Cyclotron Frequency Interactive Calculator

When you’re working on a particle accelerator, tuning a mass spectrometer, or looking at plasma in a lab, knowing how fast a charged particle orbits in a magnetic field is more than an academic detail—it’s make-or-break for your device. Use the Cyclotron Frequency Interactive Calculator below to get angular frequency, radius, period, velocity, kinetic energy, or the necessary field strength, given the particle’s charge, mass, and your chosen field. Small errors here can break frequency matching and kill performance, especially in things like isotope production, plasma studies, or ICR mass spectrometry. You’ll find the full equations, a real-world worked example, practical engineering background, and some honest FAQ on limits like relativistic effects and how things work in the lab.

What is cyclotron frequency?

Cyclotron frequency tells you how fast a charged particle completes an orbit in a steady magnetic field. It depends only on the charge-to-mass ratio and how strong your field is. Speed of the particle doesn’t change it—until relativity catches up.

Simple Explanation

Picture spinning a ball attached to a string: pull harder, make it go faster, the loop gets wider, but the time to finish each circle stays the same. Swap your ball for a charged particle and your string for a magnetic field and you’re looking at cyclotron motion. The speed changes the orbit’s size, but not the rate at which the particle goes around. That rate is the cyclotron frequency.

📐 Browse all 1000+ Interactive Calculators

Visual Diagram

Cyclotron Frequency Calculator

How to Use This Calculator

1. Pick a calculation mode—frequency, field, radius, period, velocity, or kinetic energy.
2. Enter the particle charge and mass directly, or use a quick-select option to fill these from presets.
3. Type in your required input values: this changes depending on the calculation (e.g., you’ll need a velocity or a field for certain modes).
4. Press Calculate to get the result.

<> Embed this calculator → <> Embed this animation calculator →

Fundamental Equations

Here are the equations you’ll actually use for cyclotron frequency and related parameters.

Simple Example

An electron (q = 1.602 × 10⁻¹⁹ C, m = 9.109 × 10⁻³¹ kg) moves through a 0.5 T magnetic field.

• Angular frequency: ω = (1.602 × 10⁻¹⁹ × 0.5) / 9.109 × 10⁻³¹ = 8.794 × 10¹⁰ rad/s
• Linear frequency: f = 8.794 × 10¹⁰ / (2π) = 1.400 × 10¹⁰ Hz (14.0 GHz)
• Orbital period: T = 1 / f = 7.14 × 10⁻¹¹ s (71.4 ps)

Theory & Practical Applications

Cyclotron frequency is the basic rate at which a charged particle will go around when placed in a uniform magnetic field that's perpendicular to its velocity. It comes from balancing the Lorentz force with the centripetal force. As long as things are non-relativistic, this frequency depends only on charge, mass, and field—not on speed. That’s why it’s so widely used in particle accelerators and mass specs.

Fundamental Physics of Cyclotron Motion

A charged particle moving perpendicular to a steady magnetic field feels a force at a right angle to both its velocity and the field (F = qv × B). That pins it to a circular path. Setting this magnetic force equal to the required centripetal force (qvB = mv²/r), you get the orbital radius: r = mv/(qB).

Work out the angular frequency: ω = v/r = qB/m. The velocity cancels; that means all particles with the same charge-to-mass ratio have the same cyclotron frequency in a given field, regardless of speed. If particles go faster, the orbit gets larger, but the period—one loop—doesn’t change. Once you get close to relativistic speeds, use γm₀ instead of m for mass. Cyclotron frequency then drops as energy increases—this is the hard stop for classical cyclotrons.

In accelerators, you usually drive particles with an RF electric field at this frequency. The frequency match is the whole trick: you accelerate only when the driving field stays locked to the cyclotron frequency as the particle gains energy and radius.

Mass Spectrometry and Analytical Applications

Ion cyclotron resonance mass specs (ICR-MS) hang their resolution on how tight you can measure cyclotron frequency. Ions are trapped and pushed out to a bigger orbit with an RF field at their cyclotron frequency. The induced signal is detected and converted using Fourier transform; the frequencies give you mass: m/q = B/(2πf). Because it’s based on frequency, you can get extremely high resolution—good enough to sort molecules differing by fractions of an atomic mass unit. Fields run 7-21 Tesla in these machines, and you need a solid vacuum to keep ions from damping out.

Space Plasma Physics and Magnetospheric Dynamics

In Earth's magnetic field or elsewhere in space, cyclotron motion is what makes charged particles spiral around. For electrons in geospace (say 0.3–3.0 μT), their frequencies are 8 Hz to 80 Hz; for protons, much lower. These aren't arbitrary—they match the frequencies of naturally occurring EM waves, which can scatter and shift particle paths, resulting in things like the aurora. Understanding these frequencies lets you predict when and where energetic particles get dumped into the atmosphere.

