A vessel of depth 'd' is half filled with oil of refractive index n_1 and the other half is filled with water of refractive index n_2 . The apparent depth of this vessel when viewed from above will be-

Options

A. `(dn_1n_2)/(2(n_1+n_2))`
B. `(2d(n_1+n_2))/(n_1n_2)`
C. `(dn_1n_2)/((n_1+n_2))`
D. `(d(n_1+n_2))/(2n_1n_2)`

Answer

To find the apparent depth of a vessel that is half-filled with oil and half-filled with water, we can use the concept of apparent depth in optics. The apparent depth can be calculated using the formula for refraction at the interface of two media.

### Step-by-Step Solution:

1. **Identify the Depths and Refractive Indices**:
- The vessel has a total depth \( D \).
- The oil occupies the top half of the vessel, so its depth is \( \frac{D}{2} \) with refractive index \( n_1 \).
- The water occupies the bottom half of the vessel, so its depth is also \( \frac{D}{2} \) with refractive index \( n_2 \).

2. **Calculate the Apparent Depth for Oil**:
- The apparent depth \( d_1 \) of the oil layer can be calculated using the formula:
\[ d_1 = \frac{h_1}{n_1} \]
where \( h_1 = \frac{D}{2} \) is the actual depth of the oil. Thus,
\[ d_1 = \frac{\frac{D}{2}}{n_1} = \frac{D}{2n_1} \]

3. **Calculate the Apparent Depth for Water**:
- The apparent depth \( d_2 \) of the water layer can be calculated similarly:
\[ d_2 = \frac{h_2}{n_2} \]
where \( h_2 = \frac{D}{2} \) is the actual depth of the water. Thus,
\[ d_2 = \frac{\frac{D}{2}}{n_2} = \frac{D}{2n_2} \]

4. **Combine the Apparent Depths**:
- The total apparent depth \( d_{apparent} \) of the vessel when viewed from above is the sum of the apparent depths of the oil and water:
\[ d_{apparent} = d_1 + d_2 = \frac{D}{2n_1} + \frac{D}{2n_2} \]

5. **Factor Out Common Terms**:
- We can factor out \( \frac{D}{2} \):
\[ d_{apparent} = \frac{D}{2} \left( \frac{1}{n_1} + \frac{1}{n_2} \right) \]

6. **Final Expression**:
- Therefore, the final expression for the apparent depth of the vessel is:
\[ d_{apparent} = \frac{D}{2} \left( \frac{1}{n_1} + \frac{1}{n_2} \right) \]