Introduction:
The magnetic field at the center of a current-carrying wire can be determined using Ampere's law. This law relates the magnetic field around a closed loop to the current passing through the loop.

Ampere's Law:
Ampere's law states that the line integral of the magnetic field around a closed loop is equal to the product of the current passing through the loop and the permeability of free space.

Derivation:
To determine the magnetic field at the center of a current-carrying wire, we consider a circular loop centered at the wire. The radius of the loop is equal to the distance between the wire and its center.

Applying Ampere's Law:
Using Ampere's law, we can write the equation as follows:
∮B·dl = μ₀Ienc

Where:
- ∮B·dl is the line integral of the magnetic field around the loop,
- μ₀ is the permeability of free space,
- Ienc is the enclosed current passing through the loop.

Symmetry in the Current-Carrying Wire:
Due to the symmetry of the current-carrying wire, the magnetic field will be the same at every point on the circular loop. This allows us to simplify the integral by considering the magnetic field as constant along the loop.

Magnetic Field Magnitude:
Since the magnetic field is constant along the loop, the line integral simplifies to B∮dl, where B is the magnitude of the magnetic field and ∮dl is the circumference of the loop.

Circumference of the Loop:
The circumference of the loop is given by 2πr, where r is the radius of the loop.

Final Equation:
Plugging in the values, we have:
B(2πr) = μ₀Ienc

Simplifying the equation, we can solve for the magnetic field magnitude:
B = (μ₀Ienc) / (2πr)

Conclusion:
The magnetic field magnitude at the center of a current-carrying wire is given by the equation B = (μ₀Ienc) / (2πr), where B is the magnetic field magnitude, μ₀ is the permeability of free space, Ienc is the enclosed current passing through the loop, and r is the distance between the wire and the center of the loop.