Cyclotron
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Direct answer
A cyclotron accelerates charged particles to high energy by looping them through the same small potential difference thousands of times: two hollow D-shaped electrodes ("dees") sit in a uniform magnetic field B, and an alternating voltage of frequency f_c = qB/2πm is applied across the gap. Each crossing of the gap adds energy qV, the magnetic field bends the faster particle into a slightly larger semicircle, and because the cyclotron frequency qB/2πm is independent of the particle's speed, the particle arrives at the gap in step with the oscillator at every radius — the resonance that makes the machine work. The particle spirals outward and exits with kinetic energy K_max = q^2B^2R^2/2m, set by the dees' radius R, not by the voltage. The design fails for electrons: at comparable energies their speed becomes relativistic, the mass increase shifts their frequency, and synchronism is lost.
What you must remember
- Construction picture: two semicircular dees with a small gap, enclosed in a vacuum chamber, all inside a strong uniform B perpendicular to the dee plane; an RF oscillator drives the gap.
- Cyclotron frequency: f_c = qB/2πm (typically megahertz), independent of speed and radius — the resonance condition the oscillator must match.
- Energy gain mechanism: each gap crossing adds qV (gap voltage), so after N crossings K = NqV; a modest 10^4 V gap crossed a thousand times yields mega-electron-volt energies.
- Exit energy: K_max = q^2B^2R^2/2m = ½m v_max^2 with v_max = qBR/m — only B and R matter for the final energy, not the gap voltage.
- Inside the dees: the metal shields the interior from electric fields, so within a dee the particle coasts in a semicircle under B alone; all acceleration happens in the gap.
- Time structure: each semicircle takes the same time T/2 = πm/qB regardless of radius, so equal-gap particles stay synchronised while spiralling out.
- Limitations: uncharged particles (neutrons) cannot be accelerated at all; electrons desynchronise quickly (relativistic mass growth at small energies); very high energies need synchrocyclotrons or synchrotrons.
A worked energy calculation
A cyclotron with dees of radius 0.6 m operates at B = 1.2 T accelerating protons (q = 1.6 × 10^-19 C, m = 1.67 × 10^-27 kg). The required oscillator frequency is f_c = qB/2πm = (1.6 × 10^-19 × 1.2)/(2π × 1.67 × 10^-27) ≈ 1.83 × 10^7 Hz, about 18 MHz — squarely the short-wave radio band. The exit speed is v_max = qBR/m = (1.6 × 10^-19 × 1.2 × 0.6)/1.67 × 10^-27 ≈ 6.9 × 10^7 m/s, nearly a quarter of light speed, and K_max = ½mv_max^2 = ½ × 1.67 × 10^-27 × (6.9 × 10^7)^2 ≈ 4.0 × 10^-12 J, which is about 25 MeV. Now the examiner's follow-up: doubling the gap voltage does nothing to K_max (it only halves the number of revolutions needed), whereas doubling B or R quadruples it. One more check worth quoting: at 25 MeV a proton's speed is relativistic enough (v/c ≈ 0.23) that the mass correction is still only a few per cent, so classical resonance survives — for electrons the same energy means v ≈ 0.3c already at keV scale, which is why electron machines are linacs, not cyclotrons.
Where students slip
The commonest confusion is assigning the energy gain to the magnetic field: B only steers, and the speeding up happens solely in the gap from the electric field — an assertion-reason staple where "magnetic field increases the particle's kinetic energy" is the planted false reason. Second, students quote K_max as NqV and miss that the exam usually asks the maximum attainable energy, which the geometry (R) fixes; the two formulae answer different questions. Third, the frequency question cuts both ways: f_c depends on q, B and m, and candidates forget the m in the denominator when comparing deuterons and protons in the same machine (a deuteron, roughly twice the proton mass, resonates at half the frequency and gains half the K_max at the same B and R). Finally, "why not neutrons?" is a one-mark favourite: no charge means no electric acceleration and no magnetic steering — the machine has nothing to grip.
Frequently asked questions
What is the working principle of a cyclotron?
A charged particle is repeatedly accelerated by an alternating electric field across a gap between two dees while a perpendicular magnetic field bends it into semicircles, resonance persisting because the cyclotron frequency is independent of speed.
Why must the oscillator frequency equal qB/2πm?
Only then does the particle's gap arrival stay in phase with the alternating voltage, gaining qV at every crossing; any mismatch would alternate acceleration and deceleration.
What limits the maximum energy a cyclotron can deliver?
The dee radius and field: K_max = q^2B^2R^2/2m, since the particle exits when its spiral reaches the rim; gap voltage affects only the number of turns needed.
Why are electrons not accelerated in a cyclotron?
At quite modest energies electrons move relativistically, their effective mass rises, the orbital frequency drops below the fixed oscillator frequency, and resonance fails early.
What role does the magnetic field play inside a dee?
Inside a conducting dee the electric field is shielded out, so the magnetic field alone bends the particle into a semicircle of radius r = mv/qB at constant speed.