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

Answer

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

### Step 1: Understand the Magnetic Moment
The magnetic moment \( M \) of a wire is given by the product of the pole strength and the distance between the poles. For a straight wire, the magnetic moment can be expressed as:
\[ M = m \cdot L \]
where \( m \) is the pole strength and \( L \) is the length of the wire.

### Step 2: Determine the Radius of the Semicircular Arc
When the wire is bent into a semicircular shape, the length of the wire remains the same. The length of a semicircular arc can be expressed as:
\[ L = \pi r \]
where \( r \) is the radius of the semicircle. From this, we can solve for \( r \):
\[ r = \frac{L}{\pi} \]

### Step 3: Calculate the New Magnetic Moment
In the semicircular configuration, the distance between the two poles (the ends of the semicircle) is the diameter, which is:
\[ \text{Diameter} = 2r = 2 \left(\frac{L}{\pi}\right) = \frac{2L}{\pi} \]
The magnetic moment for the semicircular arc can now be expressed as:
\[ M' = m \cdot \text{Diameter} = m \cdot \frac{2L}{\pi} \]

### Step 4: Substitute the Expression for \( m \)
Since we know that \( m = \frac{M}{L} \) (from the original straight wire configuration), we can substitute this into the equation for the new magnetic moment:
\[ M' = \left(\frac{M}{L}\right) \cdot \frac{2L}{\pi} \]
Simplifying this gives:
\[ M' = \frac{2M}{\pi} \]

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