Fundamentals of Plasma Physics

84 Chapter 3. Motion of a single plasma particle

particle orbitarea contracts

in inverse proportion

to the field strength

after compression offield,

particle orbit links same number

offield lines as before

field lines of

time-dependent

magnetic field Bz
t

tw o particles atsame position, Larmor orbit

but having different gyrocenters

increase magnetic field strength

E

E

Figure 3.6: Illustration showing how conservation offlux linked by an orbit is equivalent to
frozen-in field;also increasing magnetic field results in magnetic compression.

3.5.5 Magnetic mirrors- a consequence ofμconservation

Consider a charged particle moving in a static, but spatially nonuniform magnetic field.

The non-uniformity is such that the field strength varies in the direction of the field line so

that∂B/∂s
= 0wheresis the distance along a field line. Such a field cannot be straight

because if it were and so had the formB=Bz(z)ˆz,it would necessarily have a non-zero

divergence, i.e., it would have∇·B=∂Bz/∂z
= 0.Because magnetic fields must have

zero divergence there must be another component besidesBzand this other component

must be spatially non-uniform also in order to contribute to the divergence.Hence the field

must be curved if the field strength varies along the direction of the field.

This curvature is easy to see by sketching field lines, as shown in Fig.3.7. The density

of field lines is proportional to the strength of the magnetic field and so a gradient of field

strength along the field means that the field lines squeeze together as the field becomes

stronger. Because magnetic field lines have zero divergence they are endless and so must

bend as they squeeze together. This means that if∂Bz/∂z
= 0there must also be a field