A student designs an electromagnetic system consisting of a closely wound toroid with inner radius \(a\), outer radius \(b\), and \(N\) tightly wound turns carrying a steady current \(I\). A magnetic field sensor is mounted at the geometric center of the toroid, located on the central axis of symmetry in the plane of the coils (\(r = 0\)). The student then replaces the toroid with a modified version having \(2N\) turns and an inner radius of \(a/2\) (with outer radius \(b > a/2\)), while maintaining the same current \(I\). By what factor does the magnitude of the magnetic field detected by the sensor change?