The deflection magnetometer is most sensitive when the defiection ` theta` is

Options

A. Nearly `0^(@)`
B. Nearly `30^(@)`
C. Nearly `45^(@)`
D. Nearly `90^(@)`

Answer

To determine the condition under which a deflection magnetometer is most sensitive, we need to analyze the relationship between the deflection angle (θ) and the torque acting on the magnetic needle.

### Step-by-Step Solution:

1. **Understanding the Deflection Magnetometer**:
- A deflection magnetometer is an instrument used to measure the magnetic moment of magnets and to compare the pole strengths of two bar magnets. It consists of a magnetic needle that can rotate freely in the horizontal plane.

2. **Torque on the Magnetic Needle**:
- When a magnetic needle is placed in a magnetic field, it experiences a torque (τ) given by the formula:
\[ \tau = m \cdot B \cdot \sin(\theta) \]
where:
- \( m \) is the magnetic moment of the needle,
- \( B \) is the magnetic field strength,
- \( \theta \) is the angle of deflection from the magnetic meridian.

3. **Sensitivity of the Magnetometer**:
- The sensitivity of the deflection magnetometer is related to how much the needle deflects for a given torque. The greater the change in deflection for a small change in torque, the more sensitive the instrument is.

4. **Finding the Maximum Sensitivity**:
- To find the angle θ at which the sensitivity is maximum, we need to consider the derivative of the torque with respect to the angle:
\[ \frac{d\tau}{d\theta} = m \cdot B \cdot \cos(\theta) \]
- The sensitivity is maximum when this derivative is maximum. This occurs when \( \cos(\theta) \) is maximum.

5. **Maximizing the Cosine Function**:
- The cosine function reaches its maximum value of 1 when \( \theta = 0^\circ \). However, for practical measurements, we look for a balance where the deflection is not too small (which would make it hard to read) and not too large (which would lead to inaccuracies).
- The sensitivity is often considered to be optimal around \( \theta = 30^\circ \) to \( 45^\circ \).

6. **Conclusion**:
- Therefore, the deflection magnetometer is most sensitive when the deflection angle \( \theta \) is around \( 30^\circ \) to \( 45^\circ \).