If charged particles are to be kept moving on a fixed circular path while their speed (and hence momentum) keeps increasing, the magnetic field must:

remain constant everywhere
increase gradually in proportion to the kinetic energy of the particle
increase gradually in proportion to the speed of the particle
none of these

Correct answer: increase gradually in proportion to the speed of the particle

Solution

From r = m v/(q B), keeping r fixed needs B = m v/(q r), i.e. B proportional to v (the momentum). This is the principle of the synchrotron, where the field rises in step with the particle's speed/momentum.

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