Two bar magnets are placed at right angle to each other with the north pole of one touching the south pole of the other . If both the magnets have pole strength m and length 2L, then the mgnetic moment of the combination will be .

Options

A. 2 m l
B. `sqrt(2) ml`
C. `m l`
D. `2 sqrt(2) ml`

Explanation

To solve the problem of finding the magnetic moment of the combination of two bar magnets placed at right angles to each other, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Magnetic Moment of a Single Magnet**:
The magnetic moment \( M \) of a single bar magnet is given by the formula:
\[ M = m \cdot L \]
where \( m \) is the pole strength and \( L \) is the length of the magnet.

2. **Calculate the Magnetic Moment for Each Magnet**:
Given that both magnets have a pole strength \( m \) and a length \( 2L \), the magnetic moment for each magnet will be:
\[ M_1 = m \cdot (2L) = 2mL \]
\[ M_2 = m \cdot (2L) = 2mL \]

3. **Determine the Configuration**:
The two magnets are placed at right angles to each other, with the north pole of one touching the south pole of the other. This means that the angle \( \theta \) between the two magnetic moments is \( 90^\circ \).

4. **Use the Formula for Resultant Magnetic Moment**:
The resultant magnetic moment \( M_R \) when two magnetic moments \( M_1 \) and \( M_2 \) are at an angle \( \theta \) is given by:
\[ M_R = \sqrt{M_1^2 + M_2^2 + 2M_1M_2 \cos \theta} \]
Since \( \theta = 90^\circ \), \( \cos 90^\circ = 0 \). Thus, the formula simplifies to:
\[ M_R = \sqrt{M_1^2 + M_2^2} \]

5. **Substitute the Values**:
Substituting \( M_1 \) and \( M_2 \):
\[ M_R = \sqrt{(2mL)^2 + (2mL)^2} \]
\[ M_R = \sqrt{4m^2L^2 + 4m^2L^2} = \sqrt{8m^2L^2} = \sqrt{8} \cdot mL \]

6. **Simplify the Result**:
\[ M_R = 2\sqrt{2} mL \]

7. **Final Answer**:
Therefore, the magnetic moment of the combination of the two bar magnets is:
\[ M_R = 2\sqrt{2} mL \]