Why Do Charged Particles Spiral in a Magnetic Field? Cyclotron Motion Explained

Fire an electron into a uniform magnetic field and it will not fly straight. It curves into a circle and, if it had any speed along the field, it corkscrews along a helix. Physicists call this cyclotron motion, and our sticker Gyrolla is that endless loop drawn in pixels. This post builds on our explainer How Are Electricity and Magnetism Connected? and explains where the circle comes from, how fast it turns, and why one surprising property of it built a whole branch of physics.

What force does a magnetic field put on a moving charge?

A charge q moving with velocity v through a magnetic field B feels the magnetic part of the Lorentz force:

F = q v × B

Here q is the charge in coulombs (C), v the velocity in metres per second (m/s), B the magnetic flux density in tesla (T), and F the force in newtons (N). The cross product means the force always points at right angles to both the velocity and the field. A charge at rest feels no magnetic force at all, and neither does a charge moving exactly along the field lines.

Because the force is always perpendicular to the motion, it changes the direction of the velocity but never its size. The field does no work on the particle, so in a pure magnetic field the particle’s speed and kinetic energy stay constant (Feynman Lectures, Vol. II, Ch. 29^[1]).

Why does that make a circle?

A force of constant size that always points sideways to the motion is exactly what keeps something in a circle. It is the same geometry as a ball whirled on a string, with the magnetic force doing the string’s job. Setting the magnetic force equal to the force needed for circular motion gives the radius of the orbit:

r = m v_⊥ / (|q| B) = p_⊥ / (|q| B)

where m is the mass in kilograms (kg), v_⊥ the speed at right angles to the field, and p_⊥ the matching momentum. Feynman writes the same result as R = p/qB and notes that it stays true at relativistic speeds as long as p is the relativistic momentum (Feynman, Ch. 29^[1]).

Three things follow straight from the formula. Faster particles make bigger circles. Heavier particles make bigger circles at the same speed. A stronger field makes tighter circles. Positive and negative charges circle in opposite directions, which is how particle detectors tell them apart from the curl of their tracks.

Why a spiral, not just a circle?

The magnetic force has no component along B, so whatever velocity the particle has parallel to the field is left completely alone. Combine steady drift along the field with circling across it and you get a helix, a corkscrew wrapped around a field line. Feynman calls this the general motion of a charge in a uniform field: “a constant velocity parallel to B and a circular motion at right angles to B” (Feynman, Ch. 29^[1]). The distance the helix advances per turn, its pitch, is the parallel speed multiplied by the time for one loop.

That is why charged particles tend to follow magnetic field lines rather than cross them. Feynman points to the electrons and protons trapped in Earth’s Van Allen belts as one place where these orbits matter on a planetary scale.

What is the cyclotron frequency?

The time for one loop is the circumference divided by the speed, T = 2πr/v_⊥. Put in the radius from above and the speed cancels:

T = 2πm / (|q| B)    and    f = |q| B / (2πm)

T is the period in seconds and f the cyclotron frequency in hertz (Hz). The angular version is ω_c = |q|B/m in radians per second. This is the surprising part: a fast particle runs a bigger circle in exactly the same time as a slow particle runs a small one. As HyperPhysics puts it, the period “is independent of the radius” (HyperPhysics, “Cyclotron”^[2]).

Only the ratio of charge to mass matters, and both are known very precisely. Using the CODATA 2022 values of the charge-to-mass ratio, e/mₑ ≈ 1.758 820 × 10¹¹ C/kg for the electron (NIST^[3]) and e/mₚ ≈ 9.578 833 × 10⁷ C/kg for the proton (NIST^[4]), a field of 1 T gives:

• Electron: f = (1.758 820 × 10¹¹ × 1)/(2π) ≈ 2.80 × 10¹⁰ Hz, about 28 GHz, in the microwave band.
• Proton: f = (9.578 833 × 10⁷ × 1)/(2π) ≈ 1.52 × 10⁷ Hz, about 15 MHz, a radio frequency.

For a sense of size, a proton with 1 MeV of kinetic energy moves at about 1.38 × 10⁷ m/s (non-relativistic, v = √(2K/m)). In a 1 T field its orbit radius is r = v/(e/mₚ · B) ≈ 1.38 × 10⁷ / 9.58 × 10⁷ ≈ 0.14 m, about the size of a dinner plate.

How did Lawrence turn this into a particle accelerator?

