Cyclotron Frequency, Orbit Radius and the Velocity Selector

The one surprising fact about a charged particle in a magnetic field is that the time it takes to go round does not depend on how fast it is going. That is what makes a cyclotron possible. This gives you f = qB/2πm, the orbit radius, the energy a machine of a given size can reach, and the E/B of a velocity selector.

Last updated 11 Nov 2026

Cyclotron frequency, radius and crossed-field selector

f = qB/2πm · r = mv/qB · v = E/B

Magnetic flux density (T)The cyclotron’s field, between the poles of its big electromagnet. Classical cyclotrons ran at one or two tesla; a modern superconducting medical machine reaches three or four. Note that the frequency is proportional to B and the radius inversely proportional to it, so a stronger magnet means a faster, smaller machine.

Mass of the particle (atomic mass units, u)In unified atomic mass units. A proton is 1.007276 u, a deuteron 2.013553 u, an alpha particle 4.001506 u and an electron 0.000548580 u. The atomic mass constant is 1.660 539 068 92(52) × 10⁻²⁷ kg (CODATA 2022). Everything on this page depends on the charge-to-mass ratio and on nothing else about the particle.

Charge of the particle (multiples of e)1 for a proton or a deuteron, 2 for an alpha particle, 1 for an electron (the sign only reverses the direction of circulation, not the frequency). The elementary charge is exactly 1.602 176 634 × 10⁻¹⁹ C by the SI definition.

Kinetic energy of the particle (keV)In kiloelectronvolts, which is how accelerator people quote it. The energy sets the speed and therefore the radius — but not the frequency, which is the whole point of the machine. Keep an eye on the v/c row: above about a tenth of the speed of light the non-relativistic arithmetic here starts to drift.

Electric field in the velocity selector (V/m)For the second half of the page. A velocity selector crosses an electric field with a magnetic one so that the electric force qE and the magnetic force qvB cancel. They cancel for one speed only, v = E/B, and that speed does not depend on the charge or the mass at all — which is exactly what makes the device useful in front of a mass spectrometer.

On the left, the spiral path inside a cyclotron's two dees, drawn to the orbit radius your numbers give: raise the energy or weaken the field and the spiral really does grow. The particle crosses the gap between the dees twice per turn and is accelerated each time, at a frequency that never needs changing. On the right, the velocity selector: the vertical arrows are the electric field and the crossed circles are the magnetic field going into the page, and the number of each follows the strength you enter. Only one speed goes straight through.

12.1962MHzExample

A 500 keV proton (1.007276 u, charge e) in a 0.80 T field, selector at 2.5 × 10⁵ V/m

The frequency with no speed in it

f = qB/2πm · T = 2πm/qB · r = mv/qB · K_max = (qBR)²/2m · velocity selector: v = E/B

f = qB/2πm: the cyclotron frequency. It comes from setting the magnetic force qvB equal to the centripetal force mv²/r: the v cancels out of the period entirely. Everything about it is fixed by the charge-to-mass ratio and the field — a proton at 1 keV and the same proton at 10 MeV go round at the same rate, as long as both are slow compared with light
r = mv/qB: the orbit radius, from the same force balance. OpenStax gives it in this form. It is what a mass spectrometer measures and what a bubble-chamber photograph lets you read momentum from, because mv = qBr
Kmax = (qBR)²/2m: the energy a cyclotron of radius R can deliver, found by putting r = R into the radius formula and converting to kinetic energy. It goes as B² and as R², which is why accelerators grow expensive so quickly
v = E/B: the velocity selector. qE balances qvB when v = E/B, and the charge cancels, so the selected speed is the same for every particle whatever its charge or mass. The two fields and the velocity must be mutually perpendicular for this to hold
where it breaks: at relativistic speeds. The true orbital frequency is qB/2πγm, and γ grows with energy, so the particle arrives at the gap progressively late and eventually out of phase. The rows above give γ and the percentage error so you can see when it starts to matter

