A cyclotron accelerates particles of mass m and charge q. The energy of particles emerging is proportional to

In a cyclotron, the kinetic energy of the emerging particles is determined by the radius of the outermost orbit within the magnetic field. The particle moves in a circular path where the Lorentz force provides the necessary centripetal force, expressed as qvB = mv²/R. Solving for velocity gives v = qBR/m. The kinetic energy (Ek) is defined as ½mv². Substituting the velocity expression into the kinetic energy formula yields Ek = ½m(qBR/m)², which simplifies to Ek = q²B²R² / (2m). This formula demonstrates that for a cyclotron with a fixed maximum radius R and magnetic field B, the energy of the emerging particles is directly proportional to the square of the charge (q²) and inversely proportional to the mass (m) of the particle. Thus, the energy scales as q²/m.

Sources

1. [1] https://en.wikipedia.org/wiki/Cyclotron_motion