Cyclotron Frequency Calculator
Calculate orbital frequency, angular frequency, and period for a charged particle in a uniform magnetic field.
About cyclotron frequency
A charged particle moving perpendicular to a uniform magnetic field experiences the magnetic part of the Lorentz force. That force is always perpendicular to the instantaneous velocity, so it changes the direction of motion without changing speed. The resulting path is circular in the ideal nonrelativistic case. Equating magnetic force with centripetal force gives angular cyclotron frequency ωc = |q|B/m, where q is particle charge, B is magnetic-field magnitude, and m is particle mass. Ordinary frequency in cycles per second is fc = |q|B/(2πm). The orbital period is its reciprocal, T = 1/fc = 2πm/(|q|B). Frequency therefore rises linearly with magnetic-field strength and charge magnitude, while a more massive particle circles more slowly. The sign of charge does not change the frequency magnitude; instead, it reverses the direction of rotation. This calculator uses the absolute charge when reporting frequency and period. Enter charge in coulombs, magnetic field in teslas, and mass in kilograms. Scientific notation is convenient for elementary particles. An electron has charge magnitude approximately 1.602176634 × 10^-19 C and mass approximately 9.1093837 × 10^-31 kg. A proton has the same charge magnitude and much greater mass, so its cyclotron frequency in the same field is much lower. The entered field is treated as the component perpendicular to motion for the circular orbit interpretation. The formula assumes a spatially uniform, constant magnetic field and neglects electric fields, radiation losses, collisions, and field gradients. A velocity component parallel to the field produces helical motion, but the transverse rotation retains the same ideal frequency. At speeds approaching the speed of light, relativistic momentum increases the effective mass by the Lorentz factor, reducing the observed frequency. In that situation, use relativistic mass or a dedicated relativistic treatment rather than the rest mass alone. Cyclotron frequency is fundamental to cyclotrons, mass spectrometers, plasma diagnostics, ion traps, magnetic-resonance systems, and space physics. Real instruments also depend on field calibration, particle energy, synchronization, and geometry. Use the calculation as a transparent ideal estimate and preserve enough significant figures in constants for the accuracy your application requires.
Cyclotron frequency examples
Representative particles illustrate the strong charge-to-mass dependence.
| Particle and field | Approximate frequency | Context |
|---|---|---|
| Electron in 50 µT | 1.40 MHz | Comparable to Earth's surface magnetic field. |
| Proton in 1 T | 15.25 MHz | A common order of magnitude in magnetic systems. |
| Alpha particle in 1 T | 7.68 MHz | Twice the proton charge and about four times its mass. |
How to calculate cyclotron frequency
- Enter the particle's signed electric charge in coulombs.
- Enter the uniform magnetic-field magnitude in teslas.
- Enter the particle mass in kilograms.
- Select Calculate to obtain frequency, angular frequency, and period.
- Apply relativistic corrections separately when particle speed is very high.
Frequently asked questions
Does negative charge produce a negative frequency?
No, frequency is reported as a positive magnitude using absolute charge. Negative charge reverses the direction of gyration instead.
Does particle speed affect cyclotron frequency?
Not in the ideal nonrelativistic equation, where speed changes orbit radius but not frequency. Relativistic speeds increase effective inertia and reduce frequency.
What if velocity is not perpendicular to the field?
The perpendicular component produces circular motion while the parallel component continues along the field. Together they form a helix with the same ideal transverse frequency.
Why is proton frequency lower than electron frequency?
Their charge magnitudes are equal, but a proton is roughly 1,836 times more massive. Since frequency is inversely proportional to mass, the proton rotates much more slowly.
Can I use magnetic field in gauss?
Convert gauss to teslas before entering it. One gauss equals 0.0001 tesla.