Worked example

A 500 keV proton (1.007276 u, charge e) in a 0.80 T field, selector at 2.5 × 10⁵ V/m
GET THE MASS AND CHARGE IN SI. m = 1.007276 × 1.660 539 069 × 10⁻²⁷ = 1.6726 × 10⁻²⁷ kg; q = 1.602 176 634 × 10⁻¹⁹ C. Their ratio, q/m = 9.579 × 10⁷ C/kg, is the only thing about the proton that the frequency needs.
DIVIDE BY 2π. f = qB/2πm = (9.579 × 10⁷ × 0.80)/(2π) = 7.663 × 10⁷/6.2832 = 1.2196 × 10⁷ Hz = 12.1962 MHz. The period is 82.0 ns. That is the frequency the oscillator across the dees has to run at, and it does not change as the proton speeds up.
NOW THE ENERGY, WHICH THE FREQUENCY DID NOT NEED. 500 keV = 500 000 × 1.602 × 10⁻¹⁹ = 8.011 × 10⁻¹⁴ J, so v = √(2K/m) = √(9.580 × 10¹³) = 9.787 × 10⁶ m/s, which is 3.26% of the speed of light.
AND THE RADIUS, WHICH DID. r = mv/qB = (1.6726 × 10⁻²⁷ × 9.787 × 10⁶)/(1.602 × 10⁻¹⁹ × 0.80) = 0.1277 m, about 12.8 cm. Quadruple the energy and the radius doubles, because r goes as √K.
CHECK HOW BADLY RELATIVITY IS BEING IGNORED. γ = 1 + K/mc² = 1.000533, and the exact speed is 9.7832 × 10⁶ m/s. The classical figure is high by 0.040% — negligible here, and the row above will tell you when it stops being.
THE SELECTOR, WHICH IS A SEPARATE SUM. v = E/B = (2.5 × 10⁵)/0.80 = 3.125 × 10⁵ m/s. Note that this is nothing like the proton's own speed: to pass THIS proton you would need E = vB = 7.83 × 10⁶ V/m, which is well above the breakdown field of air and is why selectors run in a vacuum.

Four particles in the same 0.80 T field

The deuteron and the alpha are nearly the same frequency, because the alpha has twice the charge AND about twice the mass, and only the ratio matters. The electron is in a different world altogether: 1836 times lighter, so 1836 times faster round and a far tighter circle. The electron figures here are classical and should not be trusted — at 500 keV an electron is travelling at 0.86c and γ is near 2, which the formula on this page does not know about. The page’s own γ row will tell you so.
Particle	Mass (u)	Charge	Frequency	Radius at 500 keV
Electron	0.000548580	1 e	22.39 GHz	0.298 cm
Proton	1.007276	1 e	12.20 MHz	12.77 cm
Deuteron	2.013553	1 e	6.098 MHz	18.06 cm
Alpha particle	4.001506	2 e	6.139 MHz	12.74 cm

The period with no speed in it, and where that stops being true

Set the magnetic force equal to the centripetal force and watch the speed disappear. qvB = mv²/r gives r = mv/qB, so a faster particle simply travels a proportionally bigger circle. Divide the circumference by the speed and you get T = 2πm/qB, with no v in it at all. Every proton in the machine, slow or fast, comes back to the gap at the same moment. A fixed-frequency oscillator across the dees can therefore push all of them, every turn, for ever — which is Lawrence’s idea and the whole of the cyclotron.

The magnetic field never adds any energy. It cannot: the force is always perpendicular to the velocity, so it does no work. All the energy comes from the electric field in the gap between the dees, a few tens of kilovolts at a time, collected over hundreds of crossings. The magnetic field’s entire job is to bend the particle back round for another go. If you remember only one thing about accelerators, make it that.

And then it breaks, for an interesting reason. As the particle approaches the speed of light its momentum stops being mv, and the real orbital frequency becomes qB/2πγm. It drops. The particle starts arriving at the gap late, falls out of step with the oscillator and eventually gets decelerated instead. This page prints γ and the percentage error so you can see it coming — it is negligible for a 500 keV proton and fatal for a 500 keV electron. The fixes are to sweep the oscillator frequency (a synchrocyclotron) or to ramp the field and hold the radius fixed (a synchrotron).

The velocity selector is a different trick entirely, and simpler. Cross an electric field with a magnetic one at right angles so the electric force qE and the magnetic force qvB oppose each other. They cancel when v = E/B — and notice that q has cancelled too, and m never appeared. The device passes one speed and does not care what the particle is, which is exactly what you want in front of a mass spectrometer, where the point is to separate by mass and a spread of speeds would ruin it. For the force this is all built from, see the magnetic force on a moving charge; for how the particle got its energy in the first place, a charge accelerated through a potential difference.

Frequently asked questions

Why does the cyclotron frequency not depend on the speed?

Because the radius grows in exact proportion to the speed. r = mv/qB, so a particle going twice as fast goes round a circle of twice the circumference, and takes the same time to do it. Change the energy on this page by any factor you like and watch the frequency row refuse to move while the radius row does.

Does the sign of the charge matter?

Only for the direction of circulation, which this page does not report. A negative particle goes round the other way at the same rate. The frequency depends on |q|/m and the field, so an electron and a positron in the same field circulate in opposite senses at identical frequencies.

