Introduction
To determine the magnetic field at the center of a current-carrying circular loop, we will use the Biot-Savart law, which relates the magnetic field to the current and the distance from the current element.

Biot-Savart Law
The Biot-Savart law states that the magnetic field (dB) at a point due to a small segment of current-carrying wire is directly proportional to the current (I), the length of the segment (dl), and inversely proportional to the square of the distance (r) between the point and the segment. Mathematically, it can be represented as:

dB = (μ₀/4π) * (I * dl × r̂) / r²

Where:
- dB is the magnetic field at the point
- μ₀ is the permeability of free space (4π × 10⁻⁷ T·m/A)
- I is the current flowing through the wire
- dl is the length of the current element
- r is the distance between the current element and the point
- r̂ is the unit vector pointing from the current element towards the point

Magnetic Field at the Center of a Loop
To find the magnetic field at the center of the circular loop, we consider an elemental current segment at a distance r from the center. Since the current is distributed uniformly along the loop, the magnitude of the magnetic field at the center due to the elemental segment will be the same for all segments.

Considering a small segment dl on the loop, the distance from the segment to the center is equal to the radius of the loop (r = 3 cm). The magnetic field at the center due to this segment can be calculated using the Biot-Savart law.

Magnetic Field Calculation
Given:
- r = 3 cm (distance from the center to the segment)
- dB = 54 ûT (magnetic field at a point 4 cm from the center)

We can rearrange the Biot-Savart law equation as follows:

dB = (μ₀/4π) * (I * dl × r̂) / r²

Since the magnetic field is the same for all segments at the center, we can integrate the magnetic field contribution from each segment along the loop to find the total magnetic field at the center.

∫dB = (μ₀/4π) * I * ∫(dl × r̂) / r²

Since the magnetic field at the center is zero (r = 0), the integral becomes:

0 = (μ₀/4π) * I * ∫(dl × r̂) / 0²

The integral on the right side represents the total length of the loop, which is 2πr (circumference of the loop).

0 = (μ₀/4π) * I * (2πr)

Simplifying the equation:

0 = (μ₀/2) * I * r

Solving for I:

I = 0

Conclusion
The magnetic field at the center of the loop is zero. This is because the magnetic field contributions from each segment of the loop cancel out due to symmetry, resulting in a net magnetic field of zero at the center.