The magnetic fiedl at the cetre of a circular loop of diameter `0.1` m and carrying a current of 1 A is

Options

A. `2.5xx10^(-5)T`
B. `1.25xx10^(-5)T`
C. `3.8xx10^(-5)T`
D. `4.6xx10^(-5)T`

Explanation

To find the magnetic field at the center of a circular loop carrying a current, we can use the formula:

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

where:
- \( B \) is the magnetic field at the center of the loop,
- \( \mu_0 \) is the permeability of free space, approximately \( 4\pi \times 10^{-7} \, \text{T m/A} \),
- \( I \) is the current in amperes,
- \( R \) is the radius of the loop in meters.

### Step-by-Step Solution:

1. **Identify the given values**:
- Diameter of the loop, \( d = 0.1 \, \text{m} \)
- Current, \( I = 1 \, \text{A} \)

2. **Calculate the radius of the loop**:
\[ R = \frac{d}{2} = \frac{0.1 \, \text{m}}{2} = 0.05 \, \text{m} \]

3. **Substitute the values into the formula**:
\[ B = \frac{\mu_0 I}{2R} = \frac{(4\pi \times 10^{-7} \, \text{T m/A}) \times (1 \, \text{A})}{2 \times (0.05 \, \text{m})} \]

4. **Calculate the denominator**:
\[ 2R = 2 \times 0.05 \, \text{m} = 0.1 \, \text{m} \]

5. **Calculate the magnetic field**:
\[ B = \frac{(4\pi \times 10^{-7})}{0.1} = 4\pi \times 10^{-6} \, \text{T} \]

6. **Approximate the value of \( B \)**:
\[ B \approx 4 \times 3.14 \times 10^{-6} \, \text{T} \approx 12.56 \times 10^{-6} \, \text{T} \approx 12.56 \, \mu\text{T} \]

### Final Answer:
The magnetic field at the center of the circular loop is approximately \( 12.56 \, \mu\text{T} \).