A wire of length l meter carrying current i ampere is bent in form of circle. Magnetic moment is

Options

A. `I L^2 / 4 A m^2`
B. `I pi L^2 / 4 A m^2`
C. `2I L^2 / pi A m^2`
D. `I L^2 / 4pi A m^2`

Answer

To find the magnetic moment of a wire of length \( l \) meters carrying a current \( i \) amperes when bent in the form of a circle, we can follow these steps:

### Step 1: Understand the Formula for Magnetic Moment
The magnetic moment \( M \) of a current-carrying loop is given by the formula:
\[ M = N \cdot I \cdot A \]
where:
- \( N \) is the number of turns (which is 1 for a single loop),
- \( I \) is the current in amperes,
- \( A \) is the area enclosed by the loop.

### Step 2: Determine the Area of the Circle
Since the wire is bent into a circle, the area \( A \) of the circle can be calculated using the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circle.

### Step 3: Relate the Length of the Wire to the Radius
The length of the wire \( l \) is equal to the circumference of the circle formed:
\[ l = 2 \pi r \]
From this, we can solve for the radius \( r \):
\[ r = \frac{l}{2\pi} \]

### Step 4: Substitute the Radius into the Area Formula
Now, substitute \( r \) back into the area formula:
\[ A = \pi \left(\frac{l}{2\pi}\right)^2 \]
Calculating this gives:
\[ A = \pi \cdot \frac{l^2}{4\pi^2} = \frac{l^2}{4\pi} \]

### Step 5: Substitute the Area into the Magnetic Moment Formula
Now we can substitute the area \( A \) back into the magnetic moment formula:
\[ M = N \cdot I \cdot A = 1 \cdot I \cdot \frac{l^2}{4\pi} \]
Thus, the magnetic moment is:
\[ M = \frac{I l^2}{4\pi} \]

### Step 6: Final Result
The magnetic moment of the wire bent in the form of a circle is:
\[ M = \frac{I l^2}{4\pi} \]