Worked example: A cyclotron has B = 1.5 T and dee radius 0.5 m. Find the proton RF frequency and maximum energy. Step 1: Cyclotron frequency f = qB/2πm = 1.602×10⁻¹⁹×1.5/(2π×1.673×10⁻²⁷) = 2.29×10⁷ Hz ≈ 22.9 MHz. Step 2: Max momentum p = qBR = 1.602×10⁻¹⁹×1.5×0.5 = 1.20×10⁻¹⁹ kg·m/s. Step 3: E_max = p²/2m = (1.20×10⁻¹⁹)²/(2×1.673×10⁻²⁷) = 4.31×10⁻¹² J = 26.9 MeV. Step 4: At 27 MeV the proton's γ = 1.029 — relativistic lag is just beginning; beyond this a synchrocyclotron's frequency sweep is needed. Conclusion: f = qB/2πm sets the RF; p = qBR sets the energy ceiling — the two equations that built the accelerator age.