A circular loop of radius 3 cm is having a current of 12.5 A. The magnitude of magnetic field at a distance of 4 cm on its axis is

Options

A. `5.65 xx 10^(-5)T`
B. `5.27 xx 10^(-5)T`
C. `6.54 xx 10^(-5)T`
D. `9.20 xx 10^(-5)T`

Explanation

To find the magnitude of the magnetic field at a distance of 4 cm on the axis of a circular loop with a radius of 3 cm carrying a current of 12.5 A, we can use the formula for the magnetic field along the axis of a circular loop:

\[ B = \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}} \]

where:
- \( B \) is the magnetic field,
- \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \, \text{T m/A} \)),
- \( I \) is the current (12.5 A),
- \( R \) is the radius of the loop (3 cm = 0.03 m),
- \( x \) is the distance from the center of the loop along the axis (4 cm = 0.04 m).

### Step-by-step Solution:

1. **Convert the radius and distance to meters**:
- Radius, \( R = 3 \, \text{cm} = 0.03 \, \text{m} \)
- Distance, \( x = 4 \, \text{cm} = 0.04 \, \text{m} \)

2. **Substitute the values into the formula**:
\[ B = \frac{(4\pi \times 10^{-7} \, \text{T m/A}) \times (12.5 \, \text{A}) \times (0.03 \, \text{m})^2}{2((0.03 \, \text{m})^2 + (0.04 \, \text{m})^2)^{3/2}} \]

3. **Calculate \( R^2 \) and \( x^2 \)**:
- \( R^2 = (0.03)^2 = 0.0009 \, \text{m}^2 \)
- \( x^2 = (0.04)^2 = 0.0016 \, \text{m}^2 \)

4. **Calculate \( R^2 + x^2 \)**:
\[ R^2 + x^2 = 0.0009 + 0.0016 = 0.0025 \, \text{m}^2 \]

5. **Calculate \( (R^2 + x^2)^{3/2} \)**:
\[ (0.0025)^{3/2} = (0.0025)^{1.5} = 0.00395 \, \text{m}^3 \]

6. **Substitute back into the magnetic field formula**:
\[ B = \frac{(4\pi \times 10^{-7}) \times (12.5) \times (0.0009)}{2 \times 0.00395} \]

7. **Calculate the numerator**:
\[ \text{Numerator} = 4\pi \times 10^{-7} \times 12.5 \times 0.0009 \approx 1.41 \times 10^{-9} \]

8. **Calculate the denominator**:
\[ \text{Denominator} = 2 \times 0.00395 \approx 0.0079 \]

9. **Final calculation for \( B \)**:
\[ B = \frac{1.41 \times 10^{-9}}{0.0079} \approx 5.65 \times 10^{-5} \, \text{T} \]

### Conclusion:
The magnitude of the magnetic field at a distance of 4 cm on the axis of the circular loop is approximately \( 5.65 \times 10^{-5} \, \text{T} \).