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E×B Drift

Equation

\[\boxed{\,\mathbf{v}_E = \frac{\mathbf{E}\times\mathbf{B}}{B^2}\,}\]

Physical meaning

A charged particle in crossed fields doesn't run along \(\mathbf{E}\) — it marches sideways, perpendicular to both fields, at speed \(E_\perp/B\). During the gyration half where the particle moves with \(\mathbf{E}\) it speeds up (fatter arc); against, it slows (tighter arc). The lopsided circles add up to a steady sidestep.

Variables

\(\mathbf{E}\) — electric field (V/m) · \(\mathbf{B}\) — magnetic field (T) · \(\mathbf{v}_E\) — drift velocity (m/s), independent of \(q\), \(m\), and energy.

Assumptions

Magnetized particle (\(r_L\) small vs field scales, \(\omega_c\) fast vs field changes); \(E/B \ll c\); the drift describes the guiding center, not the instantaneous velocity.

The deepest way to see it

\(\mathbf{v}_E\) is the velocity of the reference frame in which \(\mathbf{E}_\perp\) vanishes (a Lorentz boost at \(\mathbf{E}\times\mathbf{B}/B^2\) removes the perpendicular electric field). In that frame the particle just gyrates; back in the lab frame, the gyration center translates. That is why the drift is independent of charge, mass, and energy — it's kinematics, not dynamics.

Consequences

  • No current: ions and electrons drift together — bulk plasma flow. (Currents come from the charge-dependent drifts.)
  • The cross-field cycloid benchmark's average motion — see cyclotron motion.
  • Tokamak rotation, ionospheric convection, Hall-thruster operation, and the \(\mathbf{E}\times\mathbf{B}\) velocimetry of lab plasmas.

Numerical verification

Exercise 1 of Chapter 3: integrate uniform crossed fields with the 3-D leapfrog and recover a helical cycloid drifting at exactly \(E/B\); Exercise 2 wraps the drift around a line charge — azimuthal orbits at \(v_E = \lambda/2\pi r B\).

Quiz

Q1 (computational). Solar-wind conditions: \(E = 1\) mV/m, \(B = 5\) nT. Drift speed?

Answer

\(v_E = 10^{-3}/5\times10^{-9} = 2\times10^5\) m/s = 200 km/s — solar-wind scale; E×B drifts are anything but small in space plasmas.

Q2 (MCQ). Doubling a particle's kinetic energy changes its E×B drift by:

  • (a) ×2 (b) ×√2 (c) ×4 (d) nothing
Answer

(d). Energy-independent — only the field ratio matters.