# Charged Particle Motion Simulator - Lorentz Force, Cyclotron & Mass Spectrometer

> Electrons, protons or alphas in any uniform E and B in 3D: gyroradius, period, pitch and E×B drift, plus a velocity selector, mass spectrometer and cyclotron.

- URL: https://qilingtools.com/en/math/charged-particle
- Category: Physics Tools (https://qilingtools.com/en/physics)
- Tags: [Physics](https://qilingtools.com/en/tags/physics), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Charged particles moving in electric and magnetic fields are a staple of electromagnetism, and this tool lets you set the parameters and watch the motion. In free-field mode, choose an electron, positron, proton, deuteron, alpha particle or a custom ion, set the initial speed and direction and uniform E and B fields pointing anywhere. The trajectory is drawn in 3D — drag to rotate, or switch to xy, xz or yz projections — with presets for a uniform circle, a helix, the cycloid of E×B drift, a trochoid, an accelerating helix with E parallel to B, and a relativistic electron at 0.9c. Results list the Lorentz factor, gyroradius, period, frequency, pitch angle, helix pitch and drift velocity. Four more scenes cover the classic devices: a velocity selector fires a beam of different speeds and only v = E/B makes it straight through the slit; a mass spectrometer accelerates neon, chlorine, carbon, lithium or uranium isotopes through the same voltage and separates them in a magnetic field, listing landing points and spacing; a cyclotron accelerates a particle across the gap between two dees every half turn, with the spiral orbit and an energy curve compared with (qBR)²/(2m); and a magnetic mirror shows particles bouncing between strong-field ends and the loss cone. Integration uses the relativistic Boris pusher, so there is no energy drift even over long runs.

## How to use

1. Pick a scene: free fields, velocity selector, mass spectrometer, cyclotron or magnetic mirror
2. Choose a particle and enter the speed, fields and other settings (1e5, 2.5k, 10m all work), or click a preset
3. Play the animation and scrub the timeline to read position, speed and kinetic energy; drag the 3D view to rotate it
4. Compare the gyroradius, period, drift or landing distances with the formulas below

## Features

- 3D trajectories in uniform E and B in any direction, with rotatable and projected views
- Relativistic Boris pusher: speed is conserved exactly in a pure magnetic field
- Gyroradius, period, frequency, pitch angle, helix pitch and E×B drift computed automatically
- Velocity selector with a multi-speed beam and the pass speed E/B
- Mass spectrometer: isotope landing points, spacing and required resolving power
- Cyclotron and magnetic mirror: extraction energy, turns, loss cone and turning points

## FAQ

### Why does a charged particle move in a circle in a uniform magnetic field, and what is the radius?

The Lorentz force F = qv × B is always perpendicular to the velocity, so it changes the direction but not the speed — it acts as a centripetal force. From qvB = mv²/r, r = mv/(qB) and the period T = 2πm/(qB), independent of speed. Any velocity component along B feels no force, so the particle circles while drifting along the field in a helix of pitch p = v∥T. Near light speed, replace m with γm; every formula in the tool already includes γ.

### What is E×B drift, and why do positive and negative charges drift the same way?

With E perpendicular to B, during each gyration the electric field speeds the particle up on one side and slows it down on the other, so its radius alternates between larger and smaller and the centre of the orbit walks sideways along E × B at v = E × B / B². Charge and mass cancel out of that result, so electrons and ions drift at the same speed in the same direction and a plasma moves as a whole without carrying current. Starting from rest the path is a cycloid; with an initial velocity it is a trochoid.

### Why does a velocity selector pass only one speed?

Between the plates the electric force qE and the magnetic force qvB point in opposite directions, and they cancel only when qE = qvB, that is v = E/B. Particles at that speed go straight through to the exit slit; faster ones are bent one way by the stronger magnetic force and slower ones the other way. The condition involves neither the sign of the charge nor the mass, which is why a selector is often placed in front of a mass spectrometer: first pick one speed, then separate by mass with a magnetic field.

### Can a cyclotron accelerate particles to any energy?

No. The cyclotron frequency f = qB/(2πγm) is independent of energy at low speeds, so a fixed-frequency voltage across the gap gives a kick every half turn. As the energy rises, γ grows and the frequency drops, the particle falls behind the RF phase and eventually stops gaining energy. The dee radius also caps the energy at E_max = (qBR)²/(2m). Classic cyclotrons reach a few tens of MeV for protons; beyond that you need a frequency-swept synchrocyclotron or a synchrotron. Tick the relativity option to watch the phase slip.

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