91Ó°ÊÓ

The key things to identify in the problem are the shape of the wire (circle), the current (which is constant), and the changes in radius when the wire is doubled on itself. The initial state of the wire results in a circle of radius \(R\), and the final state results in two smaller circles with radius \(r\), where \(r = R/2\). It is also worth noting that the same length of wire used to form the circle initially is used to form two smaller circles, this implies that the total length of the wire (circumference of the circle) remains invariant.

We know from the Biot-Savart Law that the magnetic field at the center of a circular current-carrying wire is given by \[ B = \frac{{\mu I}}{{2R}} \], where \(\mu\) is the permeability of free space, \(I\) is the current, and \(R\) is the radius of the circular wire. Therefore, the initial magnetic field \(B_{1}\) when the wire is a single loop with radius \(R\) is given by \[ B_{1} = \frac{{\mu I}}{{2R}} \]. After bending the wire more sharply into a double loop with smaller radius \(r = R/2\), the magnetic field \(B_{2}\) becomes \[ B_{2} = 2 \times \frac{{\mu I}}{{2r}} = 2 \times \frac{{\mu I}}{{R}} \] The factor of 2 is introduced due to two loops in parallel. Each smaller loop carries the same current and contributes equally to the magnetic field at the center.

Finally, by comparing \(B_{2}\) and \(B_{1}\), we find that \(B_{2} = 2B_{1}\). Hence, the magnetic field at the center when the wire is bent into two smaller loops is twice the magnetic field when the wire is in one larger loop.