Because the loop time does not depend on speed, you can push a particle with an alternating voltage at one fixed frequency and it will stay in step as it speeds up. Ernest Lawrence and M. Stanley Livingston built exactly that: two hollow D-shaped electrodes, the “dees,” inside a magnet, with a radio-frequency voltage across the gap between them. Each time an ion crosses the gap it gets a kick, its orbit grows, and it arrives back at the gap just as the voltage has flipped to kick it again. Their 1932 paper describes ions whose “time required for a traversal of a semi-circular path is independent of their velocities,” and reports 1,220,000-volt protons from a machine whose maximum applied voltage was only 4,000 volts (Lawrence & Livingston 1932^[5]). The particle reaches high energy through many small kicks instead of one giant voltage.

Lawrence received the 1939 Nobel Prize in Physics “for the invention and development of the cyclotron and for results obtained with it, especially with regard to artificial radioactive elements” (NobelPrize.org^[6]).

Where does the simple picture break down?

The neat result T = 2πm/(|q|B) assumes the particle is much slower than light. At relativistic speeds the momentum becomes p = γmv, with γ = 1/√(1 − v²/c²), so the angular frequency becomes ω = |q|B/(γm). As a particle gains energy, γ grows, its loops slow down, and it drifts out of step with a fixed-frequency voltage. Practical machines handle this by adjusting the magnetic field or the accelerating frequency (HyperPhysics^[2]). In 1945 Edwin McMillan proposed the synchrotron, a design built around that problem (McMillan 1945^[7]). Real accelerators also need carefully shaped fields to keep the beam from drifting up, down or sideways, which Feynman discusses as “focusing” (Feynman, Ch. 29^[1]).

The idealised circle also assumes a perfectly uniform field and no electric field. Add an electric field at right angles and the orbit drifts sideways, a motion Feynman works through in the same chapter.

Where is cyclotron motion used today?

Any time charges move in a magnetic field, the cyclotron frequency is the natural rhythm of the system, and you can push energy in by driving at that frequency. Fusion machines do exactly this. ITER heats the electrons in its plasma with a high-power beam at 170 GHz, which its engineers describe as the resonant frequency of the electrons; the electrons then share that energy with the ions through collisions (ITER^[8]). From f = eB/(2πmₑ), 170 GHz matches non-relativistic electrons in a field of roughly 6 T.

The same physics links to our other electromagnetism posts. Currents induced by moving magnets, the subject of why a magnet falls slowly through a copper pipe, are charges in conductors being pushed by the same v × B force. You can find Gyrolla alongside Lenzor and friends on the Field Work sticker sheet and across our Electromagnetism collection.

FAQ

Does a magnetic field speed up a charged particle?

No. The magnetic force is always at right angles to the motion, so it changes direction but not speed. In a cyclotron the energy comes from the electric field across the gap between the dees.

Why is the cyclotron frequency independent of speed?

A faster particle runs a larger circle. The radius grows in proportion to the speed, so the time per loop, 2πr/v, stays fixed at 2πm/(|q|B). This holds while the particle is much slower than light.

What decides the direction of the spiral?

The sign of the charge and the direction of B. Positive and negative charges circle in opposite senses around the same field line.

What happens to a particle moving along the field?

Nothing magnetic: with v parallel to B, v × B = 0, so it keeps going straight at constant speed.

References

1. R. P. Feynman, R. B. Leighton & M. Sands, The Feynman Lectures on Physics, Vol. II, Ch. 29: The Motion of Charges in Electric and Magnetic Fields (Caltech online edition). https://www.feynmanlectures.caltech.edu/II_29.html
2. C. R. Nave, HyperPhysics (Georgia State University), “Cyclotron”. http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/cyclot.html
3. NIST CODATA 2022 recommended values: electron charge to mass quotient. https://physics.nist.gov/cgi-bin/cuu/Value?esme
4. NIST CODATA 2022 recommended values: proton charge to mass quotient. https://physics.nist.gov/cgi-bin/cuu/Value?esmp
5. E. O. Lawrence & M. S. Livingston (1932), “The Production of High Speed Light Ions Without the Use of High Voltages,” Phys. Rev. 40, 19–35 (DOI 10.1103/PhysRev.40.19). https://journals.aps.org/pr/abstract/10.1103/PhysRev.40.19
6. NobelPrize.org, The Nobel Prize in Physics 1939 (Ernest O. Lawrence). https://www.nobelprize.org/prizes/physics/1939/summary/
7. E. M. McMillan (1945), “The Synchrotron—A Proposed High Energy Particle Accelerator,” Phys. Rev. 68, 143–144 (DOI 10.1103/PhysRev.68.143). https://journals.aps.org/pr/abstract/10.1103/PhysRev.68.143
8. ITER Organization, “External heating systems”. https://www.iter.org/machine/supporting-systems/external-heating-systems

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.