Gyromotion (${\Omega }_s=q_{s}B/(m_{s}c)$)

Rapid circular particle motion perpendicular to a locally uniform magnetic field, superposed on parallel streaming when the parallel velocity is nonzero. In Gaussian units, its signed cyclotron frequency is given in the title and its gyroradius is $v_{\perp }/|{\Omega }_s|$. Slowly varying fields permit a guiding center description.

For a constant perpendicular force in a uniform magnetic field, write the particle velocity as $\mathbf{v}={\mathbf{v}}_d+\mathbf{u}$. Choosing $q{\mathbf{v}}_d\times \mathbf{B}=-\mathbf{F}$ leaves the ordinary gyromotion equation for $\mathbf{u}$. The gyroperiod-averaged transverse velocity is therefore the displayed drift. An electric force gives the same drift for both charge signs; a charge-independent applied force gives opposite drifts.

For a curved field direction $\mathbf{b}=\mathbf{B}/B$, define the curvature vector $\mathbf{\kappa }=(\mathbf{b}\cdot \nabla )\mathbf{b}$. Motion along curved field lines produces a gyroaveraged drift perpendicular to both field and curvature. In a purely toroidal field this and magnetic gradient drift cause opposite vertical drifts for opposite charge signs.

A slowly varying field magnitude exerts the gyroaveraged force $-\mu \nabla B$ on a particle with magnetic moment $\mu =m{v}_{\perp }^2/(2B)$. Applying the perpendicular-force guiding-centre drift formula gives the displayed velocity. Its direction depends on the sign of the particle charge.