Guiding-centre electric and gradient drifts

Suppose an additional perpendicular force $\mathbf F$ varies little during one gyroperiod. At leading averaged order, balance $\mathbf F+q\mathbf v_F\times\mathbf B=0$. Crossing with the field gives $\mathbf v_F=\mathbf F\times\mathbf B/(qB^2)$. This is a drift of the guiding centre, superposed on fast gyration. It is not the instantaneous acceleration direction: particles still accelerate throughout each orbit even when their averaged centre moves steadily.

For electric force $\mathbf F=q\mathbf E_\perp$, the signed charge cancels. The resulting electric cross magnetic drift is identical for electrons and singly charged ions in the same slowly varying fields. A quasineutral population therefore carries no leading net current from their common drift alone. An electric field parallel to the magnetic field cannot be balanced this way; it accelerates particles along the field and must be treated separately.

In a field that strengthens across an orbit, curvature of the trajectory is greater on the stronger-field side. The two sides no longer close into an identical circle. Equivalently, gyroaveraging produces an effective force $-\mu\nabla B$. Applying the force-drift formula gives a gradient drift perpendicular to both field direction and field-strength gradient. Its denominator contains signed charge, while the scalar magnetic moment stays positive; that is where the charge dependence enters.

Particles with larger perpendicular kinetic energy have larger gradient drift at fixed charge and field geometry. Electrons and ions can therefore contribute to a transverse current even while maintaining nearly equal densities. A particle guiding-centre drift is not the entire macroscopic pressure current: assembling a fluid current also involves the gyromotion's magnetization contribution and the distribution of particles. Do not identify a single representative orbit with a complete fluid transport law.

Field-line curvature produces another magnetic drift proportional to parallel energy and inverse signed charge. A changing perpendicular electric field produces a polarization correction, which depends on mass and the rate of electric-field change. These effects are absent from a uniform static electric-drift example, not prohibited in general. Comparing their orders requires the field-variation scale, gyrofrequency, parallel speed, and electric-field timescale rather than a list of drift names alone.

A practical direction check starts with a right-handed Cartesian basis. Compute the electric cross product before dividing by the positive squared field magnitude. For gradient drift, compute the unit-field cross gradient, then include the sign of charge. Finally compare the drift speed with the fast perpendicular speed and the orbit radius with the gradient scale. If these assumptions fail, a numerical orbit calculation or kinetic treatment may be needed instead of another term in the same truncated formula.