A magnetic field which varies spatially in strength in the direction of the field is of considerable importance in various mirror configurations-Fig. 4.2b. We suggest such a general case in Fig.5.12 and assess its effect on ion motion.

Cylindrical coordinates, (r,9,z) in Fig. 5.12, will serve us well for which Maxwell’s Second Equation, Eq.(5.63), gives

rdrK ’ г дв dz

Axial symmetry is generally expected to hold so that ЭВ9/Э6 = 0 and we obtain

(5.65)

with the boundary condition Br (r = 0) = 0 and dBjdz taken to be independent of r. Then, assuming Be = 0, we write

В = ВгЄг+ Bzez (5.66)

where er and ez are the radial and axial unit vectors in cylindrical geometry, respectively, Fig. 5.12. The corresponding equation of motion of a charged particle in this field is

m<^- = q{vXBzez) + q{vXBrer) — (5-67)

dt

The first term is recognized as the force responsible for particle gyration about Bz and also as that which leads to a grad-B drift, e. g. Eq.(5.51), when the dependence of Bz on r becomes significant. The second term, explicitly written as

q(v x Br er) = — q ve Brez + q vz Br ее (5.68)

contains a force parallel or antiparallel to the field on the axis and an azimuthal force. While the latter urges the particles to drift in the radial direction thus rendering their guiding centres to follow the specific magnetic field lines, the parallel force component will accelerate or, respectively, decelerate ions and electrons along the z-axis depending on the direction of their motion. With the aid of Eq. (5.65) this force component can be expressed as

d R

Fn = jqyer^. (5.69)

dz

We may now recognize the associated effect on the guiding centre by taking again an average Fy over a gyroperiod. Specifically taking the particles to gyrate around the central В-line at r = 0 renders the relation

v0 = — sign(q)v± (5.70)

so that for r = rg, we obtain

= (5.71)

Upon substitution of rg by Eq. (5.17), a generalization of this force can be shown to be given by

F|| (5.72)

Thus a net force acts on charged particles-independent of their sign-in a direction opposite to that of increasing B.