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Why does the magnetic field in a synchrotron increase as particles gain energy?

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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 synchrotron, the magnetic field strength is increased as particles gain energy to keep the radius of their orbit constant.12

Unlike a cyclotron, which uses a constant magnetic field and causes particles to spiral outward as they accelerate, a synchrotron is designed to guide particles along a fixed circular path. Because the momentum of the particles increases as they are accelerated by radio-frequency voltages, a stronger magnetic field is required to maintain the same orbital radius. By synchronizing the increase in magnetic field strength with the increase in particle energy, the machine can effectively steer the beam within a ring of constant dimensions.21