An iron rod of length L and magnetic moment M is bent in the form of a semicircle. Now its magnetic moment will be

Options

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

Explanation

To find the new magnetic moment of an iron rod of length \( L \) that has been bent into the shape of a semicircle, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Initial Magnetic Moment**:
The magnetic moment \( M \) of the straight rod can be expressed as:
\[ M = m \cdot L \]
where \( m \) is the pole strength and \( L \) is the length of the rod.

2. **Determine the New Shape**:
When the rod is bent into a semicircle, its length remains the same, but its shape changes. The length of the rod \( L \) will now correspond to the arc length of the semicircle.

3. **Relate Length to Radius**:
The length of the semicircle can be expressed in terms of its radius \( R \):
\[ L = \pi R \]
This is because the circumference of a full circle is \( 2\pi R \), and for a semicircle, it is half of that.

4. **Solve for Radius**:
From the equation \( L = \pi R \), we can solve for \( R \):
\[ R = \frac{L}{\pi} \]

5. **Calculate the New Magnetic Moment**:
The magnetic moment of the semicircular rod can be expressed as:
\[ M' = m \cdot (2R) \]
Here, \( 2R \) is the diameter of the semicircle.

6. **Substitute for Radius**:
Substitute \( R \) into the equation for \( M' \):
\[ M' = m \cdot (2 \cdot \frac{L}{\pi}) = \frac{2mL}{\pi} \]

7. **Relate New Magnetic Moment to Original**:
Since \( mL = M \), we can substitute this into our equation for \( M' \):
\[ M' = \frac{2M}{\pi} \]

### Final Result:
The new magnetic moment \( M' \) of the iron rod bent into the shape of a semicircle is:
\[ M' = \frac{2M}{\pi} \]