Now that we have obtained the various covariant forms of the Lorentz force law, we can easily determine the trajectories of charged particles in various fixed fields. In fact, we could have done this weeks ago (if not years) even without knowing the covariant forms.

In a static magnetic field, the equations of motion are:

(17.52) | |||

(17.53) |

for the energy and momentum, respectively (arranged like pieces of a four vector for clarity). Clearly the speed of the particle is constant since the force is perpendicular to the motion and does no work. is therefore also constant. Thus

(17.54) |

(17.55) |

This is too droll for words (and in fact you have probably already taught it
to your kids in kiddy physics) but it does yield one important result. The
magnitude of the momentum perpendicular to is

(17.56) |

Sections 12.2-12.4 are too simple to waste time on. 12.5-12.6 are interesting but important only to plasma people. 12.7 is redundant of things we will do correctly later. Thus we skip to 12.8, leaving you to read any or all of the intermediate material on your own. We will skip 12.9. Finally, we will do 12.10-12.11 to complete chapter 12.