Following figures show the arrangement of bar magnets in different configurations. Each magnet has magnetic dipole moment (m). Which configuration has highest value of magnetic dipole moment?

Options

A.
B.
C.
D.

Answer

To determine which configuration of bar magnets has the highest value of magnetic dipole moment, we will analyze the magnetic dipole moment in each configuration using the formula for the net magnetic moment.

### Step-by-Step Solution:

1. **Understanding the Formula**: The net magnetic moment \( M_{net} \) for two magnetic dipoles \( m_1 \) and \( m_2 \) is given by:
\[ M_{net} = \sqrt{m_1^2 + m_2^2 + 2 m_1 m_2 \cos \theta} \]
where \( \theta \) is the angle between the two magnetic moments.

2. **Configuration 1**:
- **Angle**: \( \theta = 90^\circ \)
- **Calculation**:
\[ M_{net} = \sqrt{m^2 + m^2 + 2m^2 \cos 90^\circ} = \sqrt{m^2 + m^2 + 0} = \sqrt{2m^2} = m\sqrt{2} \]

3. **Configuration 2**:
- **Angle**: \( \theta = 180^\circ \)
- **Calculation**:
\[ M_{net} = \sqrt{m^2 + m^2 + 2m^2 \cos 180^\circ} = \sqrt{m^2 + m^2 - 2m^2} = \sqrt{0} = 0 \]

4. **Configuration 3**:
- **Angle**: \( \theta = 30^\circ \)
- **Calculation**:
\[ M_{net} = \sqrt{m^2 + m^2 + 2m^2 \cos 30^\circ} = \sqrt{m^2 + m^2 + 2m^2 \cdot \frac{\sqrt{3}}{2}} = \sqrt{2m^2 + \sqrt{3}m^2} = m\sqrt{2 + \sqrt{3}} \]

5. **Configuration 4**:
- **Angle**: \( \theta = 60^\circ \)
- **Calculation**:
\[ M_{net} = \sqrt{m^2 + m^2 + 2m^2 \cos 60^\circ} = \sqrt{m^2 + m^2 + 2m^2 \cdot \frac{1}{2}} = \sqrt{m^2 + m^2 + m^2} = \sqrt{3m^2} = m\sqrt{3} \]

6. **Comparing Values**:
- Configuration 1: \( m\sqrt{2} \)
- Configuration 2: \( 0 \)
- Configuration 3: \( m\sqrt{2 + \sqrt{3}} \)
- Configuration 4: \( m\sqrt{3} \)

To find the maximum, we need to evaluate \( m\sqrt{2 + \sqrt{3}} \) and \( m\sqrt{3} \):
- Since \( \sqrt{3} \approx 1.732 \) and \( 2 + \sqrt{3} \approx 3.732 \), we find \( \sqrt{2 + \sqrt{3}} \approx 1.93 \), which is less than \( \sqrt{3} \).

7. **Conclusion**: The configuration with the highest magnetic dipole moment is Configuration 4, which has a net magnetic moment of \( m\sqrt{3} \).

### Final Answer:
The configuration with the highest value of magnetic dipole moment is **Configuration 4**.