A microscope is focussed 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 ? (Refration index of water is `4//3`)

Options

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

Answer

To solve the problem, we need to determine how deep the water should be poured into the beaker so that the coin is again in focus after the microscope has been raised by 1 cm.

### Step-by-Step Solution:

1. **Understanding the Situation**:
- Initially, the microscope is focused on a coin at the bottom of the beaker.
- When the microscope is raised by 1 cm, the coin will appear blurred because the distance from the microscope to the coin has increased.

2. **Apparent Shift of the Coin**:
- To bring the coin back into focus, we need to account for the apparent shift caused by the refraction of light in water.
- The apparent height (H') of the coin when viewed through water is given by the formula:
\[ H' = H \left(1 - \frac{1}{n}\right) \]
where \( H \) is the actual depth of the coin and \( n \) is the refractive index of water.

3. **Setting Up the Equation**:
- Since the microscope is raised by 1 cm, we want the apparent height of the coin to increase by 1 cm as well:
\[ H' = 1 \text{ cm} \]
- Therefore, we can set up the equation:
\[ 1 = H \left(1 - \frac{1}{n}\right) \]

4. **Substituting the Refractive Index**:
- The refractive index of water is given as \( n = \frac{4}{3} \).
- Substituting this into the equation gives:
\[ 1 = H \left(1 - \frac{1}{\frac{4}{3}}\right) \]
- Simplifying the term inside the parentheses:
\[ 1 - \frac{1}{\frac{4}{3}} = 1 - \frac{3}{4} = \frac{1}{4} \]

5. **Solving for H**:
- Now substituting this back into the equation:
\[ 1 = H \left(\frac{1}{4}\right) \]
- To find \( H \), we rearrange the equation:
\[ H = 1 \times 4 = 4 \text{ cm} \]

6. **Conclusion**:
- Therefore, the depth of water that should be poured into the beaker to bring the coin back into focus is **4 cm**.

### Final Answer:
The depth of water to be poured into the beaker is **4 cm**.

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