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 will be-

Options

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

Answer

To solve the problem of finding the resultant magnetic moment of two identical concentric coils with perpendicular planes, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
We have two identical coils that are concentric (sharing the same center) and their planes are perpendicular to each other. Each coil has a magnetic moment denoted by \( M \).

2. **Identifying the Magnetic Moments**:
Let’s denote the magnetic moment of the first coil as \( \vec{M_1} \) and the second coil as \( \vec{M_2} \). Since the coils are perpendicular, the angle \( \theta \) between the two magnetic moments is \( 90^\circ \).

3. **Using the Formula for Resultant Magnetic Moment**:
The resultant magnetic moment \( \vec{M_R} \) of two vectors can be calculated using the formula:
\[ |\vec{M_R}| = \sqrt{|\vec{M_1}|^2 + |\vec{M_2}|^2 + 2 |\vec{M_1}| |\vec{M_2}| \cos(\theta)} \]
Since \( |\vec{M_1}| = M \) and \( |\vec{M_2}| = M \), and \( \theta = 90^\circ \) (where \( \cos(90^\circ) = 0 \)), the formula simplifies to:
\[ |\vec{M_R}| = \sqrt{M^2 + M^2} \]

4. **Calculating the Resultant**:
Substitute the values into the equation:
\[ |\vec{M_R}| = \sqrt{M^2 + M^2} = \sqrt{2M^2} = \sqrt{2} \cdot M \]

5. **Final Result**:
Therefore, the resultant magnetic moment of the two coils is:
\[ |\vec{M_R}| = \sqrt{2} M \]

### Conclusion:
The resultant magnetic moment of the two coils is \( \sqrt{2} M \).