A magnetic needle suspended by a silk thread is vibrating in the earth's magnetic field. If temperature of the needle is increased to `1000^(@)C`, then

Options

A. the time period decreases
B. the time period remains unchanged
C. the time period increases
D. the needle stops vibrating

Answer

To solve the problem, we need to analyze the effect of increasing the temperature of a magnetic needle on its time period of vibration in the Earth's magnetic field.

### Step-by-Step Solution:

1. **Understanding the Magnetic Needle**:
- A magnetic needle is a small magnet that can rotate freely and is suspended by a silk thread. It aligns itself with the Earth's magnetic field and can vibrate about its equilibrium position.

2. **Effect of Temperature on Magnetic Properties**:
- When the temperature of the magnetic needle is increased, the thermal energy causes the magnetic domains within the needle to become less aligned. This results in a decrease in the magnetic properties of the needle.

3. **Magnetic Moment**:
- The magnetic moment (m) of the needle is a measure of its strength and orientation in a magnetic field. As the temperature increases, the magnetic moment decreases due to the loss of alignment of the magnetic domains.

4. **Time Period of Vibration**:
- The time period (T) of the vibrating magnetic needle is given by the formula:
\[ T = 2\pi \sqrt{\frac{I}{m \cdot k}} \]
where:
- \(I\) is the moment of inertia of the needle (which remains constant),
- \(m\) is the magnetic moment (which decreases with increasing temperature),
- \(k\) is the restoring torque constant (which remains constant as well).

5. **Analyzing the Relationship**:
- Since the time period \(T\) is inversely proportional to the square root of the magnetic moment \(m\):
\[ T \propto \frac{1}{\sqrt{m}} \]
- If the magnetic moment \(m\) decreases, then \(\sqrt{m}\) also decreases, leading to an increase in the time period \(T\).

6. **Conclusion**:
- Therefore, when the temperature of the magnetic needle is increased to \(1000^\circ C\), the time period of vibration increases.

### Final Answer:
The time period of the magnetic needle increases when the temperature is raised to \(1000^\circ C\).

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