The magnetic field due to a current carrying circular loop of radius `3 m` at as point on the axis at a distance of `4m` from the centre is `54 muT`. What will be its value at the centre of the loop.

Options

A. `250muT`
B. `150muT`
C. `125muT`
D. `75muT`

Explanation

To find the magnetic field at the center of a current-carrying circular loop, we can use the information given about the magnetic field at a point on the axis of the loop.

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Radius of the loop, \( r = 3 \, \text{m} \)
- Distance from the center to the point on the axis, \( x = 4 \, \text{m} \)
- Magnetic field at point P (on the axis), \( B_P = 54 \, \mu T \)

2. **Use the Formula for Magnetic Field on the Axis**:
The magnetic field \( B \) at a distance \( x \) from the center of a circular loop of radius \( r \) carrying current \( I \) is given by:
\[ B_P = \frac{\mu_0 I r^2}{2(r^2 + x^2)^{3/2}} \]
Here, \( \mu_0 \) is the permeability of free space.

3. **Set Up the Equation**:
We can express \( B_P \) in terms of \( I \):
\[ 54 \times 10^{-6} = \frac{\mu_0 I (3^2)}{2((3^2) + (4^2))^{3/2}} \]

4. **Calculate the Denominator**:
Calculate \( r^2 + x^2 \):
\[ r^2 + x^2 = 3^2 + 4^2 = 9 + 16 = 25 \]
Thus, \( (r^2 + x^2)^{3/2} = 25^{3/2} = 125 \).

5. **Substitute Values**:
Substitute \( r \) and \( (r^2 + x^2)^{3/2} \) into the equation:
\[ 54 \times 10^{-6} = \frac{\mu_0 I (9)}{2(125)} \]

6. **Rearranging the Equation**:
Rearranging gives:
\[ \mu_0 I = \frac{54 \times 10^{-6} \times 250}{9} \]

7. **Calculate \( \mu_0 I \)**:
\[ \mu_0 I = \frac{13500 \times 10^{-6}}{9} = 1500 \times 10^{-6} = 1.5 \times 10^{-3} \, \text{T m} \]

8. **Magnetic Field at the Center**:
The magnetic field \( B_C \) at the center of the loop is given by:
\[ B_C = \frac{\mu_0 I}{2r} \]
Substitute \( \mu_0 I \) and \( r \):
\[ B_C = \frac{1.5 \times 10^{-3}}{2 \times 3} = \frac{1.5 \times 10^{-3}}{6} = 0.25 \times 10^{-3} = 250 \, \mu T \]

9. **Final Answer**:
Therefore, the magnetic field at the center of the loop is:
\[ B_C = 250 \, \mu T \]