A 100 turn closely wound circular coil of radius `10cm` carries a current of `3*2A`. (i) What is the field at the centre of the coil? (ii) What is the magnetic moment of this arrangement?

Answer

To solve the problem, we will follow these steps:

### Given Data:
- Number of turns (N) = 100
- Radius (r) = 10 cm = 0.1 m
- Current (I) = 3.2 A

### (i) Finding the Magnetic Field at the Center of the Coil:

1. **Formula for Magnetic Field (B) at the Center of a Circular Coil**:
The magnetic field at the center of a circular coil is given by the formula:
\[ B = \frac{\mu_0 N I}{2R} \]
where:
- \( \mu_0 \) (permeability of free space) = \( 4\pi \times 10^{-7} \, \text{T m/A} \)
- \( N \) = number of turns
- \( I \) = current in amperes
- \( R \) = radius of the coil in meters

2. **Substituting the Values**:
\[ B = \frac{(4\pi \times 10^{-7}) \times 100 \times 3.2}{2 \times 0.1} \]

3. **Calculating the Magnetic Field**:
- First, calculate the numerator:
\[ 4\pi \times 10^{-7} \times 100 \times 3.2 = 4\pi \times 320 \times 10^{-7} \]
- Now, calculate \( 4\pi \approx 12.56 \):
\[ 12.56 \times 320 \times 10^{-7} = 4.0192 \times 10^{-4} \]
- Now divide by the denominator:
\[ B = \frac{4.0192 \times 10^{-4}}{0.2} = 2.0096 \times 10^{-3} \, \text{T} \approx 2 \times 10^{-3} \, \text{T} \]

### (ii) Finding the Magnetic Moment of the Arrangement:

1. **Formula for Magnetic Moment (M)**:
The magnetic moment is given by:
\[ M = NIA \]
where:
- \( A \) = area of the circular coil = \( \pi R^2 \)

2. **Calculating the Area (A)**:
\[ A = \pi R^2 = \pi (0.1)^2 = \pi \times 0.01 \approx 3.14 \times 0.01 = 0.0314 \, \text{m}^2 \]

3. **Substituting Values into the Magnetic Moment Formula**:
\[ M = N \cdot I \cdot A = 100 \cdot 3.2 \cdot 0.0314 \]

4. **Calculating the Magnetic Moment**:
- First, calculate \( 100 \cdot 3.2 = 320 \):
\[ M = 320 \cdot 0.0314 \approx 10.048 \, \text{A m}^2 \approx 10 \, \text{A m}^2 \]

### Final Answers:
- (i) The magnetic field at the center of the coil is approximately \( 2 \times 10^{-3} \, \text{T} \).
- (ii) The magnetic moment of the arrangement is \( 10 \, \text{A m}^2 \).