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.

Explanation

To determine which configuration of bar magnets has the highest value of magnetic dipole moment, we will analyze each configuration step by step.

### Step 1: Analyze Configuration A
In Configuration A, we have two magnets positioned at a right angle (90 degrees) to each other.

- Each magnet has a magnetic dipole moment \( m \).
- The net magnetic moment \( M_{\text{net}} \) can be calculated using the formula for the resultant of two vectors at an angle:

\[ M_{\text{net}} = \sqrt{m^2 + m^2 + 2 \cdot m \cdot m \cdot \cos(90^\circ)} \]

Since \( \cos(90^\circ) = 0 \):

\[ M_{\text{net}} = \sqrt{m^2 + m^2} = \sqrt{2m^2} = m\sqrt{2} \approx 1.414m \]

### Step 2: Analyze Configuration B
In Configuration B, the two magnets are aligned in opposite directions.

- The magnetic moments are \( m \) and \( -m \).
- The net magnetic moment \( M_{\text{net}} \) is:

\[ M_{\text{net}} = m - m = 0 \]

### Step 3: Analyze Configuration C
In Configuration C, the two magnets are positioned at an angle of 30 degrees to each other.

- The net magnetic moment \( M_{\text{net}} \) can be calculated as follows:

\[ M_{\text{net}} = \sqrt{m^2 + m^2 + 2 \cdot m \cdot m \cdot \cos(30^\circ)} \]

Since \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \):

\[ M_{\text{net}} = \sqrt{m^2 + m^2 + 2m^2 \cdot \frac{\sqrt{3}}{2}} = \sqrt{2m^2 + \sqrt{3}m^2} = m\sqrt{2 + \sqrt{3}} \approx 1.93m \]

### Step 4: Analyze Configuration D
In Configuration D, the two magnets are aligned in the same direction.

- The net magnetic moment \( M_{\text{net}} \) is:

\[ M_{\text{net}} = m + m = 2m \]

### Step 5: Compare the Results
Now we will compare the net magnetic moments calculated for each configuration:

- Configuration A: \( M_{\text{net}} \approx 1.414m \)
- Configuration B: \( M_{\text{net}} = 0 \)
- Configuration C: \( M_{\text{net}} \approx 1.93m \)
- Configuration D: \( M_{\text{net}} = 2m \)

### Conclusion
The configuration with the highest value of magnetic dipole moment is Configuration D, with a net magnetic moment of \( 2m \).