A magnetic wire of dipole moment `4pi Am^(2)` is bent in the form of semicircle. The new magnetic moment is

Options

A. `4piAm^(2)`
B. `8piAm^(2)`
C. `4Am^(2)`
D. none of thse

Answer

To solve the problem, we need to determine the new magnetic moment of a wire that has been bent into the shape of a semicircle. We start with the given dipole moment and follow through the steps logically.

### Step-by-Step Solution:

1. **Identify the Initial Magnetic Moment:**
The initial magnetic moment \( M \) of the wire is given as:
\[ M = 4\pi \, \text{A m}^2 \]

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

3. **Determine the Initial Length:**
Let the initial length of the wire be \( L \).

4. **Bend the Wire into a Semicircle:**
When the wire is bent into a semicircle, the length of the semicircle is equal to the initial length \( L \):
\[ \text{Circumference of semicircle} = \pi R = L \]
From this, we can find the radius \( R \):
\[ R = \frac{L}{\pi} \]

5. **Calculate the New Length:**
The total length of the wire when bent into a semicircle is:
\[ L' = 2R = 2 \left(\frac{L}{\pi}\right) = \frac{2L}{\pi} \]

6. **Determine the New Magnetic Moment:**
The new magnetic moment \( M' \) can be calculated using the pole strength \( m \) which remains constant:
\[ M' = m \cdot L' \]
Since \( m = \frac{M}{L} = \frac{4\pi}{L} \), we substitute this into the equation:
\[ M' = \left(\frac{4\pi}{L}\right) \cdot \left(\frac{2L}{\pi}\right) \]

7. **Simplify the Expression:**
Simplifying the expression gives:
\[ M' = \frac{4\pi \cdot 2L}{L \cdot \pi} = \frac{8\pi}{\pi} = 8 \, \text{A m}^2 \]

8. **Final Answer:**
Therefore, the new magnetic moment when the wire is bent into a semicircle is:
\[ M' = 8 \, \text{A m}^2 \]