Two bar magnets of the same mass, length and breadth but magnetic moment M and 2M respectively, when placed in same position, time period is 3 sec. What will be the time period when they are placed in different position?

Options

A. `sqrt(3)sec`
B. `3sqrt(3)sec`
C. 3 sec
D. 6 sec

Answer

To solve the problem, we need to find the time period of two bar magnets with different magnetic moments when placed in different positions. Let's break it down step by step.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- We have two bar magnets with magnetic moments \( M \) and \( 2M \).
- When placed in the same position, the time period \( T_1 \) is given as 3 seconds.

2. **Formula for Time Period**:
- The time period \( T \) for two magnets in a magnetic field is given by:
\[ T = 2\pi \sqrt{\frac{I_1 + I_2}{M_{\text{total}} \cdot B}} \]
- Where \( I_1 \) and \( I_2 \) are the moments of inertia of the magnets, \( M_{\text{total}} \) is the total magnetic moment, and \( B \) is the magnetic field strength.

3. **Case 1: Magnets in Same Position**:
- When the magnets are placed in the same position, the total magnetic moment \( M_{\text{total}} \) is:
\[ M_{\text{total}} = M + 2M = 3M \]
- Thus, the time period can be expressed as:
\[ T_1 = 2\pi \sqrt{\frac{I_1 + I_2}{3M \cdot B}} \]
- Given \( T_1 = 3 \) seconds, we can set up the equation:
\[ 3 = 2\pi \sqrt{\frac{I_1 + I_2}{3M \cdot B}} \quad \text{(Equation 1)} \]

4. **Case 2: Magnets in Different Positions**:
- When the magnets are placed in different positions, the total magnetic moment \( M_{\text{total}} \) becomes:
\[ M_{\text{total}} = 2M - M = M \]
- The time period for this case can be expressed as:
\[ T_2 = 2\pi \sqrt{\frac{I_1 + I_2}{M \cdot B}} \quad \text{(Equation 2)} \]

5. **Finding the Relationship Between Time Periods**:
- We can divide Equation 2 by Equation 1 to find the relationship between \( T_2 \) and \( T_1 \):
\[ \frac{T_2}{T_1} = \frac{2\pi \sqrt{\frac{I_1 + I_2}{M \cdot B}}}{2\pi \sqrt{\frac{I_1 + I_2}{3M \cdot B}}} \]
- Simplifying this gives:
\[ \frac{T_2}{3} = \sqrt{\frac{3M}{M}} = \sqrt{3} \]
- Therefore:
\[ T_2 = 3\sqrt{3} \]

6. **Final Answer**:
- The time period when the magnets are placed in different positions is:
\[ T_2 = 3\sqrt{3} \text{ seconds} \]