Cyclotron Frequency Calculator - Charge, Mass, and Field

A cyclotron frequency calculator finds the frequency at which a charged particle orbits when a uniform magnetic field bends its path into a circle. Enter charge, magnetic field, and particle mass, and the calculator returns angular frequency in rad/s, linear frequency in Hz, period in seconds, and orbit speed in m/s when a radius is given.

• • Particle Accelerator Sizing: Pick the rf-field switching frequency for a cyclotron or synchrocyclotron by computing qB over 2 pi m for the chosen particle and field.
• • Mass Spectrometer Calibration: Match a measured resonance frequency to a known charge to mass ratio to identify the ion hitting the detector.
• • Plasma and Fusion Calculations: Estimate the gyrofrequency of ions and electrons in a tokamak or magnetic confinement plasma before plugging it into transport models.
• • Homework and Lab Verification: Check the numerical answer to a cyclotron frequency problem and see how the orbit period scales with charge, mass, and field strength.

The cyclotron frequency only depends on the charge to mass ratio and the magnetic field, not on the particle speed or orbit radius. That independence is what lets a cyclotron keep accelerating the same particle on every pass: the switching frequency of the electric field can stay fixed while the particle spirals outward.

When the cyclotron result is reported in rad/s, the Angular Frequency Calculator gives the same omega to Hz conversion for any pure rotation or simple harmonic motion.

The calculator reads charge q, magnetic field B, mass m, and an optional radius r, then applies omega equals qB over m. The linear cyclotron frequency, period, and orbit speed follow from omega in one step.

• f: Linear cyclotron frequency in hertz (cycles per second).
• omega_c: Angular cyclotron frequency in radians per second.
• q: Charge of the moving particle in coulombs.
• B: Strength of the uniform magnetic field perpendicular to the motion in tesla.
• m: Rest mass of the moving particle in kilograms.
• r: Optional orbit radius in meters, used only for the velocity row.

The preset selector fills charge and mass with the matching CODATA constant for a proton, electron, alpha particle, or deuteron. Choosing Custom frees both fields so any charged particle can be entered by hand.

The orbital radius is optional. When it stays at zero the calculator reports only the frequency outputs; when it is greater than zero the orbit speed v equals omega times r is reported in m/s.

According to NIST CODATA elementary charge, the elementary charge is exactly 1.602176634 x 10 to the negative 19 coulombs, which sets the charge used for the proton, positron, and deuteron presets

According to Hyperphysics cyclotron frequency, a moving charge in a uniform magnetic field orbits at angular frequency omega equals qB over m, which gives the cyclotron frequency f equals qB over 2 pi m

Because the cyclotron formula balances the magnetic force against m v squared over r, the Centrifugal Force Calculator is the natural place to read off the matching centripetal acceleration for the same orbit.

Four ideas make every cyclotron frequency result easier to interpret: Lorentz force, charge to mass ratio, gyrofrequency, and the magnetic field as centripetal force.

These four concepts are the same ones used in plasma physics, mass spectrometry, and the derivation of cyclotron resonance. They are the building blocks that the cyclotron frequency calculator turns into a numerical result on screen.

The angular cyclotron frequency omega_c is the same kind of quantity as the angular velocity in rotational kinematics, so the Angular Velocity Calculator handles the rad/s reading when the problem is phrased in revolutions per minute.

Pick a particle preset, enter the magnetic field, optionally enter an orbit radius, and read the cyclotron frequency in Hz and rad/s along with the orbital period and speed.

1. 1 Pick a particle preset: Use the Particle Preset dropdown to choose Proton, Electron, Alpha, or Deuteron. Charge and Mass fill with CODATA values. Choose Custom to enter your own.
2. 2 Enter the magnetic field: Type the uniform magnetic field strength B in tesla. Most textbook problems use 0.1 T to 2 T.
3. 3 Override charge or mass if needed: Switch the preset to Custom and edit Charge or Mass to handle heavier ions, isotopes, or hypothetical particles.
4. 4 Add an orbit radius for the velocity row: Enter the orbit radius r in meters when you need the cyclotron velocity. Leave it at 0 to skip the velocity row.
5. 5 Read omega, f, T, and v: The primary result is omega in rad/s. The same motion appears as f in Hz, T in seconds, and v in m/s.

For a proton in a 1 T field, leave the preset on Proton, type 1 in the magnetic field box, leave the radius at 0, and read omega equals 9.579e7 rad/s, f equals 15.24 MHz, and T equals 65.6 ns.

When the same particle is treated as a satellite in a central field instead of a charge in a uniform magnetic field, the Orbital Period Calculator returns the matching orbital period for a given mean radius.

The calculator removes the unit math, applies the same formula to any charged particle, and reports the frequency in both rad/s and Hz so it drops straight into the next physics step.

• • CODATA Presets: Proton, electron, alpha, and deuteron presets use NIST CODATA values, so the cyclotron frequency comes out at the same precision published in accelerator physics tables.
• • Custom Ions Supported: Switching to Custom lets you enter any charge and mass, so heavier ions, exotic isotopes, and homework hypotheticals are not blocked by the four defaults.
• • Both Frequency Units: The calculator shows omega in rad/s and f in Hz together. SHM and AC circuit formulas want rad/s; accelerator and rf-system formulas want Hz, so both are kept on screen at once.
• • Orbit Velocity From Radius: Entering an orbit radius adds the cyclotron velocity v equals omega times r in m/s, which is the next quantity needed when the kinetic energy is being estimated.
• • Inline Relativistic Caveat: When the orbit radius and field push v above the speed of light, the calculator surfaces a caveat next to the velocity row so the classical formula is not silently misapplied.

The cyclotron frequency calculator combines CODATA presets, both frequency units, and the optional radius row, which makes this the right tool when the next step is an accelerator design, a mass spectrometer check, or an undergraduate homework problem.

Plasma physics extends the cyclotron formula to magnetohydrodynamics, and the Alfven Velocity Calculator covers the Alfven wave speed that shows up alongside the cyclotron frequency in fusion and space plasma problems.

Three things drive the cyclotron frequency: the charge to mass ratio, the magnetic field strength, and whether the orbit speed stays non-relativistic. The orbit radius never enters the frequency formula itself.

• • The classical cyclotron formula ignores relativistic effects. When the orbit velocity implied by the entered radius and field exceeds about one tenth of the speed of light, treat the result as an upper bound.
• • The magnetic field is treated as perpendicular to the velocity and perfectly uniform. A field with a parallel component, or one that varies across the orbit, changes the resonance frequency in a way this calculator does not model.

The right way to interpret a cyclotron frequency is as the resonance frequency of the classical circular motion, with relativistic corrections applied when the particle energy gets close to its rest energy.

When the orbit velocity exceeds the speed of light, the calculator still reports the numerical value but flags it as a sign that relativistic effects need to be added.

According to NIST CODATA proton mass, the proton mass is 1.67262192595 x 10 to the negative 27 kilograms, which sets the scale for every proton cyclotron frequency the calculator returns

Accelerators combine cyclotron frequency with conservation of momentum at every crossing, so the Conservation of Momentum Calculator is the right place to check the momentum transfer for the same charge and field.