A microscope is focused on a coin lying at the bottom of a beaker. The microscope is now raised up by `1 cm`. To what depth should the water be poured into the beaker so that coin is again in focus ? (Refractive index of water is `4//3`)

Options

A. `1 cm`
B. `4//3 cm`
C. `3 cm`
D. `4 cm`

Explanation

To solve the problem, we need to determine how deep we should pour water into the beaker so that the coin, which is initially in focus, remains in focus after the microscope is raised by 1 cm.

### Step-by-Step Solution:

1. **Understanding the Situation**:
- Initially, the microscope is focused on a coin at a certain depth (let's call it \( d \)).
- The microscope is raised by 1 cm, meaning the new height of the microscope is \( d + 1 \) cm above the bottom of the beaker.

2. **Refractive Index**:
- The refractive index of water (\( \mu \)) is given as \( \frac{4}{3} \).

3. **Apparent Depth Formula**:
- The apparent depth (\( d_{des} \)) when viewed through a medium (water in this case) can be calculated using the formula:
\[ d_{des} = \frac{d}{\mu} \]
- Here, \( d \) is the actual depth of the coin, and \( \mu \) is the refractive index of water.

4. **Setting Up the Equation**:
- We want the apparent depth to be such that when the microscope is raised by 1 cm, the coin is still in focus. This means:
\[ d_{des} = d - 1 \]
- Substituting the formula for apparent depth:
\[ \frac{d}{\mu} = d - 1 \]

5. **Substituting the Refractive Index**:
- Plugging in the value of \( \mu \):
\[ \frac{d}{\frac{4}{3}} = d - 1 \]

6. **Clearing the Fraction**:
- Multiply both sides by \( \frac{4}{3} \):
\[ d = \frac{4}{3}(d - 1) \]

7. **Distributing the Right Side**:
- Expanding the right side:
\[ d = \frac{4}{3}d - \frac{4}{3} \]

8. **Rearranging the Equation**:
- Bringing all terms involving \( d \) to one side:
\[ d - \frac{4}{3}d = -\frac{4}{3} \]
- This simplifies to:
\[ -\frac{1}{3}d = -\frac{4}{3} \]

9. **Solving for \( d \)**:
- Multiply both sides by -3:
\[ d = 4 \]

10. **Conclusion**:
- Therefore, the depth of water that should be poured into the beaker is **4 cm**.