Two immiscible liquids of refractive index `sqrt 2` and `2sqrt2` are filled with equal height h in a vessel. Then apparent depth of bottom surface of the container given that outside medium is air

Options

A. `(3sqrt2 h)/4`
B. `(3h)/4`
C. `(3h)/2`
D. `(3h)/(4sqrt2)`

Explanation

To solve the problem of finding the apparent depth of the bottom surface of a container filled with two immiscible liquids of different refractive indices, we can follow these steps:

### Step 1: Understand the Setup
We have two immiscible liquids with refractive indices:
- Liquid 1: \( n_1 = \sqrt{2} \)
- Liquid 2: \( n_2 = 2\sqrt{2} \)

Both liquids are filled to the same height \( h \) in the container. The outside medium is air, which has a refractive index of \( n_0 = 1 \).

### Step 2: Apply the Formula for Apparent Depth
The apparent depth \( d \) when looking through a medium can be calculated using the formula:
\[ d = \frac{h}{n} \]
where \( h \) is the actual depth and \( n \) is the refractive index of the medium.

### Step 3: Calculate the Apparent Depth for Each Liquid
1. **For Liquid 1 (Refractive Index \( n_1 = \sqrt{2} \))**:
- The apparent depth due to this liquid is:
\[ d_1 = \frac{h}{n_1} = \frac{h}{\sqrt{2}} \]

2. **For Liquid 2 (Refractive Index \( n_2 = 2\sqrt{2} \))**:
- The apparent depth due to this liquid is:
\[ d_2 = \frac{h}{n_2} = \frac{h}{2\sqrt{2}} \]

### Step 4: Combine the Apparent Depths
Since both liquids are stacked on top of each other, the total apparent depth \( d \) from the top of the first liquid to the bottom of the container is the sum of the apparent depths from both liquids:
\[ d = d_1 + d_2 = \frac{h}{\sqrt{2}} + \frac{h}{2\sqrt{2}} \]

### Step 5: Simplify the Expression
To combine the fractions, we need a common denominator:
\[ d = \frac{h}{\sqrt{2}} + \frac{h}{2\sqrt{2}} = \frac{2h}{2\sqrt{2}} + \frac{h}{2\sqrt{2}} = \frac{2h + h}{2\sqrt{2}} = \frac{3h}{2\sqrt{2}} \]

### Step 6: Final Expression
To express this in a more simplified form, we can multiply and divide by \( \sqrt{2} \):
\[ d = \frac{3h}{2\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{3\sqrt{2}h}{4} \]

### Conclusion
Thus, the apparent depth of the bottom surface of the container is:
\[ d = \frac{3\sqrt{2}}{4} h \]