How does a cyclotron use a magnetic field to move charged particles in a circle?

A cyclotron uses a uniform magnetic field to guide charged particles in a circular path by leveraging the Lorentz force. When a charged particle moves with velocity $v$ perpendicular to a magnetic field $B$, the magnetic force acts as a centripetal force, causing the particle to follow a circular trajectory with a radius defined by $r=mv/qB$.123

Inside the cyclotron, particles are placed in an evacuated chamber between the poles of an electromagnet. The device features two hollow, D-shaped electrodes called dees, separated by a narrow gap. An alternating electric field is applied across this gap.12

As particles cross the gap, the electric field accelerates them, increasing their speed and kinetic energy. Once inside the dees, the magnetic field forces them into a semicircular path. Because the particles gain speed with each crossing, the radius of their orbit continually increases, resulting in a spiral-like path. The electric field is carefully timed to reverse polarity so that it always accelerates the particles as they return to the gap.23