Introduction:
When a current flows through a long wire, it creates a magnetic field around it. The strength of this magnetic field depends on various factors, including the shape and size of the wire. In this scenario, the wire is first shaped as a circular coil of one turn and then transformed into a coil of two turns of smaller radius. We need to determine whether the magnetic field at the center of the coil will change and, if so, by how much.

Explanation:

Magnetic Field of a Circular Coil:
When a current passes through a circular coil, it creates a magnetic field at its center. The magnitude of this magnetic field can be determined using Ampere's law. The formula to calculate the magnetic field at the center of a circular coil is given by:

B = μ₀ * (n * I) / (2 * R)

Where:
- B is the magnetic field
- μ₀ is the permeability of free space
- n is the number of turns in the coil
- I is the current passing through the coil
- R is the radius of the coil

Effect of Changing the Coil Configuration:
When the circular coil with one turn is transformed into a coil with two turns of smaller radius, it means that the number of turns (n) has increased while the radius (R) has decreased. Let's analyze the effect of these changes on the magnetic field at the center of the coil.

Increased Number of Turns:
As the number of turns in the coil increases, the magnetic field at the center also increases. This is because the magnetic field created by each turn of the coil adds up to produce a stronger overall magnetic field. Therefore, when the coil is transformed from one turn to two turns, the magnetic field at the center will increase.

Decreased Radius:
As the radius of the coil decreases, the magnetic field at the center also decreases. This is because the magnetic field produced by each turn of the coil spreads out over a smaller area, resulting in a weaker magnetic field at the center. Therefore, when the coil is transformed into a coil of smaller radius, the magnetic field at the center will decrease.

Net Effect:
Considering both the increased number of turns and the decreased radius, the net effect on the magnetic field at the center of the coil is not clear. It depends on the relative magnitude of these two effects. If the increase in the number of turns is significant enough to compensate for the decrease in radius, the magnetic field at the center may increase. However, if the decrease in radius dominates, the magnetic field at the center may decrease.

Conclusion:
In summary, when a circular coil of one turn is transformed into a coil of two turns of smaller radius, the magnetic field at the center of the coil may change. The net effect depends on the relative magnitude of the increase in the number of turns and the decrease in radius. Without specific values for the number of turns and the radius, it is not possible to determine the exact change in the magnetic field at the center.