The Lorentz Force: How Electric and Magnetic Fields Steer a Charge

One equation, four kinds of motion

A charged particle in electric and magnetic fields obeys a single vector law, the Lorentz force: F = q(E + v×B). The electric term qE pushes the particle in a straight line, parallel to the field, and does work on it — it speeds the particle up or slows it down. The magnetic term qv×B is different in kind: because it is a cross product with the velocity, it is always perpendicular to v, so it can never change the particle's speed, only its direction. Magnetic fields steer; they never accelerate a charge along its own path.

Set E = 0 and drop a charge into a uniform B and you get the simplest case: circular motion. The magnetic force supplies exactly the centripetal force needed for a circle, qvB = mv²/r, which rearranges to the cyclotron radius r = mv/(qB) — often called the Larmor radius in a plasma-physics context. The angular frequency of that circle, ω = qB/m, is the cyclotron frequency, and remarkably it does not depend on the particle's speed at all: fast particles just trace bigger circles at the same frequency. That speed-independence is why cyclotrons work as particle accelerators — a fixed-frequency oscillating voltage stays in sync with the particle as it spirals outward and gains energy on every crossing.

Adding gravity or a second field: cycloids and drift

If you add a uniform E field perpendicular to B (or, equivalently, a uniform force like gravity), the circular orbit stops closing on itself. The particle's own instantaneous velocity keeps changing the magnetic force it feels, and the combination traces a cycloid — a spiral of loops that drifts steadily sideways. The remarkable result is that this drift velocity, v_drift = E×B / B², is completely independent of the particle's charge, mass and initial velocity: electrons and protons drift together at the same speed and in the same direction. This E×B drift is the mechanism that confines and rotates plasma in a magnetron sputtering source and, at a vastly larger scale, sets the rotation of plasma in the Van Allen radiation belts and in magnetically confined fusion devices.

v_drift = (E × B) / |B|^2      // charge- and mass-independent
r_L     = m v_perp / (|q| B)   // Larmor (cyclotron) radius
omega_c = |q| B / m            // cyclotron frequency, independent of v

Integrating the motion: why RK4 and not Euler

The equations of motion are a coupled, velocity-dependent ODE — the acceleration at the next instant depends on the velocity you are trying to solve for, because v×B feeds back on itself. A plain Euler step (advance velocity with the old acceleration, then position with the old velocity) systematically injects or removes energy from a magnetic orbit, and a circular path visibly spirals outward or inward over a few dozen turns even though the true physics conserves the particle's kinetic energy exactly. The simulation on this page integrates F = q(E + v×B) with classical fourth-order Runge–Kutta, sampling the force four times per step and blending the estimates, which keeps a cyclotron orbit closed and its energy flat over thousands of revolutions at a modest step size. (A dedicated plasma code would more likely use the Boris algorithm, a specialised leapfrog-like scheme that is not fourth-order accurate but is exactly energy-conserving for pure magnetic rotation and cheaper per step than RK4 — a good example of a symplectic method beating a higher-order one for a specific physics problem.)

Where this shows up

The same three regimes — pure rotation, cycloidal drift and free ballistic motion when both fields vanish — describe the aurora (electrons spiralling down Earth's field lines and gyrating around them), the mass spectrometer (measuring m/q from the radius of a known B and v), the cathode-ray tube and CRT electron guns, and the confinement scheme of a tokamak, where the E×B drift is deliberately used to rotate the plasma column away from wall contact.

Frequently asked questions

Try it live

Everything above runs in your browser — open Lorentz Force and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.