Calculate the magnetic field at the centre of a 100 turn circular coil of radius 10 cm which carries a current of 3. 2 A.

Options

A. `2.01 xx 10^(- 3)`T
B. `5.64 xx 10^(-3)` T
C. `2.64 xx 10^(-4)`T
D. `5.64 xx 10^(-4)` T

Explanation

To calculate the magnetic field at the center of a circular coil, we can use the formula:

\[ B = \frac{\mu_0 \cdot n \cdot I}{2 \cdot R} \]

Where:
- \( B \) is the magnetic field at the center of the coil,
- \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \, \text{T m/A} \)),
- \( n \) is the number of turns per unit length (in this case, the total number of turns),
- \( I \) is the current in amperes,
- \( R \) is the radius of the coil in meters.

### Step 1: Identify the given values
- Number of turns, \( N = 100 \)
- Radius of the coil, \( R = 10 \, \text{cm} = 0.1 \, \text{m} \)
- Current, \( I = 3.2 \, \text{A} \)

### Step 2: Substitute the values into the formula
We substitute the values into the formula for the magnetic field:

\[ B = \frac{4\pi \times 10^{-7} \, \text{T m/A} \cdot 100 \cdot 3.2 \, \text{A}}{2 \cdot 0.1 \, \text{m}} \]

### Step 3: Calculate the numerator
Calculating the numerator:

\[ 4\pi \times 10^{-7} \cdot 100 \cdot 3.2 = 4\pi \times 320 \times 10^{-7} \]

### Step 4: Calculate the denominator
Calculating the denominator:

\[ 2 \cdot 0.1 = 0.2 \]

### Step 5: Combine the results
Now, we can combine the results:

\[ B = \frac{4\pi \times 320 \times 10^{-7}}{0.2} \]

### Step 6: Simplify the expression
Simplifying further:

\[ B = \frac{4\pi \times 320 \times 10^{-7}}{0.2} = 4\pi \times 1600 \times 10^{-7} \]

### Step 7: Calculate the final value
Using \( \pi \approx 3.14 \):

\[ B \approx 4 \times 3.14 \times 1600 \times 10^{-7} \approx 4 \times 3.14 \times 1.6 \times 10^{-4} \]

Calculating this gives:

\[ B \approx 20.096 \times 10^{-4} \, \text{T} = 2.0096 \times 10^{-3} \, \text{T} = 2.01 \, \text{mT} \]

### Final Answer
The magnetic field at the center of the coil is approximately \( 2.01 \, \text{mT} \).

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