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A cyclotron with dee radius R is used to speed up protons of mass m. An alternating electric field of frequency ν is applied across the dees. For the cyclotron to operate, the magnetic field B and the kinetic energy K of the emerging proton beam are related by:

1. B = mν / e and K = 2mπ²ν²R²
2. B = 2πmν / e and K = m²πνR²
3. B = 2πmν / e and K = 2mπ²ν²R²
4. B = mν / e and K = m²πνR²

Correct answer: B = 2πmν / e and K = 2mπ²ν²R²

Solution

Cyclotron resonance: nu = eB/(2 pi m) so B = 2 pi m nu/e. v_max = 2 pi nu R, so K = (1/2)m(2 pi nu R)^2 = 2 m pi^2 nu^2 R^2.

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