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`

Answer

To find the new magnetic moment of an iron rod bent into the shape of a semicircle, we can follow these steps:

### Step 1: Understand the initial conditions
- We have an iron rod of length \( L \) and magnetic moment \( M \).
- The rod is bent into the shape of a semicircle.

### Step 2: Determine the radius of the semicircle
- The length of the semicircle is equal to the length of the rod, which is \( L \).
- The length of a semicircle is given by the formula:
\[ L = \pi r \]
where \( r \) is the radius of the semicircle.
- Rearranging this gives:
\[ r = \frac{L}{\pi} \]

### Step 3: Calculate the new magnetic moment
- The magnetic moment \( m' \) of a magnetic dipole is given by the formula:
\[ m' = n \cdot d \]
where \( n \) is the pole strength and \( d \) is the distance between the poles.
- Initially, the distance between the poles was \( L \).
- After bending the rod into a semicircle, the distance between the two poles becomes the diameter of the semicircle, which is:
\[ d = 2r = 2 \cdot \frac{L}{\pi} = \frac{2L}{\pi} \]

### Step 4: Relate the new magnetic moment to the original
- Since the pole strength \( n \) remains constant, we can express the new magnetic moment as:
\[ m' = n \cdot d = n \cdot \frac{2L}{\pi} \]
- The original magnetic moment \( M \) can also be expressed as:
\[ M = n \cdot L \]
- Therefore, we can substitute \( n \) from the original magnetic moment into the equation for \( m' \):
\[ m' = \frac{M}{L} \cdot \frac{2L}{\pi} = \frac{2M}{\pi} \]

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