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 is brought near it, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formula for the time period (T)**:
The time period of a bar magnet oscillating in a magnetic field is given by the formula:
\[ T = 2\pi \sqrt{\frac{I}{M \cdot B}} \]
where:
- \( T \) is the time period,
- \( I \) is the moment of inertia of the magnet,
- \( M \) is the magnetic moment of the magnet,
- \( B \) is the magnetic field strength (which is constant in this case).

2. **Identify the initial conditions**:
Initially, we have a bar magnet with a magnetic moment \( M \) oscillating in the Earth's magnetic field. The time period in this case is given as \( T \).

3. **Introduce the second bar magnet**:
A second bar magnet with a magnetic moment of \( 2M \) is brought near the first magnet. Since magnetic moments are vector quantities, we can add them directly when they are aligned.

4. **Calculate the new net magnetic moment**:
The net magnetic moment when both magnets are considered together is:
\[ M_{\text{net}} = M + 2M = 3M \]

5. **Determine the new time period**:
Now, substituting the new magnetic moment into the time period formula, we get:
\[ T' = 2\pi \sqrt{\frac{I}{M_{\text{net}} \cdot B}} = 2\pi \sqrt{\frac{I}{3M \cdot B}} \]

6. **Relate the new time period to the old time period**:
We can express the new time period \( T' \) in terms of the original time period \( T \):
\[ T' = T \cdot \sqrt{\frac{M}{3M}} = T \cdot \frac{1}{\sqrt{3}} = \frac{T}{\sqrt{3}} \]

### Final Answer:
The new time period \( T' \) of the bar magnet after bringing the second magnet near it is:
\[ T' = \frac{T}{\sqrt{3}} \]