A particle with mass \(m\) and positive charge \(q\) travels with constant speed \(v\) in the \(+x\)-direction. At time \(t = 0\), the particle enters a region of uniform magnetic field of magnitude \(B\) directed perpendicular to its velocity, bounded by the line \(x = 0\). The particle follows a semicircular trajectory inside the field before exiting back across the boundary at \(x = 0\). Which of the following correctly pairs the radius \(r\) of the particle's path with the total time \(\Delta t\) the particle spends inside the magnetic field region?