Time period of a bar magnet oscillating in earth's magnetic field is T. Its magnetic moment is M. Now, a bar magnet of magnetic moment 2M is brought near it. The new time period will be

Options

A. T
B. `(T)/(sqrt(3))`
C. `T sqrt(3)`
D. `3T`

Answer

To find the new time period of a bar magnet oscillating in the Earth's magnetic field when another bar magnet with double the magnetic moment is brought near it, we can follow these steps:

### Step 1: Understand the formula for the time period of a bar magnet
The time period \( T \) of a bar magnet oscillating in a magnetic field is given by the formula:
\[ T = 2\pi \sqrt{\frac{I}{mB}} \]
where:
- \( I \) is the moment of inertia of the magnet,
- \( m \) is the magnetic moment of the magnet,
- \( B \) is the magnetic field strength.

### Step 2: Identify the initial conditions
Let:
- The initial magnetic moment \( m_1 = M \),
- The initial time period \( T_1 = T \).

### Step 3: Determine the new magnetic moment when another magnet is brought near
When a second bar magnet with magnetic moment \( m_2 = 2M \) is brought near the first magnet, the effective magnetic moment \( M_f \) becomes:
\[ M_f = M + 2M = 3M \]

### Step 4: Write the new time period in terms of the new magnetic moment
The new time period \( T_2 \) can be expressed as:
\[ T_2 = 2\pi \sqrt{\frac{I}{M_f B}} = 2\pi \sqrt{\frac{I}{3MB}} \]

### Step 5: Relate the new time period to the initial time period
Since we know that the initial time period \( T_1 \) is:
\[ T_1 = 2\pi \sqrt{\frac{I}{MB}} \]
We can compare the two time periods:
\[ T_2 = 2\pi \sqrt{\frac{I}{3MB}} = \frac{T_1}{\sqrt{3}} \]
Thus, we can express \( T_2 \) in terms of \( T \):
\[ T_2 = \frac{T}{\sqrt{3}} \]

### Conclusion
The new time period when a bar magnet of magnetic moment \( 2M \) is brought near the first magnet is:
\[ T_2 = \frac{T}{\sqrt{3}} \]

### Final Answer
The correct answer is \( \frac{T}{\sqrt{3}} \).

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