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Cyclotron protons, charge-to-mass ratio

Short answer

Charge-to-mass ratio q/m = v/(BR) ≈ 9.62×10⁷ C/kg. Frequency f= v/(2πR) ≈ 7.65×10⁶ Hz = 7.65 MHz.

A cyclotron which has the dees of radius 42cm and magnetic field of flux density 0.5 weber/m² is employed to accelerate protons. If the final velocity of the proton is 2.02×10⁷ m/sec, calculate the charge to mass ratio for the proton and the frequency of the alternating potential between the dees.

At the final radius R (the dee radius) the proton moves with v = (q/m) B R, and its orbital frequency is f = qB/(2πm) = v/(2πR).

Given: R = 42 cm = 0.42 m, B = 0.5 Wb/m², v = 2.02×10⁷ m/s.

Charge-to-mass ratio

q/m = v / (B R) = 2.02×10⁷ / (0.5 × 0.42)
    = 2.02×10⁷ / 0.21
    = 9.62×10⁷ C/kg

(The accepted value for the proton is 9.58×10⁷ C/kg.)

Frequency of the alternating potential

f = (q/m) B / (2π) = (9.62×10⁷ × 0.5) / (2π)
  = 4.81×10⁷ / 6.283
  = 7.65×10⁶ Hz
(check: v/(2πR) = 2.02×10⁷ / (2π × 0.42) = 7.65×10⁶ Hz)

Answer: q/m ≈ 9.6×10⁷ C/kg; f ≈ 7.65 MHz

✓ Answer checked by Claude Opus 5.5 (high effort)

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