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 it is bent into the shape of a circle, we can follow these steps:

### Step 1: Relate the length of the wire to the radius of the circle
When the wire is bent into a circle, the length of the wire \( L \) is equal to the circumference of the circle. The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle. Therefore, we can write:
\[ L = 2\pi r \]
From this, we can express the radius \( r \) in terms of the length \( L \):
\[ r = \frac{L}{2\pi} \]

### Step 2: Calculate the area of the circle
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
Substituting the expression for \( r \) we found in Step 1:
\[ A = \pi \left(\frac{L}{2\pi}\right)^2 \]
This simplifies to:
\[ A = \pi \cdot \frac{L^2}{4\pi^2} = \frac{L^2}{4\pi} \]

### Step 3: Calculate the magnetic moment
The magnetic moment \( M \) of a current-carrying loop is given by the formula:
\[ M = I \cdot A \]
Substituting the expression for area \( A \) from Step 2:
\[ M = I \cdot \frac{L^2}{4\pi} \]
Thus, the magnetic moment of the wire bent into a circle is:
\[ M = \frac{I L^2}{4\pi} \text{ ampere meter}^2 \]

### Final Answer
The magnetic moment of the wire bent into a circle is:
\[ M = \frac{I L^2}{4\pi} \text{ ampere meter}^2 \]

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