Cyclotron frequency calculator
A charge of 1 coulomb with 1 kilogram of mass in a 1 tesla field circles at about 0.159 hertz. Enter charge, mass and field strength to get the frequency, the angular frequency and the time for one loop. Free in your browser.
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Run — free
Worked example
- Charge
- 1
- Mass
- 1
- Field strength
- 1
- Abs charge
- 1
- Frequency
- 0.1591549431
- Angular frequency
- 1
- Frequency unit
- Hz
The cyclotron frequency of a charged particle is how many gyration cycles it completes each second while the Lorentz force bends its velocity around a uniform magnetic field. This calculator accepts three SI numbers—charge in coulombs, rest mass in kilograms, and magnetic field strength in tesla—and returns the ordinary cyclotron frequency in hertz from the closed-form identity f equals the absolute value of q times B divided by two pi m. Angular cyclotron frequency omega equals the absolute value of q times B over m is returned in radians per second, together with the gyration period when frequency is positive. Charge may be signed so electrons, protons, and ions share one contract; frequency uses the absolute charge so the result is never negative.
What cyclotron frequency means for a charged particle in a magnetic field
When a charged particle moves in a uniform magnetic field, the magnetic part of the Lorentz force is always perpendicular to velocity, so kinetic energy is unchanged and the trajectory in the plane perpendicular to B is a circle, or a helix if there is also a parallel velocity component. The time to complete one loop is then the circumference over that speed, which simplifies to T equals two pi m over |q| B, independent of kinetic energy in the non-relativistic limit.
Ordinary frequency in hertz is the reciprocal: f equals |q| B over two pi m. That independence of speed is why a classical cyclotron can keep a constant radio-frequency dee voltage while ions spiral outward. Angular frequency omega equals |q| B over m is the same quantity in radians per second and is the symbol plasma physicists often call the cyclotron frequency or gyrofrequency.
Doubling B doubles f; doubling mass halves f; an electron gyrates roughly eighteen hundred times faster than a proton in the same field because of the mass ratio.
What to enter and what you get back
If mass is zero, negative, or so small that it is treated as non-positive, the task fails because f equals |q| B over two pi m would diverge. If field strength is not positive, the task fails with a message that field strength must be positive. A pedagogical check: charge one coulomb, mass one kilogram, field strength one tesla yields frequency one over two pi hertz, about 0.159 hertz.
An electron-scale check with charge 1.602 times ten to the minus nineteen coulombs, mass 9.109 times ten to the minus thirty-one kilograms, and one tesla yields tens of gigahertz, the well-known electron cyclotron frequency near twenty-eight gigahertz in a one-tesla field. Failed validation is not billed.
Units, laboratory numbers, and what this calculator is not
Keep units strictly SI: convert grams to kilograms, atomic mass units to kilograms, gauss to tesla (one tesla is ten thousand gauss), and elementary charges to coulombs before you call, because this calculator does not expand prefixes or unit symbols. For a proton, charge is plus the elementary charge and mass is about 1.673 times ten to the minus twenty-seven kilograms; for an electron, charge is minus the elementary charge and mass is about 9.109 times ten to the minus thirty-one kilograms.
The sign of the charge does not change f, but the mass ratio does, which is why electron cyclotron resonance sits in the microwave band while proton cyclotron resonance in the same tesla-scale field sits in the radio band. Relativistic cyclotron motion replaces m with gamma m, which this non-relativistic identity does not apply; if your particle is a sizable fraction of the speed of light, multiply mass by gamma yourself or use a dedicated relativistic tool.
Typical lab numbers: a one-tesla field gives an electron cyclotron frequency near twenty-eight gigahertz and a proton cyclotron frequency near fifteen megahertz, figures used in electron-cyclotron resonance heating, magnetron design, Penning traps, and Fourier-transform ion cyclotron resonance mass spectrometry. This landing is for people searching cyclotron frequency, gyrofrequency, or f equals q B over two pi m; it is not a cyclotron radius calculator, not a Lorentz force magnitude tool, not an LC resonant-frequency calculator, and not a converter from gauss to tesla.
Prefer this capability when you already have q, m, and B and only need the gyration frequency in hertz.
What you can do with it
Plasma and fusion gyrofrequency checks
Compute electron or ion cyclotron frequencies in tesla-scale tokamak and stellarator fields before comparing them with heating-system bands.
