A vessel of depth 2d cm is half filled with a liquid of refractive index `mu_(1)` and the upper half with a liquid of refractive index `mu_(2)`. The apparent depth of the vessel seen perpendicularly is

Options

A. `d[(mu_(1)mu_(2))/(mu_(1)+mu_(2))]`
B. `d[(1)/(mu_(1))+(1)/(mu_(2))]`
C. `2d[(1)/(mu_(1))+(1)/(mu_(2))]`
D. `2d[(1)/(mu_(1)mu_(2))]`

Explanation

To find the apparent depth of a vessel that is half-filled with two different liquids, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Depths and Refractive Indices**:
- The total depth of the vessel is \(2d\) cm.
- The vessel is half-filled, so the depth of each liquid is \(d\) cm.
- Let the refractive index of the lower liquid be \(\mu_1\) and the upper liquid be \(\mu_2\).

2. **Apparent Depth Formula**:
- The apparent depth of a medium is given by the formula:
\[ \text{Apparent Depth} = \frac{\text{Real Depth}}{\text{Refractive Index}} \]
- For the lower liquid (depth \(d\)):
\[ h_1 = \frac{d}{\mu_1} \]
- For the upper liquid (depth \(d\)):
\[ h_2 = \frac{d}{\mu_2} \]

3. **Total Apparent Depth**:
- The total apparent depth \(h\) of the vessel is the sum of the apparent depths of both liquids:
\[ h = h_1 + h_2 = \frac{d}{\mu_1} + \frac{d}{\mu_2} \]

4. **Factor Out Common Terms**:
- We can factor out \(d\) from the equation:
\[ h = d \left( \frac{1}{\mu_1} + \frac{1}{\mu_2} \right) \]

5. **Final Expression for Apparent Depth**:
- Thus, the final expression for the apparent depth of the vessel is:
\[ h = d \left( \frac{1}{\mu_1} + \frac{1}{\mu_2} \right) \]

### Conclusion:
The apparent depth of the vessel seen perpendicularly is given by:
\[ h = d \left( \frac{1}{\mu_1} + \frac{1}{\mu_2} \right) \]