Deriva E×B

Select a particle or example, set the fields and initial velocity components, and calculate. With a nonzero magnetic field, Cycles sets the time window in complete-period units, including fractional cycles. With B = 0, Duration sets a separate window in nanoseconds. Move the time probe or enter an exact percentage from 0 to 100%; both views show that same time. The initial probe is 25%. Example changes keep the probe percentage. Clear result keeps the inputs.

The initial position is (0, 0). E = (E, 0, 0) and B = (0, 0, B): positive x points right and positive y points up. Positive B points out of the drawing plane (⊙), negative B into it (⊗). The field symbols indicate direction only, not vector magnitudes. The particles are an electron with q = −e, a proton with q = +e, or an alpha particle with q = +2e.

The Lorentz equation is m·v′ = q(E + v×B), with v×B = (vyB, −vxB, 0). For B ≠ 0, the signed angular frequency is ω = qB/m and the period is 2π/|ω|. The mean drift is vD = (0, −E/B), independent of charge and mass. Reversing charge or B reverses gyration; reversing E or B reverses the drift. Drift is the average over complete periods, not necessarily over an arbitrary shorter trajectory segment.

Relative to the drift, the initial velocity is w₀ = (vx₀, vy₀ + E/B). It rotates through angle −ωt. The gyroradius is |w₀|/|ω|, determined by relative velocity rather than total laboratory speed. The moving circle center is C(t) = ((vy₀ + E/B)/ω, −vx₀/ω − Et/B). With relative position ρ = (−wy/ω, wx/ω), the laboratory position is P = C + ρ. The pure-drift example has w₀ = 0: the laboratory path is straight and the relative circle has zero radius.

The laboratory panel shows the whole calculated path and a dashed line for the moving centers, with an arrow in the drift direction. P is the current particle, C its center and O the fixed origin; the red connection is the radius. The other panel shows the full relative circle about zero and the same instantaneous ρ. Each panel has equal x and y scales, but the panels can use different scales and SI length units. Axis labels consistently use nm, µm, mm or m; the numerical results remain in meters, seconds, m/s and eV. A stationary center has no drift arrow.

At B = 0, x = vx₀t + qEt²/(2m), y = vy₀t, vx = vx₀ + qEt/m and vy = vy₀. There is no E×B drift, gyroradius, gyroperiod or relative-circle panel. If E is also zero, the particle moves uniformly or remains at rest when its initial velocity is zero. Kinetic energy is m|v|²/2, displayed in electronvolts; its change equals the electric work qEΔx.

Motion is evaluated analytically; the time probe is independent of the drawing samples. The square frames include the path and centers, or the relative circle, with 20% total margin. At rest the minimum axis span is 1.2 nm. The displayed laboratory path uses 32–1024 segments with B ≠ 0 and 256 at B = 0. There is no time-step integration or automatic animation.

This is a nonrelativistic test-particle model in constant uniform fields, without collisions, space charge or radiation. The input limits keep speeds below 0.05 times the speed of light. E ranges from −8000 to 8000 V/m; B is zero or 0.005 ≤ |B| ≤ 0.2 T; each initial velocity component ranges from −100000 to 100000 m/s. The window is 0.25–8 cycles or, for B = 0, 0.001–10 ns. All calculations stay in your browser.

Constants from CODATA 2022: e = 1.602176634×10⁻¹⁹ C exactly, electron mass = 9.1093837139×10⁻³¹ kg, proton mass = 1.67262192595×10⁻²⁷ kg and alpha-particle mass = 6.6446573450×10⁻²⁷ kg. The alpha option is a doubly charged helium nucleus, not a singly ionized helium atom.

Sources: Richard Fitzpatrick, Motion in Uniform Fields; NIST, CODATA 2022 complete constants table.