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?

Answer

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: Understanding Magnetic Dipole Moment
The magnetic dipole moment (m) of a bar magnet is a vector quantity that represents the strength and orientation of a magnet's magnetic field. When multiple magnets are arranged in different configurations, the resultant magnetic dipole moment can be calculated based on their orientations.

### Step 2: Analyze Configuration A
In the first configuration (let's denote it as A), we have two magnets arranged perpendicular to each other. The resultant magnetic dipole moment (M1) can be calculated using the Pythagorean theorem:
\[ M1 = \sqrt{m^2 + m^2} = \sqrt{2m^2} = m\sqrt{2} \]

### Step 3: Analyze Configuration B
In the second configuration (B), the two magnets are aligned in opposite directions. Therefore, the resultant magnetic dipole moment (M2) is:
\[ M2 = m - m = 0 \]
This means there is no net magnetic dipole moment in this configuration.

### Step 4: Analyze Configuration C
In the third configuration (C), the two magnets are arranged at an angle of 30 degrees. The resultant magnetic dipole moment (M3) can be calculated using the formula:
\[ M3 = \sqrt{m^2 + m^2 + 2m \cdot m \cdot \cos(30^\circ)} \]
Substituting \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\):
\[ M3 = \sqrt{m^2 + m^2 + 2m^2 \cdot \frac{\sqrt{3}}{2}} \]
\[ M3 = \sqrt{2m^2 + m^2\sqrt{3}} = m\sqrt{2 + \sqrt{3}} \]
Calculating this gives approximately \( M3 \approx 1.93m \).

### Step 5: Analyze Configuration D
In the fourth configuration (D), the two magnets are arranged at an angle of 60 degrees. The resultant magnetic dipole moment (M4) is calculated as:
\[ M4 = \sqrt{m^2 + m^2 + 2m \cdot m \cdot \cos(60^\circ)} \]
Substituting \(\cos(60^\circ) = \frac{1}{2}\):
\[ M4 = \sqrt{m^2 + m^2 + 2m^2 \cdot \frac{1}{2}} \]
\[ M4 = \sqrt{2m^2 + m^2} = m\sqrt{3} \]
Calculating this gives approximately \( M4 \approx 1.732m \).

### Step 6: Compare the Results
Now we compare the values of the magnetic dipole moments from each configuration:
- \( M1 = m\sqrt{2} \approx 1.414m \)
- \( M2 = 0 \)
- \( M3 \approx 1.93m \)
- \( M4 \approx 1.732m \)

### Conclusion
The configuration with the highest value of magnetic dipole moment is configuration C, with a value of approximately \( 1.93m \).