The cyclotron frequency calculator above answers two related questions about a charged particle in a magnetic field: how many times per second it goes round, and how big the circle is. The first depends only on the particle's charge-to-mass ratio and the field strength. The second depends on how fast it is going as well. Keeping those two facts separate is most of the understanding this topic requires.
Arb Digital publishes free physics calculators that show their working. This page prints the angular frequency, the period, the gyroradius and the speed expressed as a fraction of the speed of light, so you can see at a glance whether the non-relativistic treatment it uses is still safe for your numbers. When it is not, the note under the results says so explicitly rather than quietly returning a wrong answer.
What This Cyclotron Frequency Calculator Does
A charged particle moving through a magnetic field feels a force perpendicular to both its velocity and the field. A force that is always perpendicular to the motion does no work and cannot change the speed, so it can only change the direction. The result is circular motion, and the magnetic force supplies exactly the centripetal force that circular motion requires.
Setting those two equal gives the radius, and dividing the circumference by the speed gives the period. When you do the algebra the speed cancels out of the period entirely. Faster particles travel further round a larger circle in exactly the same time. That cancellation is the entire reason the cyclotron works as an accelerator: a fixed-frequency alternating voltage stays in step with the particle no matter how much energy it has already gained.
The same frequency turns up under other names in other fields. Plasma physicists call it the gyrofrequency and use it to describe how tightly particles are tied to field lines. Spectroscopists meet it as the resonance condition in electron spin resonance and ion cyclotron resonance mass spectrometry. It is one equation with several vocabularies.
How to Use It
- Choose the particle or enter a custom charge and mass. For an ion, use the charge state in units of the elementary charge and the mass in atomic mass units.
- Enter the field in the unit your instrument reports. Laboratory magnets are usually quoted in tesla, geophysical fields in nanotesla or gauss.
- Set the kinetic energy for the radius. The frequency ignores it. If you only want a frequency, the energy value does not matter.
- Check the speed shown in the grid. Once it passes a few per cent of the speed of light, relativistic mass increase starts to shift the frequency and the classical answer drifts away from reality.
- Remember the perpendicular assumption. This tool treats all the velocity as perpendicular to the field. A particle with velocity along the field spirals instead, and only the perpendicular part sets the radius.
The Formula: How Cyclotron Frequency Is Calculated
Equating the magnetic force to the centripetal force gives qvB = mv²/r, so the radius is r = mv ÷ (qB). The period is the circumference divided by the speed, T = 2πr/v = 2πm ÷ (qB), in which the speed has vanished. The frequency is the reciprocal, f = qB ÷ (2πm), and the angular frequency is ωc = qB ÷ m. OpenStax University Physics Volume 2, section 11.3 on the motion of a charged particle in a magnetic field, derives both results in exactly this way and works through the helical case where the velocity has a component along the field.
Speed comes from the kinetic energy through v = √(2E/m) in the non-relativistic regime. The elementary charge is exact in the current SI at 1.602176634 × 10−19 C, and the electron's charge-to-mass ratio, which sets its cyclotron frequency directly, is given by NIST's CODATA value for the electron charge-to-mass quotient as 1.75882000838 × 1011 C/kg. The speed of light used for the fraction-of-c figure is the exact NIST CODATA value of 299,792,458 m/s.
Work the defaults through by hand. An electron in 0.1 T has f = (1.602176634 × 10−19 × 0.1) ÷ (2π × 9.1093837139 × 10−31) = 1.602177 × 10−20 ÷ 5.72359 × 10−30 = 2.799 × 109 Hz, that is 2.80 GHz. The period is 357 ps. A kinetic energy of 1 keV gives a speed of √(2 × 1.602177 × 10−16 ÷ 9.1094 × 10−31) = 1.8755 × 107 m/s, which is 6.26 per cent of the speed of light, and a radius of 9.1094 × 10−31 × 1.8755 × 107 ÷ 1.602177 × 10−20 = 1.066 mm.
Why the Frequency Stops Being Constant at High Energy
The independence of frequency from speed holds only while mass is constant. In special relativity the momentum is γmv rather than mv, so the effective inertia rises with speed and the cyclotron frequency falls as f = qB ÷ (2πγm). At one per cent of the speed of light the correction is about half a part in ten thousand and nobody cares. At half the speed of light γ is 1.155 and the frequency is 13 per cent low, which is enough to lose synchronisation with a fixed accelerating voltage entirely.
This is precisely why the classical cyclotron has an energy ceiling and why the machines that replaced it do something about it. A synchrocyclotron sweeps the driving frequency downwards as the particles gain energy. An isochronous cyclotron shapes the magnetic field so that B rises with radius, keeping the ratio B/γ constant. A synchrotron keeps the radius fixed and ramps both field and frequency together. Every one of those designs exists because of the γ in that denominator.