Medical Applications: Cyclotron-Based Isotope Production

Medical cyclotrons make radioisotopes for PET scans using moderate-energy protons and fields around 1.5–2.0 Tesla. The cyclotron frequency (for instance, 27.5 MHz for 1.8 T with protons) sets your radio frequency drive. Reliability depends on maintaining resonance as the protons spiral outward and pick up energy. Most commercial systems use negative hydrogen ions and strip both electrons for easy extraction, sidestepping complex beam-switching solutions and letting one machine serve multiple targets.

Worked Example: Electron in a Mass Spectrometer

Problem: An electron beam is injected perpendicular to a 0.875 Tesla magnetic field in an ICR cell, after acceleration through 247 V. Compute (a) cyclotron frequency; (b) orbital radius; (c) orbital period; (d) kinetic energy, in J and eV.

Given:

• Electron charge: q = -1.602 × 10⁻¹⁹ C (use absolute value)
• Electron mass: m = 9.109 × 10⁻³¹ kg
• B = 0.875 T
• V = 247 V

Part (a): Cyclotron Frequencies

Angular frequency is:

ω = qB / m = (1.602 × 10⁻¹⁹ × 0.875) / (9.109 × 10⁻³¹)

ω = 1.402 × 10⁻¹⁹ / 9.109 × 10⁻³¹

ω = 1.539 × 10¹¹ rad/s

Linear frequency:

f = ω / (2π) = 1.539 × 10¹¹ / (2π)

f = 2.449 × 10¹⁰ Hz = 24.49 GHz

This frequency puts you in the microwave band—electronics for direct pickup need to handle this range.

Part (b): Orbital Radius

First: KE from the acceleration voltage:

KE = qV = (1.602 × 10⁻¹⁹ C)(247 V) = 3.957 × 10⁻¹⁷ J

From KE = ½mv², solve for v:

v = √(2KE / m) = √(2 × 3.957 × 10⁻¹⁷ / 9.109 × 10⁻³¹)

v = √(8.687 × 10¹³) = 9.321 × 10⁶ m/s

That's about 3.1% of c, so non-relativistic math is reasonable here.

Now work out r:

r = mv / (qB) = (9.109 × 10⁻³¹)(9.321 × 10⁶) / [(1.602 × 10⁻¹⁹)(0.875)]

r = 8.490 × 10⁻²⁴ / 1.402 × 10⁻¹⁹

r = 6.057 × 10⁻⁵ m = 60.57 μm

This is a very small orbit by human standards but typical in this kind of setup.

Part (c): Orbital Period

T = 2π / ω = 2π / (1.539 × 10¹¹)

T = 4.083 × 10⁻¹¹ s = 40.83 ps

Or, T = 1/f:

T = 1 / (2.449 × 10¹⁰ Hz) = 4.083 × 10⁻¹¹ s ✓

This electron orbits about 24.5 billion times per second.

Part (d): Kinetic Energy (check)

We already have KE = 3.957 × 10⁻¹⁷ J. In eV:

KE (eV) = KE (J) / (1.602 × 10⁻¹⁹)

KE = 3.957 × 10⁻¹⁷ / 1.602 × 10⁻¹⁹

KE = 247 eV

So the calculations check out and match the accelerating voltage input.

Meaning: Orbits here (∼60 micron radius) are small enough to keep plenty of electrons in a small device. The very high cyclotron frequency means your detection electronics need to work at microwave or millimeter-wave frequencies. For ions in real FT-ICR work, things slow down because the masses are much larger, so more approachable MHz-range electronics can be used.

Relativistic Corrections and High-Energy Limitations

Once your particle gets above about 10% of the speed of light, you have to use relativistic mass—m replaced by γm₀, where γ = 1/√(1 - v²/c²). This lowers the frequency as energy increases. Protons get into trouble above about 25 MeV; cyclotrons have to back off or use frequency ramps. Synchrotrons and synchrocyclotrons adjust the RF frequency or field as you go. With electrons, relativistic effects are an issue even below 1 MeV. That’s why electron-specific cyclotrons switch to ECR or linear acceleration at higher energies.

Practical Considerations for Laboratory Systems

For precise work, magnetic field uniformity is the main pain point. High-end mass specs require one part per million or better field homogeneity—superconducting magnets plus careful shimming. Even tiny field errors will shift the cyclotron frequency and blur your results. If your field drifts with temperature, so does your frequency. High-spec setups use active temperature and field stabilization, and sometimes compensate for tiny distortions caused by space charge or other effects. For reference or extra calculations, see the main calculator library.

Frequently Asked Questions