A steel wire of length l has a magnetic moment M. It is bent into a semicircular arc. What is the new magnetic moment?

Options

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

Answer

To find the new magnetic moment when a steel wire of length \( l \) is bent into a semicircular arc, we can follow these steps:

### Step 1: Understand the Magnetic Moment
The magnetic moment \( M \) of a straight wire can be expressed as:
\[ M = m \cdot l \]
where \( m \) is the magnetic dipole moment per unit length, and \( l \) is the effective length of the wire.

### Step 2: Determine the Length of the Semicircular Arc
When the wire is bent into a semicircular arc, the length of the arc is given by:
\[ \text{Length of semicircular arc} = \pi r \]
where \( r \) is the radius of the semicircle. Since the total length of the wire is \( l \), we can express the radius in terms of \( l \):
\[ l = \pi r \implies r = \frac{l}{\pi} \]

### Step 3: Find the Distance Between End Points
The distance between the endpoints of the semicircular arc is the diameter \( D \), which can be calculated as:
\[ D = 2r = 2 \left(\frac{l}{\pi}\right) = \frac{2l}{\pi} \]

### Step 4: Calculate the New Magnetic Moment
The new magnetic moment \( M' \) when the wire is bent into a semicircular arc can be expressed as:
\[ M' = m \cdot D \]
Substituting for \( D \):
\[ M' = m \cdot \left(\frac{2l}{\pi}\right) \]

### Step 5: Relate \( m \) to the Original Magnetic Moment \( M \)
Since the magnetic dipole moment \( m \) does not change when the shape of the wire changes, we can express \( m \) in terms of the original magnetic moment \( M \):
\[ m = \frac{M}{l} \]
Substituting this into the equation for \( M' \):
\[ M' = \left(\frac{M}{l}\right) \cdot \left(\frac{2l}{\pi}\right) \]
The \( l \) cancels out:
\[ M' = \frac{2M}{\pi} \]

### Conclusion
Thus, the new magnetic moment when the steel wire is bent into a semicircular arc is:
\[ M' = \frac{2M}{\pi} \]