Mass spectrometry and Penning traps
Obtain the ion cyclotron frequency |q|B/(2πm) that FT-ICR and Penning-trap experiments use to convert measured f into mass-to-charge ratio.
Classroom and homework oracles
Let students submit q, m, and B and verify f in hertz, including deliberate zero-mass and zero-field rejections, against a deterministic result.
FAQ
How is cyclotron frequency calculated from charge, mass, and magnetic field?
The ordinary cyclotron frequency in hertz is f = |q| B / (2 π m), with q in coulombs, B in tesla, and m in kilograms. Angular frequency ω = |q| B / m is returned in rad/s as well.
Why must mass and field strength be positive?
A zero or negative mass would send f = |q|B/(2πm) through a non-physical denominator, and a non-positive field is not a gyration.
Does the sign of the charge change the cyclotron frequency?
No. Frequency uses the absolute charge |q|, so electrons (negative q) and protons (positive q) return a non-negative f. Sense of rotation is not reported.
What units should I send for q, m, and B?
SI only: charge in coulombs, mass in kilograms, field strength in tesla.
Does the cyclotron frequency depend on how fast the particle moves?
No, as long as the speed is not relativistic. A faster particle runs a wider circle in the same time, so the frequency stays fixed. That is why a classical cyclotron can keep a constant radio-frequency voltage while ions spiral outward.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/phys/cyclotron-frequency \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"charge":1.60217662e-19,"mass":9.1093837e-31,"field_strength":1}'const res = await fetch("https://api.kit.forhosting.com/phys/cyclotron-frequency", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"charge": 1.60217662e-19,
"mass": 9.1093837e-31,
"field_strength": 1
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/phys/cyclotron-frequency",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"charge": 1.60217662e-19,
"mass": 9.1093837e-31,
"field_strength": 1
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/phys/cyclotron-frequency", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"charge":1.60217662e-19,"mass":9.1093837e-31,"field_strength":1}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"charge":1.60217662e-19,"mass":9.1093837e-31,"field_strength":1}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/phys/cyclotron-frequency", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"charge": 1.60217662e-19,
"mass": 9.1093837e-31,
"field_strength": 1
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "phys.cyclotron_frequency",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
Mass and field strength must be strictly positive: a zero or negative mass would send the formula through a non-physical denominator, and a non-positive field is rejected the same way with invalid_input instead of Infinity.
What cyclotron frequency means for a charged particle in a magnetic field
The centripetal requirement that mass times perpendicular speed squared over radius equals the magnitude of q times perpendicular speed times B cancels the speed and leaves the gyroradius equal to m v_perp over |q| B. This endpoint uses SI throughout and returns the hertz form the query asked for, not a relativistic gamma correction and not a laboratory radio frequency in megahertz unless you scale the result yourself. Zero or negative mass, and zero or negative field strength, leave the classical model, so they fail with invalid_input instead of returning NaN or a signed nonsense frequency that a grader might treat as success.
Input contract, response shape, and error behaviour
Send a JSON body with three required finite numbers. charge, alias q, is the particle charge in coulombs; it may be positive, negative, or zero. mass, alias m, is the rest mass in kilograms and must be strictly positive. field_strength, aliases B, magnetic_field, tesla, and b, is the magnetic field magnitude in tesla and must be strictly positive. Missing fields, empty strings, non-numeric values, and non-finite numbers fail with invalid_input naming the field. On success the body echoes the cleaned charge, mass, and field_strength, sets abs_charge to the absolute charge used in the formula, frequency to the ordinary cyclotron frequency in hertz as the primary result, angular_frequency to |q| B over m in radians per second, period to one over frequency in seconds when frequency is positive (the period key is omitted when charge is zero and frequency is therefore zero), formula to the string f = |q| B / (2 π m), and SI unit labels for every quantity. Successful API calls cost $0.002 per request.
Units, laboratory numbers, and what this calculator is not
Round-trip checks are easy: angular_frequency should equal two pi times frequency, and period should equal one over frequency when frequency is positive. Invalid_input tasks do not consume prepaid balance, which keeps exploratory scripts cheap when learners deliberately set mass or field to zero.
Technical questions
What does the cyclotron frequency calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.