An air bubble inside water. The refractive index of water is 4/3 . At what distance from the air bubble should a point object be placed so as to form a real image at the same distance from the bubble:-

Options

A. 2R
B. 3R
C. 4R
D. The air bubble cannot form a real image

Explanation

To solve the problem of determining the distance from an air bubble in water at which a point object should be placed to form a real image at the same distance from the bubble, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have an air bubble in water. The refractive index of water (n_water) is given as \( \frac{4}{3} \).
- The air bubble can be treated as a spherical surface where light will refract.

2. **Identify the Nature of the Image**:
- The problem states that we want to form a real image at the same distance from the bubble. This implies that the object distance (u) and the image distance (v) from the bubble must be equal in magnitude but opposite in sign, i.e., \( |u| = |v| \).

3. **Apply the Lens Maker's Formula**:
- For a spherical interface, we can use the formula for refraction at a spherical surface:
\[ \frac{n_1}{u} + \frac{n_2}{v} = \frac{n_2 - n_1}{R} \]
- Here, \( n_1 \) is the refractive index of air (approximately 1), \( n_2 \) is the refractive index of water \( \left(\frac{4}{3}\right) \), and \( R \) is the radius of curvature of the bubble (which we will assume to be positive since we are considering the refraction from air to water).

4. **Substituting Values**:
- Since we want \( |u| = |v| \), we can denote \( u = -d \) and \( v = d \) (where d is the distance from the bubble).
- Substituting into the formula gives:
\[ \frac{1}{-d} + \frac{\frac{4}{3}}{d} = \frac{\frac{4}{3} - 1}{R} \]

5. **Simplifying the Equation**:
- This simplifies to:
\[ -\frac{1}{d} + \frac{4}{3d} = \frac{\frac{1}{3}}{R} \]
- Combine the left side:
\[ \frac{-3 + 4}{3d} = \frac{1}{3d} = \frac{1}{3R} \]

6. **Finding the Distance**:
- From the equation, we can see that the left side simplifies to \( \frac{1}{3d} = \frac{1}{3R} \).
- Thus, we find that \( d = R \).

7. **Conclusion**:
- Therefore, the distance from the air bubble at which the object should be placed to form a real image at the same distance from the bubble is equal to the radius of curvature of the bubble.

### Final Answer:
The object should be placed at a distance equal to the radius of curvature of the air bubble.