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

Explanation

To find the new magnetic moment of a steel wire bent into a semicircular arc, we can follow these steps:

### Step 1: Understand the Definition of Magnetic Moment
The magnetic moment \( M \) of a straight wire is defined as:
\[ M = m \cdot l \]
where \( m \) is the pole strength and \( l \) is the length of the wire.

### Step 2: Determine the Pole Strength
From the definition, we can express the pole strength \( m \) as:
\[ m = \frac{M}{l} \]

### Step 3: Analyze the Bending of the Wire
When the wire is bent into a semicircular arc, the length of the wire remains the same, which is \( l \). The length of the semicircular arc can be expressed in terms of its radius \( R \):
\[ \text{Length of semicircular arc} = \pi R \]
Setting this equal to the original length \( l \):
\[ \pi R = l \implies R = \frac{l}{\pi} \]

### Step 4: Calculate the New Magnetic Moment
For a semicircular arc, the magnetic moment can be expressed as:
\[ M' = m \cdot d \]
where \( d \) is the distance between the two endpoints of the semicircular arc. The distance \( d \) for a semicircle is equal to the diameter, which is:
\[ d = 2R \]
Substituting the value of \( R \):
\[ d = 2 \left(\frac{l}{\pi}\right) = \frac{2l}{\pi} \]

### Step 5: Substitute Values to Find the New Magnetic Moment
Now substituting \( m \) and \( d \) into the equation for the new magnetic moment:
\[ M' = \left(\frac{M}{l}\right) \cdot \left(\frac{2l}{\pi}\right) \]
Simplifying this gives:
\[ M' = \frac{M \cdot 2l}{l \cdot \pi} = \frac{2M}{\pi} \]

### Final Answer
The new magnetic moment of the wire when bent into a semicircular arc is:
\[ M' = \frac{2M}{\pi} \]