The tool warns when the computed speed exceeds a few per cent of c, because past that point the classical frequency it prints is an idealisation rather than a measurement you would expect to reproduce.
The Gyroradius and Why It Matters in Plasmas
The orbit radius, called the gyroradius or Larmor radius, is the length scale on which a magnetic field controls a particle. If the gyroradius is small compared with the size of the device, the particle is effectively glued to a field line and drifts along it, which is what magnetic confinement relies on. If it is comparable to the device, the particle wanders across field lines and is lost.
Because radius goes as mv/(qB), a proton at the same energy as an electron has a gyroradius about 43 times larger — the square root of the mass ratio — even though its charge magnitude is identical. In a fusion device this asymmetry between electron and ion orbit sizes drives a great deal of the transport physics. In the Earth's magnetosphere it is why energetic ions and electrons populate different regions.
Two particles of opposite sign in the same field circle in opposite senses at the same rate. That is the origin of the sign conventions that make cyclotron resonance able to distinguish charge polarity, and it is the basis of ion cyclotron resonance mass spectrometry, where measuring a frequency in a known field yields a mass with extraordinary precision.
How This Sits Next to the Other Charged-Particle Tools
This page covers circular motion in a static magnetic field. For the field itself, the solenoid magnetic field calculator and the magnetic field of a wire calculator give the two standard geometries, and the magnetic field converter handles tesla, gauss and amperes per metre. For acceleration by an electric field rather than deflection by a magnetic one, use the charge acceleration calculator and the electric field calculator.
To convert an accelerating voltage into an energy, the electron volt calculator does the arithmetic, the electron speed calculator turns that energy into a velocity, and the kinetic energy calculator covers the general case. The frequency period calculator and the angular velocity calculator handle the rotational bookkeeping, and in a plasma context the Debye length calculator gives the other length scale that matters alongside the gyroradius.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Expecting energy to change the frequency — it changes the radius only, until relativistic effects appear.
- Using the classical formula at high energy — the frequency falls as one over gamma, and by half the speed of light the error is more than a tenth.
- Feeding in the full speed when the motion is helical — only the component perpendicular to the field sets the radius; the parallel component just carries the particle along.
- Mixing gauss and tesla — a factor of ten thousand separates them, and geophysical and laboratory literature use different conventions as a matter of course.
- Assuming a uniform field — real fields have gradients and curvature, which add drift motions on top of the circular orbit that this calculation does not describe.
Related Free Tools From Arb Digital
Work out the field first with the solenoid magnetic field calculator or the magnetic field of a wire calculator, converting units with the magnetic field converter. Get the particle's energy from the electron volt calculator, its speed from the electron speed calculator or the kinetic energy calculator, and its behaviour in an electric field from the charge acceleration calculator and the electric field calculator. For plasma work, pair the gyroradius with the Debye length calculator. The frequency period calculator and angular velocity calculator round out the set on the free online tools hub.
Frequently Asked Questions
Because a faster particle travels a proportionally larger circle. The radius rises in exact proportion to the speed, so the circumference and the time to cover it scale together and cancel. The period depends only on mass, charge and field, which is the property that made the fixed-frequency cyclotron possible.
They are the same quantity under different names. Accelerator physicists say cyclotron frequency, plasma physicists say gyrofrequency or Larmor frequency. Some texts reserve the angular form, in radians per second, for one name and the ordinary frequency in hertz for the other, so check which one a formula expects before substituting.
The frequency falls by one over gamma, so the error is around half a part in ten thousand at one per cent of the speed of light, about one per cent at fourteen per cent of c, and thirteen per cent at half of c. For most laboratory electron work above a few tens of keV the correction is already worth carrying.
It sets the plane of the orbit and the sense of rotation. Positive and negative charges circle in opposite directions in the same field, at the same rate. Only the velocity component perpendicular to the field produces circular motion; any component along the field is unaffected, which turns the circle into a helix.
It is the length scale over which a magnetic field controls a particle. When it is small compared with the size of a device, particles follow field lines closely, which is the basis of magnetic confinement. When it is comparable to the device, particles cross field lines and escape, so the ratio is a design constraint in fusion and in accelerator beam optics.
Yes. Select the custom option and enter the charge state in units of the elementary charge and the mass in atomic mass units. A singly ionised argon atom, for example, has charge 1 and mass about 40. The frequency scales with charge over mass, so heavy ions circle far more slowly than electrons in the same field.
Because frequency is proportional to charge divided by mass, and a proton is about 1,836 times heavier than an electron while carrying the same magnitude of charge. In the same field an electron therefore circles roughly 1,836 times faster, which is why electron resonance sits in the microwave range while proton resonance sits far lower.
This tool is provided for educational use. It assumes a uniform static magnetic field, motion entirely perpendicular to that field, and non-relativistic dynamics, and it does not model gradient or curvature drifts, collisions or radiation losses.
