Calculate the magnetic moment of a closely wound coil of 200 turns having a radius 15 cm and carrying a current of 4 A.

Options

A. `36.5 Am^(2)`
B. `56.5 A m^(2)`
C. `66.5 A m^(2)`
D. `108 A m^(2)`

Explanation

To calculate the magnetic moment \( \mu \) of a closely wound coil, we can use the formula:

\[ \mu = n \cdot I \cdot A \]

where:
- \( n \) is the number of turns per unit length (in this case, the total number of turns since the coil is closely wound),
- \( I \) is the current flowing through the coil,
- \( A \) is the area of the coil.

### Step 1: Identify the given values
- Number of turns \( N = 200 \)
- Radius of the coil \( r = 15 \, \text{cm} = 0.15 \, \text{m} \) (convert cm to m)
- Current \( I = 4 \, \text{A} \)

### Step 2: Calculate the area \( A \) of the coil
The area \( A \) of a circular coil is given by the formula:

\[ A = \pi r^2 \]

Substituting the value of \( r \):

\[ A = \pi (0.15)^2 \]

Calculating \( (0.15)^2 \):

\[ (0.15)^2 = 0.0225 \, \text{m}^2 \]

Now substituting this value into the area formula:

\[ A = \pi \cdot 0.0225 \]

Using \( \pi \approx 3.14 \):

\[ A \approx 3.14 \cdot 0.0225 \approx 0.07065 \, \text{m}^2 \]

### Step 3: Calculate the magnetic moment \( \mu \)
Now substituting the values of \( N \), \( I \), and \( A \) into the magnetic moment formula:

\[ \mu = N \cdot I \cdot A \]

\[ \mu = 200 \cdot 4 \cdot 0.07065 \]

Calculating this step by step:

1. \( 200 \cdot 4 = 800 \)
2. \( 800 \cdot 0.07065 \approx 56.52 \, \text{A m}^2 \)

### Final Result
Thus, the magnetic moment \( \mu \) of the coil is approximately:

\[ \mu \approx 56.52 \, \text{A m}^2 \]

### Answer
The magnetic moment of the coil is \( 56.52 \, \text{A m}^2 \).

---