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Why was the cyclotron limited as an accelerator?

Lawrence's cyclotron was developed in the 1930's and improved, accelerating particles to higher and higher energies using a constant radio frequency accelerating field. The basic cyclotron could produce 25MeV protons.

By equating the 'centripetal force' and the 'force on a moving charge' expressions for F,

and also using

it is easy to prove that the cyclotron frequency must be

However above this energy (25MeV protons) the particles were moving fast enough for the relativistic increase in mass to become significant, and this increasing mass meant that the constant radio frequency created no significant further acceleration. Increasing the mass means that, other things being equal, the applied frequency must be reduced, so the reversal of the electric field gets 'out of sync' with the motion of the particles unless the frequency is changed. This technique of synchronising the electric field oscillations was tried successfully in the synchrocyclotron.

In this way 200MeV protons could be produced, but the monster 184" diameter machine completed by Lawrence at Berkley in 1946 shows the other flaw in the design; it requires a uniform field to be applied over the entire area as the protons spiral outwards, a demanding specification, and one that the synchrotron minimises by confining the particles to a circle of constant radius and increasing - synchronising - the radio frequency accelerating field as the particles get faster and faster.