What this calculator does
A charged particle moving through a magnetic field travels in a circle, and the rate at which it goes round does not depend on how fast it is moving. A faster particle simply traces a proportionally larger circle in the same time.
That independence is what made the cyclotron possible. A fixed accelerating frequency keeps working as the particle speeds up and spirals outward, which is why the design was such a breakthrough in the 1930s.
The formula
The angular frequency is charge times magnetic field divided by mass. Dividing by 2π converts it to cycles per second. This holds while the particle remains non-relativistic.
| Term | Meaning |
|---|---|
| Cyclotron frequency | The rotation rate of a charged particle in a magnetic field, independent of its speed. |
| Lorentz force | The force on a moving charge in a magnetic field, always perpendicular to the motion, which is what curves the path into a circle. |
| Non-relativistic | The condition for this simple form. At relativistic speeds the effective mass rises and the frequency drops. |
The inputs explained
| Field | What to enter |
|---|---|
| Particle charge (electron ≈ 1.602176634e-19) (C) | The particle charge in coulombs. An electron or proton carries about 1.602 × 10⁻¹⁹. |
| Magnetic field strength (T) | The magnetic field strength in tesla. |
| Particle mass (electron ≈ 9.1093837015e-31) (kg) | The particle mass in kilograms. |
When to use it
Designing a cyclotron
The accelerating voltage must alternate at exactly this frequency to keep pushing the particle each half turn.
Understanding mass spectrometry
Ions of different mass circle at different frequencies in the same field, which is one way of separating them.
Analysing plasma behaviour
Charged particles in a magnetised plasma gyrate at this frequency, which governs much of the plasma's response.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does field strength change the orbit rate?
The same particle in fields of increasing strength.
| Magnetic field | Cyclotron frequency | Orbital period |
|---|---|---|
| 0.5 T | 1.3996e+10 Hz | 7.1448e-11 s |
| 1 T | 2.7992e+10 Hz | 3.5724e-11 s |
| 2 T | 5.5985e+10 Hz | 1.7862e-11 s |
Questions
Why does speed not affect the frequency?
Because a faster particle experiences a proportionally larger magnetic force, which curves it into a proportionally larger circle. The larger path and the higher speed cancel exactly, leaving the time per orbit unchanged.
Why did cyclotrons eventually hit a limit?
Relativity. As particles approach light speed their effective mass rises, which lowers the cyclotron frequency and puts them out of step with a fixed accelerating voltage. Synchrocyclotrons and synchrotrons were developed to handle this.
Does the direction of the field matter?
It sets the plane of the circle and the direction of rotation, but not the frequency. Only motion perpendicular to the field is curved; any component along the field continues unaffected, producing a helical path.
Do heavier particles circle more slowly?
Yes, inversely with mass. A proton is around 1,836 times heavier than an electron, so in the same field it circles 1,836 times more slowly.
For the field a current produces, see the solenoid magnetic field calculator. For angular motion generally, see the angular velocity calculator.