Which one of the following modifications may increase the senstivity of moving coil galvanometer?
`1^(st)` Way : By using spring of smaller torsion constant.
`2^(nd)` Way : By using a smaller coil
`3^(rd)` Way : By using a stronger magnet
`4^(th)` Way: By using a coil having fewer number of turns.

Options

A. `1^(st)` and `4^(th)` ways only
B. `1^(st)` and `3^(rd)` ways only
C. `1^(st)` and `3^(rd)` ways
D. `2^(nd)` and `4^(th)` only

Explanation

To determine which modifications may increase the sensitivity of a moving coil galvanometer, we need to analyze the factors that affect its sensitivity. The sensitivity (S) of a moving coil galvanometer is given by the formula:

\[ S = \frac{B \cdot N \cdot A}{K} \]

Where:
- \( S \) = Sensitivity
- \( B \) = Magnetic field strength
- \( N \) = Number of turns of the coil
- \( A \) = Area of the coil
- \( K \) = Torsion constant of the spring

Now, let's evaluate each option provided:

1. **Using a spring of smaller torsion constant (K)**:
- Sensitivity is inversely proportional to the torsion constant \( K \). Therefore, if we decrease \( K \) (use a spring with a smaller torsion constant), the sensitivity \( S \) will increase.
- **Conclusion**: This modification **increases sensitivity**.

2. **Using a smaller coil**:
- The area \( A \) of the coil will decrease if we use a smaller coil. Since sensitivity is directly proportional to the area, a smaller area will lead to a decrease in sensitivity.
- **Conclusion**: This modification **decreases sensitivity**.

3. **Using a stronger magnet**:
- The sensitivity is directly proportional to the magnetic field strength \( B \). By using a stronger magnet, we increase \( B \), which in turn increases the sensitivity.
- **Conclusion**: This modification **increases sensitivity**.

4. **Using a coil having fewer number of turns (N)**:
- Sensitivity is directly proportional to the number of turns \( N \). If we reduce the number of turns, the sensitivity will decrease.
- **Conclusion**: This modification **decreases sensitivity**.

### Summary of Findings:
- **Option 1**: Using a spring of smaller torsion constant - **Increases sensitivity**.
- **Option 2**: Using a smaller coil - **Decreases sensitivity**.
- **Option 3**: Using a stronger magnet - **Increases sensitivity**.
- **Option 4**: Using a coil having fewer number of turns - **Decreases sensitivity**.

### Final Answer:
The modifications that may increase the sensitivity of the moving coil galvanometer are:
- **1st Way**: By using a spring of smaller torsion constant.
- **3rd Way**: By using a stronger magnet.

Thus, the correct options are **1 and 3 only**.

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