An arc of a circle of radius `R` subtends an angle `(pi)/2` at the centre. It carriers a current `i`. The magnetic field at the centre will be

Options

A. `(mu_(0)i)/(2R)`
B. `(mu_(0)i)/(8R)`
C. `(mu_(0)i)/(4R)`
D. `(2mu_(0) i)/(5R)`

Explanation

To find the magnetic field at the center of an arc of a circle of radius \( R \) that subtends an angle \( \frac{\pi}{2} \) at the center and carries a current \( i \), we can use the Biot-Savart law. Here are the steps to derive the solution:

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- The arc subtends an angle \( \theta = \frac{\pi}{2} \) at the center of the circle.
- The radius of the arc is \( R \).

2. **Using Biot-Savart Law**:
- The Biot-Savart law states that the magnetic field \( B \) at a point due to a current-carrying conductor is given by:
\[ B = \frac{\mu_0}{4\pi} \int \frac{i \, dL \times \hat{r}}{r^2} \]
- For a circular arc, we can simplify this using the formula for a circular loop.

3. **Magnetic Field Due to a Full Circular Loop**:
- For a full circular loop of radius \( R \) carrying a current \( i \), the magnetic field at the center is given by:
\[ B_{\text{full}} = \frac{\mu_0 i}{2R} \]

4. **Adjusting for the Arc**:
- Since we only have an arc that subtends an angle \( \theta = \frac{\pi}{2} \), we need to find the proportion of the magnetic field due to the full loop.
- The proportion of the angle is:
\[ \frac{\theta}{2\pi} = \frac{\frac{\pi}{2}}{2\pi} = \frac{1}{4} \]

5. **Calculating the Magnetic Field for the Arc**:
- Therefore, the magnetic field at the center due to the arc is:
\[ B_{\text{arc}} = B_{\text{full}} \times \frac{1}{4} = \frac{\mu_0 i}{2R} \times \frac{1}{4} = \frac{\mu_0 i}{8R} \]

6. **Final Result**:
- The magnetic field at the center of the arc is:
\[ B = \frac{\mu_0 i}{8R} \]

### Summary:
The magnetic field at the center of an arc of a circle of radius \( R \) that subtends an angle \( \frac{\pi}{2} \) and carries a current \( i \) is given by:
\[ B = \frac{\mu_0 i}{8R} \]