A circular coil of `300` turns and diameter `14cm` carries a current of `15A`. What is the magnetic moment of the loop?

Explanation

To find the magnetic moment of a circular coil, we can use the formula:

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

where:
- \( M \) is the magnetic moment,
- \( N \) is the number of turns,
- \( I \) is the current in amperes,
- \( A \) is the area of the coil in square meters.

### Step 1: Identify the given values
- Number of turns, \( N = 300 \)
- Diameter of the coil = 14 cm
- Current, \( I = 15 \, A \)

### Step 2: Convert the diameter to radius in meters
The radius \( r \) is half of the diameter. First, convert the diameter from centimeters to meters:
\[ \text{Diameter} = 14 \, \text{cm} = 0.14 \, \text{m} \]
\[ r = \frac{\text{Diameter}}{2} = \frac{0.14 \, \text{m}}{2} = 0.07 \, \text{m} \]

### Step 3: 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 \( r \):
\[ A = \pi (0.07 \, \text{m})^2 = \pi (0.0049 \, \text{m}^2) \approx 0.0154 \, \text{m}^2 \]

### Step 4: Substitute the values into the magnetic moment formula
Now substitute \( N \), \( I \), and \( A \) into the magnetic moment formula:
\[ M = N \cdot I \cdot A = 300 \cdot 15 \cdot 0.0154 \]

### Step 5: Calculate the magnetic moment
Calculating the above expression:
\[ M = 300 \cdot 15 \cdot 0.0154 \approx 69.3 \, \text{Joule Tesla}^{-1} \]

### Final Answer
The magnetic moment of the circular coil is approximately \( 69.3 \, \text{Joule Tesla}^{-1} \).

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