Motion in Uniform Fields

Let us, first of all, consider the motion of a particle of mass $m$ and charge $e$ in spatially and temporally uniform electromagnetic fields. The particle's equation of motion takes the form

$$m\,\frac{d{\bf v}}{dt} = e\,({\bf E} + {\bf v}\times {\bf B}).$$ (2.1)

The component of this equation parallel to the magnetic field,

$$\frac{d v_\parallel} {dt} = \frac{e}{m} \,E_\parallel,$$ (2.2)

predicts uniform acceleration along magnetic field-lines. Consequently, plasmas close to equilibrium generally have either small or vanishing $E_\parallel$.

As can easily be verified by substitution, the perpendicular (to the magnetic field) component of Equation (2.1) yields

$${\bf v}_\perp = \frac{{\bf E}\times{\bf B}}{B^{2}} + \rho\,{\mit\... ... t +\gamma_0) \,{\bf e}_1 + \cos({\mit\Omega}\, t +\gamma_0)\,{\bf e}_2\right],$$ (2.3)

where ${\mit\Omega}=eB/m$ is the gyrofrequency, $\rho$ is the gyroradius, ${\bf e}_1$ and ${\bf e}_2$ are unit vectors such that ${\bf e}_1$, ${\bf e}_2$, ${\bf B}$ form a right-handed, mutually orthogonal set, and $\gamma_0$ is the particle's initial gyrophase. The motion consists of gyration around the magnetic field at the frequency ${\mit\Omega}$, superimposed on a steady drift with velocity

$${\bf v}_E = \frac{{\bf E}\times{\bf B}}{B^{2}}.$$ (2.4)

This drift, which is termed the E-cross-B drift, is identical for all plasma species, and can be eliminated entirely by transforming to a new inertial frame in which ${\bf E}_\perp= {\bf0}$. This frame, which moves with velocity ${\bf v}_E$ with respect to the old frame, can properly be regarded as the rest frame of the plasma.

We can complete the previous solution by integrating the velocity to find the particle position. Thus,

$${\bf r}(t) = {\bf R}(t) + \mbox{\boldmath$\rho$} (t),$$ (2.5)

where

$$\mbox{\boldmath$\rho$} (t) = \rho \,[-\cos({\mit\Omega}\, t+\gamma_0)\,{\bf e}_1 +\sin({\mit\Omega}\,t + \gamma_0)\,{\bf e}_2],$$ (2.6)

and

$${\bf R}(t)=\left(v_{0\,\parallel } \,t + \frac{e}{m}\, E_\parallel \,\frac{t^{2}}{2}\right) {\bf b} + {\bf v}_E \,\,t.$$ (2.7)

Here, ${\bf b} \equiv {\bf B}/B$. Of course, the trajectory of the particle describes a spiral. The gyrocenter, ${\bf R}$, of this spiral, which is termed the guiding center, drifts across the magnetic field with the velocity ${\bf v}_E$, and also accelerates along field-lines at a rate determined by the parallel electric field.

The concept of a guiding center gives us a clue as to how to proceed. Perhaps, when analyzing charged particle motion in nonuniform electromagnetic fields, we can somehow neglect the rapid, and relatively uninteresting, gyromotion, and focus, instead, on the far slower motion of the guiding center? In order to achieve this goal, we need to somehow average the equation of motion over gyrophase, so as to obtain a reduced equation of motion for the guiding center.