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 step by step, 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 1: Understand the system
A magnetic needle suspended by a silk thread is vibrating in the Earth's magnetic field. The time period of this vibration is influenced by the magnetic properties of the needle.

### Step 2: Identify the effect of temperature on magnetic properties
When the temperature of the magnetic needle is increased to 1000°C, the magnetic properties of the needle will decrease. This is because high temperatures can cause thermal agitation, which disrupts the alignment of magnetic domains within the material, leading to a reduction in its magnetic moment (denoted as M).

### Step 3: Recall the formula for the time period
The time period (T) of a magnetic needle in a magnetic field is given by the formula:
\[ T = 2\pi \sqrt{\frac{I}{M \cdot B}} \]
where:
- \( T \) = time period
- \( I \) = moment of inertia of the needle (which remains constant)
- \( M \) = magnetic moment of the needle (which decreases with temperature)
- \( B \) = magnetic field strength (which remains constant)

### Step 4: Analyze the relationship between time period and magnetic moment
From the formula, we can see that the time period \( T \) is inversely proportional to the square root of the magnetic moment \( M \):
\[ T \propto \frac{1}{\sqrt{M}} \]
This means that if the magnetic moment \( M \) decreases, the time period \( T \) will increase.

### Step 5: Conclude the effect on the time period
Since the magnetic moment \( M \) decreases when the temperature of the needle is increased to 1000°C, the time period \( T \) will increase.

### Final Answer
Therefore, the time period of the magnetic needle will **increase** when the temperature is raised to 1000°C.

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