The component of this equation parallel to the magnetic field,

(2.2) |

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

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

(2.3) |

where is the gyrofrequency, is the gyroradius, and are unit vectors such that , , form a right-handed, mutually orthogonal set, and is the particle's initial

This drift, which is termed the

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

(2.5) |

where

(2.6) |

and

(2.7) |

Here, . Of course, the trajectory of the particle describes a spiral. The

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.