A circular coil is made from a wire of length 2m. Its radius is `4/pi` cm. When a current of 1A passes through it, it’s magnetic moment is

Options

A. `piAm^(2)`
B. `4/(100pi)Am^(2)`
C. `16/(pi)Am^(2)`
D. `1/piAm^(2)`

Answer

To find the magnetic moment of a circular coil made from a wire of length 2 m and with a radius of \( \frac{4}{\pi} \) cm when a current of 1 A passes through it, we can follow these steps:

### Step 1: Convert the radius to meters
The radius \( r \) is given as \( \frac{4}{\pi} \) cm. To convert this to meters:
\[ r = \frac{4}{\pi} \text{ cm} = \frac{4}{\pi} \times \frac{1}{100} \text{ m} = \frac{4}{100\pi} \text{ m} \]

### Step 2: Calculate the circumference of the coil
The circumference \( C \) of a circular coil is given by the formula:
\[ C = 2\pi r \]
Substituting the value of \( r \):
\[ C = 2\pi \left(\frac{4}{100\pi}\right) = \frac{8}{100} = 0.08 \text{ m} \]

### Step 3: Find the number of turns in the coil
The total length of the wire \( L \) is given as 2 m. The number of turns \( n \) in the coil can be calculated using the formula:
\[ L = n \cdot C \]
Rearranging gives:
\[ n = \frac{L}{C} = \frac{2}{0.08} = 25 \]

### Step 4: Calculate the area of the coil
The area \( A \) of the circular coil is given by:
\[ A = \pi r^2 \]
Substituting the value of \( r \):
\[ A = \pi \left(\frac{4}{100\pi}\right)^2 = \pi \cdot \frac{16}{10000\pi^2} = \frac{16}{10000\pi} = \frac{4}{2500\pi} \text{ m}^2 \]

### Step 5: Calculate the magnetic moment
The magnetic moment \( m \) of the coil is given by:
\[ m = n \cdot I \cdot A \]
Where \( I \) is the current (1 A). Substituting the values:
\[ m = 25 \cdot 1 \cdot \frac{4}{2500\pi} = \frac{100}{2500\pi} = \frac{1}{25\pi} \text{ A m}^2 \]

### Final Answer
The magnetic moment of the coil is:
\[ m = \frac{1}{25\pi} \text{ A m}^2 \]

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