Can a cyclotron accelerate electrons?

Barely. An electron reaches a tenth of the speed of light at only about 2.6 keV, so γ departs from 1 almost immediately and the electron falls out of step within a few turns. Electron machines are linear accelerators, betatrons or synchrotrons instead. Put the electron mass into this page and the relativistic rows will make the point for you.

Why does the selected speed not depend on the particle?

Because the charge cancels. The electric force is qE and the magnetic force is qvB; setting them equal gives qE = qvB, and q divides out, leaving v = E/B. The mass never enters at all, because neither force depends on it. A selector in front of a mass spectrometer therefore delivers a beam of one speed containing everything, and the spectrometer then separates it by m/q.

What is magnetic rigidity, in the row called Bρ?

It is p/q, the product of the field and the radius a particle of that momentum would follow — measured in tesla metres. Accelerator physicists use it because it describes how hard a particle is to bend without having to say what the particle is. Two particles with the same Bρ follow the same path through the same magnet whatever their mass.

Related calculators

References

University Physics Volume 2, chapters 11 and 14, OpenStax, Rice University (openstax.org/books/university-physics-volume-2, key-equations pages read 11 November 2026). An openly licensed university text, cited here because the coaching sites that dominate searches for this topic are not citable and mostly copy one another. Chapter 11 gives the radius of a charged particle’s path as r = mv/(qB), its period as T = 2πm/(qB), the Hall potential as V = IBl/(neA), the drift velocity in crossed fields as v_d = E/B, the magnetic dipole moment as μ = NIA and the torque as τ = μ × B. Chapter 14 gives L_solenoid = μ₀N²A/l, NΦ = LI, ε = −L dI/dt, M = N₂Φ₂₁/I₁ = N₁Φ₁₂/I₂ and U = ½LI². Neither key-equations page lists a cyclotron frequency or a field energy density, so those two are stated here from the definitions and not attributed to this source.
Fundamental Physical Constants — Frequently used constants (2022 CODATA wallet card), NIST Committee on Data for Science and Technology, physics.nist.gov/cuu/pdf/wallet_2022.pdf (read 11 November 2026). Every constant on this page comes from here, and the thing worth noticing is which numbers have a bracket after them. The elementary charge is printed 1.602 176 634 × 10⁻¹⁹ C and the speed of light 299 792 458 m s⁻¹, both with no bracket at all, because since 2019 they are exact by definition — the coulomb and the metre are now defined from them. But the vacuum electric permittivity is printed 8.854 187 8188(14) × 10⁻¹² F m⁻¹ and the vacuum magnetic permeability 1.256 637 061 27(20) × 10⁻⁶ N A⁻²: both now carry an uncertainty of about 1.6 parts in 10¹⁰, where before 2019 μ₀ was exactly 4π × 10⁻⁷ and ε₀ exactly 1/μ₀c². Almost every school book still prints the old exact forms. At three significant figures nothing changes; which constants are definitions changed completely.
Annual Syllabus, Class XII, Physics, Code 42, Session 2025-26, Central Board of Secondary Education, edustud.nic.in/edu/SYLLABUS_2025_26/12/12_Physics_EM.pdf (read 11 November 2026). The board’s own document, cited and not endorsed by it. Unit I Chapter 1 asks for “Electric flux, statement of Gauss’s theorem and its applications to find field due to infinitely long straight wire, uniformly charged infinite plane sheet and uniformly charged thin spherical shell (field inside and outside)”. Unit III Chapter 4 asks for “force on a moving charge in uniform magnetic and electric fields”, “torque experienced by a current loop in uniform magnetic field” and “Current loop as a magnetic dipole and its magnetic dipole moment”. Unit IV Chapter 6 is one line and covers three of this batch’s records: “Electromagnetic induction; Faraday’s laws, induced EMF and current; Lenz’s Law, Self and mutual induction”. The Hall effect is not in this document.
The International System of Units (SI), 9th edition 2019, version V4.01 of June 2026, Bureau International des Poids et Mesures, bipm.org/documents/d/guest/si-brochure-9-en-pdf (read 11 November 2026). Cited for the units, from Table 4 of coherent derived units with special names: magnetic flux, weber (Wb) = kg m² s⁻² A⁻¹ = V s; magnetic flux density, tesla (T) = kg s⁻² A⁻¹ = Wb/m²; inductance, henry (H) = kg m² s⁻² A⁻² = Wb/A. Those three equalities are worth more than they look: a weber is a volt-second, so a flux change of one weber in one second is one volt — which is Faraday’s law written as a unit. And a henry is a weber per ampere, so an inductance is just how much flux linkage one ampere produces.