Hong Life CalculatorsUnderstanding the cyclotron frequency calculator — concept and principle
A charged particle moving across a magnetic field travels in a circle at a frequency that does not depend on its speed. This calculator finds that cyclotron frequency, the period and, for a given kinetic energy, the particle's speed and orbit radius.
How it is calculated
- Magnetic force qvB provides the centripetal force mv² ÷ r.
- Radius r = mv ÷ (|q|B).
- Period T = 2πm ÷ (|q|B); cyclotron frequency f = |q|B ÷ (2πm); angular frequency ω = |q|B ÷ m.
- Speed from kinetic energy: v = √(2K ÷ m) (non-relativistic).
- Constants: e = 1.602176634 × 10⁻¹⁹ C; mp = 1.67262192595 × 10⁻²⁷ kg (CODATA 2022).
The principle
Faster particles travel larger circles, but in exactly the same time, because the radius grows in proportion to speed. That speed independence is what made Ernest Lawrence's cyclotron possible: an alternating voltage at the fixed cyclotron frequency kicks the particle every half-turn, and it spirals outwards as it gains energy. At a few per cent of the speed of light, relativistic mass increase lowers the frequency, which synchrocyclotrons and isochronous cyclotrons correct for.
Worked example
Default inputs: proton, B = 1.5 T, kinetic energy 10 keV.
- f = 1.602 × 10⁻¹⁹ × 1.5 ÷ (2π × 1.6726 × 10⁻²⁷) ≈ 22.8678 MHz.
- ω = 2πf ≈ 1.437 × 10⁸ rad/s; period ≈ 43.73 ns.
- Speed = √(2 × 10,000 × 1.602 × 10⁻¹⁹ ÷ 1.6726 × 10⁻²⁷) ≈ 1.384 × 10⁶ m/s, 0.4617% of light speed.
- Radius = 1.6726 × 10⁻²⁷ × 1.384 × 10⁶ ÷ (1.602 × 10⁻¹⁹ × 1.5) ≈ 9.633 mm.
| Magnetic flux density B (T) | Cyclotron frequency (MHz) |
|---|---|
| 0.375 | 5.71694 |
| 0.75 | 11.4339 |
| 1.12 | 17.1508 |
| 1.5 | 22.8678 |
| 1.88 | 28.5847 |
| 2.25 | 34.3017 |
| 3 | 45.7356 |
How to read the result
The frequency tells you what radio frequency a cyclotron must use for that particle and field. The same physics underlies mass spectrometers (lighter ions circle faster), magnetic confinement of plasmas and the spiralling of charged particles in Earth's magnetic field. Electrons have frequencies about 1,836 times higher than protons in the same field.
Common mistakes
- Using the electron mass for a proton, or the reverse.
- Forgetting the charge multiple for ions such as alpha particles (q = 2e).
- Using the non-relativistic formula at high energy.
- Mixing up f and ω (they differ by 2π).
Frequently asked questions
What is the cyclotron frequency of a proton in 1 T?
About 15.2 MHz; it scales linearly with field strength.
Why does the frequency not depend on speed?
The radius grows in proportion to speed, so the time per turn stays the same.
What is the electron cyclotron frequency?
About 28 GHz per tesla, used in electron cyclotron resonance heating of plasmas.
Sources: Lawrence, E. O. & Livingston, M. S. (1932). The Production of High Speed Light Ions Without the Use of High Voltages. Physical Review, 40, 19. · CODATA 2022 recommended values of the fundamental physical constants, NIST