The magnetic field produced at the center of a current carrying circular coil of radius r, is

Options

A. directly proportional to r
B. inversely proportional to r
C. directly proportional to `r^(2)`
D. inversely proportional to `r^(2)`

Answer

To find the magnetic field produced at the center of a current-carrying circular coil of radius \( r \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a circular coil of radius \( r \) carrying a current \( I \). We want to find the magnetic field \( B \) at the center of this coil.

2. **Using Biot-Savart Law**:
- According to the Biot-Savart Law, the magnetic field \( dB \) produced by a small segment of current \( dl \) at a distance \( r \) is given by:
\[ dB = \frac{\mu_0}{4\pi} \frac{I \, dl \, \sin \theta}{r^2} \]
- Here, \( \mu_0 \) is the permeability of free space, \( I \) is the current, \( dl \) is the length of the current segment, and \( \theta \) is the angle between the current element \( dl \) and the line joining the element to the point where we are calculating the field.

3. **Identifying the Angle**:
- For a circular coil, at the center, the angle \( \theta \) between \( dl \) and the radius \( r \) is \( 90^\circ \). Therefore, \( \sin \theta = \sin 90^\circ = 1 \).

4. **Simplifying the Expression**:
- Substituting \( \sin \theta = 1 \) into the equation for \( dB \):
\[ dB = \frac{\mu_0}{4\pi} \frac{I \, dl}{r^2} \]

5. **Integrating Around the Coil**:
- To find the total magnetic field \( B \) at the center, we need to integrate \( dB \) around the entire coil:
\[ B = \int dB = \int_0^{2\pi} \frac{\mu_0}{4\pi} \frac{I \, dl}{r^2} \]
- The total length of the coil is \( 2\pi r \), hence \( dl \) integrates to \( 2\pi r \):
\[ B = \frac{\mu_0}{4\pi} \frac{I}{r^2} \int_0^{2\pi} dl = \frac{\mu_0}{4\pi} \frac{I}{r^2} \cdot 2\pi r \]

6. **Final Calculation**:
- Simplifying the expression:
\[ B = \frac{\mu_0 I}{4\pi} \cdot \frac{2\pi}{r} = \frac{\mu_0 I}{2r} \]

7. **Conclusion**:
- Therefore, the magnetic field at the center of a current-carrying circular coil of radius \( r \) is given by:
\[ B = \frac{\mu_0 I}{2r} \]