E x B Drift and the Cycloid

What you are seeing: a charged particle in crossed uniform fields, $\mathbf{B} = B \hat{z}$ (out of page) and $\mathbf{E} = E \hat{x}$ (right). Starting from rest, the particle traces a cycloid: it accelerates in the $+x$ direction under $\mathbf{E}$ until the resulting $\mathbf{v} \times \mathbf{B}$ curves it back, then repeats. The net motion is a uniform drift in the $\mathbf{E} \times \mathbf{B} / B^2$ direction (here $-y$ ).

The drift velocity $\mathbf{v}_d = \mathbf{E} \times \mathbf{B} / B^2$ is independent of the particle's charge and mass; positive and negative particles drift the same way. The cycloid amplitude is $E / B^2$ and its period equals the cyclotron period $T_c = 2 \pi m / (q B)$ . Vary $E$ to make the drift faster; vary $B$ to tighten the cycloid loops.

Figure 1. E x B drift. Method: RK4 integration with analytic drift overlay.

E0.50

B1.00

charge q1.00

mass m1.00

speed2

WHAT TO TRY

The particle traces a cycloid, looping as it gyrates yet marching steadily sideways. That net motion is the ExB drift, v_d = E/B, perpendicular to both fields and, strikingly, independent of the charge and mass.
Raise E or lower B: the drift speeds up and the loops stretch into a flatter cycloid (the loop size scales as E/B squared). At the cusps the particle momentarily stops before the field accelerates it again.
Flip the charge sign: the gyration reverses, but the particle still drifts the same way. A whole plasma drifts together regardless of the sign of its charges, so the ExB drift carries no current.