If the planes of two identical concentric coils are perpendicular and the magnetic moment of each coil is M, then the resultant magnetic moment of the two coils wil be

Options

A. M
B. 2M
C. 3M
D. `sqrt(2)M`

Answer

To find the resultant magnetic moment of two identical concentric coils whose planes are perpendicular to each other, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Magnetic Moments**:
Let the magnetic moment of each coil be denoted as \( m \). Therefore, for the first coil, we have:
\[ m_1 = m \]
and for the second coil:
\[ m_2 = m \]

2. **Determine the Angle Between the Coils**:
Since the planes of the two coils are perpendicular to each other, the angle \( \theta \) between the magnetic moments of the two coils is:
\[ \theta = 90^\circ \]

3. **Use the Formula for Resultant Magnetic Moment**:
The resultant magnetic moment \( m_{\text{net}} \) for two vectors (in this case, the magnetic moments of the coils) that are at an angle \( \theta \) is given by:
\[ m_{\text{net}} = \sqrt{m_1^2 + m_2^2 + 2 m_1 m_2 \cos(\theta)} \]
Since \( \theta = 90^\circ \), we have \( \cos(90^\circ) = 0 \). Thus, the formula simplifies to:
\[ m_{\text{net}} = \sqrt{m_1^2 + m_2^2} \]

4. **Substituting the Values**:
Substituting \( m_1 = m \) and \( m_2 = m \) into the equation, we get:
\[ m_{\text{net}} = \sqrt{m^2 + m^2} = \sqrt{2m^2} \]

5. **Simplifying the Result**:
Taking the square root, we find:
\[ m_{\text{net}} = \sqrt{2} m \]

### Final Result:
Thus, the resultant magnetic moment of the two coils is:
\[ m_{\text{net}} = \sqrt{2} m \]

### Conclusion:
The correct answer is \( \sqrt{2} m \).

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