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Why do particles spiral outward in a cyclotron?

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Diagram of a linear proton resonance accelerator, or linac, showing protons entering from a source, passing through a series of labeled drift tubes powered by a radio-frequency source, and exiting as a beam through a cavity. — particle accelerator The accelerator is a large-diameter tube within which an electric field oscillates at a high radio frequency. Within the accelerator tube are smaller diameter metallic drift tubes, which are carefully sized and spaced to shield the protons from decelerating oscillations of the electric field. In the spaces between the drift tubes, the electric field is oriented properly to accelerate the protons in their direction of travel.

In a cyclotron, particles spiral outward because they gain kinetic energy each time they cross the gap between the two D-shaped electrodes, known as dees.12

The radius of a charged particle's circular path in a uniform magnetic field is determined by the equation $r = \frac{m v}{q B}$ , where $m$ is the mass, $v$ is the velocity, $q$ is the charge, and $B$ is the magnetic field strength. As the particle crosses the gap, an alternating electric field accelerates it, increasing its velocity ( $v$ ). Because the magnetic field and the particle's charge remain constant, this increase in velocity forces the particle to follow a path with a larger radius.2314

Consequently, the particle's trajectory becomes a series of semicircles of continually increasing radius, creating a spiral-like path as it gains energy and moves toward the periphery of the device.12