Magnetic field due to `0.1 A` current flowing through a circular coil of radius `0.1`m and `1000` turns at the centre of the coil is

Options

A. 0.2 T
B. `2xx10^(-4) T`
C. `6.28xx10^(-4)T`
D. `9.8xx10^(-4)T`

Answer

To find the magnetic field at the center of a circular coil due to a current flowing through it, we can use the formula derived from the Biot-Savart law:

### Step-by-Step Solution:

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

2. **Substitute the given values**:
- Current \( I = 0.1 \, \text{A} \)
- Number of turns \( n = 1000 \)
- Radius \( r = 0.1 \, \text{m} \)

Plugging these values into the formula gives:
\[ B = \frac{(4\pi \times 10^{-7}) \times 1000 \times 0.1}{2 \times 0.1} \]

3. **Simplify the equation**:
- The \( 0.1 \) in the numerator and denominator cancels out:
\[ B = \frac{(4\pi \times 10^{-7}) \times 1000}{2} \]

4. **Calculate the magnetic field**:
- Now simplify further:
\[ B = \frac{4\pi \times 10^{-7} \times 1000}{2} = 2\pi \times 10^{-4} \]

5. **Substitute the value of \( \pi \)**:
- Using \( \pi \approx 3.14 \):
\[ B = 2 \times 3.14 \times 10^{-4} = 6.28 \times 10^{-4} \, \text{T} \]

6. **Final answer**:
- The magnetic field at the center of the coil is:
\[ B = 6.28 \times 10^{-4} \, \text{T} \]

### Conclusion:
The correct answer is \( 6.28 \times 10^{-4} \, \text{T} \).