A magnet of magnetic `'M'` is in the form of a quadrant of a circle. If it is strightened, its new magnetic moment will be

Options

A. `(Mpi)/(sqrt2)`
B. `(M)/(sqrt2)`
C. `(sqrt2M)/(pi)`
D. `(Mpi)/(2sqrt2)`

Explanation

To solve the problem of finding the new magnetic moment of a magnet that was initially in the form of a quadrant of a circle and then straightened, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Initial Configuration**:
- The magnet is in the shape of a quadrant of a circle.
- Let's denote the radius of the circle as \( R \).

2. **Calculate the Length of the Magnet**:
- The length \( L \) of the arc of the quadrant can be calculated as:
\[ L = \frac{1}{4} \times 2\pi R = \frac{\pi R}{2} \]

3. **Determine the Distance Between the Poles**:
- The distance between the north and south poles of the magnet, which are located at the ends of the radius, is given by:
\[ d = R\sqrt{2} \]
- This is derived from the Pythagorean theorem since the two radii form a right triangle.

4. **Calculate the Initial Magnetic Moment**:
- The magnetic moment \( M \) of the magnet can be expressed as:
\[ M = \text{(Pole Strength)} \times d = \text{(Pole Strength)} \times (R\sqrt{2}) \]
- Let the pole strength be denoted as \( m \). Thus, we have:
\[ M = m \cdot R\sqrt{2} \]

5. **Express Radius in Terms of Length**:
- From the earlier expression for \( L \):
\[ R = \frac{2L}{\pi} \]

6. **Substitute \( R \) into the Magnetic Moment**:
- Substitute \( R \) into the expression for \( M \):
\[ M = m \cdot \left(\frac{2L}{\pi}\right) \sqrt{2} = \frac{2mL\sqrt{2}}{\pi} \]

7. **Straighten the Magnet**:
- When the magnet is straightened, its new length becomes \( L \), and the distance between the poles remains \( L \).
- The new magnetic moment \( M' \) can be expressed as:
\[ M' = \text{(Pole Strength)} \times L = m \cdot L \]

8. **Relate the New Magnetic Moment to the Old One**:
- From the earlier expression for \( m \):
\[ m = \frac{M\pi}{2L\sqrt{2}} \]
- Substitute \( m \) back into the expression for \( M' \):
\[ M' = \left(\frac{M\pi}{2L\sqrt{2}}\right) \cdot L = \frac{M\pi}{2\sqrt{2}} \]

9. **Final Expression for the New Magnetic Moment**:
- Therefore, the new magnetic moment \( M' \) is given by:
\[ M' = \frac{M\pi}{2\sqrt{2}} \]

### Conclusion:
The new magnetic moment when the magnet is straightened is:
\[ M' = \frac{M\pi}{2\sqrt{2}} \]