Motion of a Point Charge in a Static Magnetic Field

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:

$$\frac{dE}{dt} = 0$$ (17.52)

$$\frac{d\vec{p}}{dt} = \frac{q}{c} \vec{v} \times \vec{B}$$ (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. $\gamma$ is therefore also constant. Thus

$$\frac{d \vec{v}}{dt} = \vec{v} \times \vec{\omega_B}$$ (17.54)

where

$$\vec{\omega_B} = \frac{q \vec{B}}{\gamma mc} = \frac{q c \vec{B}}{E}$$ (17.55)

is the gyration or precession (cyclotron) frequency. The motion described by this equation is a circular motion perpendicular to $\vec{B}$ coupled to a uniform motion parallel to $\vec{B}$.

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 $\vec{B}$ is

$$cp_\perp = q B a$$ (17.56)

where $a$ is the radius of the circular helix. From this (in, for example, a bubble chamber, where the track can be photographed) and a knowledge (or guess) as the the charge, the transverse momentum can be measured. Measuring other things (like the rate of change of the curvature of the track) can yield the mass of the particle from a knowledge of its momentum. From these humble traces the entire picture we currently have of the sub-atomic zoo has been built up.

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.