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. `2xx10^(-1)T`
B. `4.31xx10^(-2)T`
C. `6.28xx10^(-4)T`
D. `9.81xx10^(-4)T`

Explanation

To find the magnetic field at the center of a circular coil with a given current, radius, and number of turns, we can use the formula derived from the Biot-Savart law. The magnetic field \( B \) at the center of a circular coil is given by:

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

Where:
- \( B \) = magnetic field at the center of the coil (in Tesla)
- \( \mu_0 \) = permeability of free space \( = 4\pi \times 10^{-7} \, \text{T m/A} \)
- \( n \) = number of turns of the coil
- \( I \) = current flowing through the coil (in Amperes)
- \( R \) = radius of the coil (in meters)

### Step-by-Step Solution:

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

2. **Substitute the values into the formula:**
\[ B = \frac{4\pi \times 10^{-7} \cdot 1000 \cdot 0.1}{2 \cdot 0.1} \]

3. **Simplify the equation:**
- Calculate the denominator:
\[ 2 \cdot 0.1 = 0.2 \]
- Therefore, the equation becomes:
\[ B = \frac{4\pi \times 10^{-7} \cdot 1000 \cdot 0.1}{0.2} \]

4. **Calculate the numerator:**
- First, calculate \( 1000 \cdot 0.1 = 100 \)
- Then, calculate:
\[ 4\pi \times 10^{-7} \cdot 100 = 4\pi \times 10^{-5} \]

5. **Now substitute back into the equation:**
\[ B = \frac{4\pi \times 10^{-5}}{0.2} \]

6. **Calculate the final value:**
- Dividing \( 4\pi \times 10^{-5} \) by \( 0.2 \):
\[ B = 20\pi \times 10^{-5} \]
- Using \( \pi \approx 3.14 \):
\[ B \approx 20 \times 3.14 \times 10^{-5} \approx 62.8 \times 10^{-5} \, \text{T} = 6.28 \times 10^{-4} \, \text{T} \]

### Final Answer:
The magnetic field at the center of the coil is approximately \( 6.28 \times 10^{-4} \, \text{T} \).

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