent depth is \[ d_1 = \frac{\frac{x}{2}}{\mu_1} = \frac{x}{2\mu_1}. \]
Step 2: Find the apparent depth of the water layer. Depth of water layer, \[ \frac{x}{2}. \] Its apparent depth is \[ d_2 = \frac{\frac{x}{2}}{\mu_2} = \frac{x}{2\mu_2}. \]
Step 3: Calculate the total apparent depth. \[ d = d_1+d_2. \] \[ d = \frac{x}{2\mu_1} + \frac{x}{2\mu_2}. \] \[ d = \frac{x}{2} \left( \frac{1}{\mu_1} + \frac{1}{\mu_2} \right). \] \[ d = \frac{x}{2} \left( \frac{\mu_1+\mu_2}{\mu_1\mu_2} \right). \] \[ d = \frac{x(\mu_1+\mu_2)} {2\mu_1\mu_2}. \]
Step 4: Write the final answer. \[ \boxed{ d= \frac{x(\mu_1+\mu_2)} {2\mu_1\mu_2} } \] \[ \boxed{\text{Answer = (A)}} \]
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