A vessel of height 2d is half-filled with a liquid of refractive index `sqrt(2)` and the other half with a liquid of refractive index n. (The given liquids are immiscible). Then the apparent depth of the inner surface of the bottom of the vessel (neglecting the thickness of the bottom of the vessel) will be

Options

A. `(n)/(d(n+sqrt2))`
B. `(d(n+sqrt2))/(nsqrt2)`
C. `(nsqrt2)/(d(n+sqrt2))`
D. `(nd)/(d+sqrt(2n))`

Explanation

To solve the problem, we need to find the apparent depth of the inner surface of the bottom of a vessel that is half-filled with two immiscible liquids of different refractive indices.

### Step-by-Step Solution:

1. **Understand the Setup**:
The vessel has a total height of \(2d\). It is half-filled with a liquid of refractive index \(\sqrt{2}\) and the other half with a liquid of refractive index \(n\). This means each liquid occupies a height of \(d\).

2. **Identify the Layers**:
- The top layer (height \(d\)) has a refractive index of \(\sqrt{2}\).
- The bottom layer (height \(d\)) has a refractive index of \(n\).

3. **Use the Formula for Apparent Depth**:
The apparent depth \(h_a\) of an object submerged in a medium is given by the formula:
\[ h_a = \frac{h}{\mu} \]
where \(h\) is the actual depth and \(\mu\) is the refractive index of the medium.

4. **Calculate the Apparent Depth for Each Layer**:
- For the liquid with refractive index \(\sqrt{2}\):
\[ h_{a1} = \frac{d}{\sqrt{2}} \]
- For the liquid with refractive index \(n\):
\[ h_{a2} = \frac{d}{n} \]

5. **Combine the Apparent Depths**:
The total apparent depth \(H_a\) of the bottom of the vessel is the sum of the apparent depths from both layers:
\[ H_a = h_{a1} + h_{a2} = \frac{d}{\sqrt{2}} + \frac{d}{n} \]

6. **Simplify the Expression**:
To combine the two fractions, find a common denominator:
\[ H_a = d \left(\frac{1}{\sqrt{2}} + \frac{1}{n}\right) = d \left(\frac{n + \sqrt{2}}{n\sqrt{2}}\right) \]
Thus, the final expression for the apparent depth is:
\[ H_a = \frac{d(n + \sqrt{2})}{n\sqrt{2}} \]

### Final Answer:
The apparent depth of the inner surface of the bottom of the vessel is:
\[ H_a = \frac{d(n + \sqrt{2})}{n\sqrt{2}} \]