Single-Particle Motion: Drifts in E and B

A charged particle obeying $m\dot{\vec v}=q(\vec E+\vec v\times\vec B)$ , integrated with the Boris pusher, a time-reversible leapfrog that conserves the speed exactly in a pure magnetic field. In a uniform $\vec B$ the orbit is a helix at the cyclotron frequency $\omega_c=|q|B/m$ . Add a perpendicular $\vec E$ and the guiding centre drifts at $\vec v_d=\vec E\times\vec B/B^2$ , the same for every charge and mass. A gradient or curvature of $\vec B$ gives the grad-B and curvature drifts; a converging field (a magnetic bottle) reflects the particle, conserving the adiabatic invariant $\mu=mv_\perp^2/2B$ . The trail is the real 3D orbit drawn in an orthographic Canvas2D projection (deterministic and gate-robust; the 3D motion is honestly a projection, not a WebGL scene). The headline readouts are $|\vec v|$ (constant to $10^{-9}$ in pure $\vec B$ ) and $\mu$ .

Figure 1. Boris-pushed orbit of a charged particle in static E and B fields, orthographically projected: a cyclotron helix, the E x B drift, grad-B and curvature drifts, and magnetic-mirror reflection with the conserved adiabatic invariant. Method: Boris leapfrog integration of the Lorentz force (gate-tested sim.js); Canvas2D orthographic projection of the 3D trajectory; the live readouts are the conserved speed and magnetic moment.

preset

B strength1.00

charge

speed3

WHAT TO TRY

Step through the presets: pure cyclotron gyration in uniform B, the E x B drift (independent of charge and mass), grad-B and curvature drifts, and the magnetic mirror that reflects the particle where B is strong.
Flip the charge sign: the gyration reverses sense, but the E x B drift keeps the same direction, which is why it transports both signs together and drives no current.
Raise B: the cyclotron frequency omega_c climbs and the gyroradius shrinks, so the helix tightens while the guiding-centre drift carries it steadily across the field.