Consider then charged particle motion in a magnetic field В whose field lines possess a radius of curvature R. In examining such a case we specify here the curved magnetic field lines by

В = B0k + B0~ri

К

when referred to in the vicinity of the у-axis, that is for Iz/RI « 1. A graphical depiction is provided in Fig. 5.10.

Introducing this field configuration, Eq. (5.52), into the Eorentz-force equation, Eq.(5.6), yields

m— = q( x В k) + q dt V ’

For small curvatures, i. e. Iz/RI « 1, we may, as in the previous cases, approximate the product z(vxj) by the undisturbed uniform solutions given in

(v, c x k) + —[- viiі + v± cos( ft> r)k] (5.55)

at m mR L J

where we specified z0 = 0 and the initial phase angle of Eq. (5.48) as ф = 0. Further substituting qBJm by Eq. (5.12), we separate Eq. (5.55) into its Cartesian components and write

Apparently, Eq. (5.56c) can be solved straightforwardly upon knowing that vgc z(0) = V||. Averaging the resulting velocity vgcz over a gyration period will suppress the small oscillations with the cyclotron frequency cog and indicate a steady drift anti-parallel to the В-field; this velocity drift, however, is seen to depend on the initial condition, v± = v0, and is therefore not classified as a drift.

To solve for the velocity components perpendicular to В we uncouple the

mutual dependencies of Eqs. (5.56a) and (5.56b) by differentiation and subsequent substitution to obtain

d Vf’X + C02g vgc, x = — sign(q)a>g • (5.57)

dt R

The solution of this inhomogeneous differential equation is accomplished by summing of the homogeneous solution of the form Acos(COgt + a) and a stationary solution to the inhomogeneous equation, which, by inspection of Eq.(5.57), we take as (sign(q)v12/(0gR). Referring to the initial conditions vgc, x(0)=vgc y(0)=0 associated with the undisturbed field at z(0) = 0, we determine

the constants A and a and finally find

2

Vll

vgc, x = sign(q)—’!—[cos(cog t) -1 ] (5.58a)

cogR

and

2

V|.

Vgc, y = —^ [C0gt — sin((Ogt)] . (5.58b)

cogR

The resulting drifts are, as preceedingly, revealed by averaging over a gyration period and are thus

where t represents an average length of time corresponding to one half of the interval of averaging. The latter equation clearly demonstrates that the guiding centre is accelerated in the direction of j at the magnitude V||2/R which, of course, is known to be the centripetal acceleration acp required to make the particle follow the curved В-field line. The only real steady drift is then accounted for by (Eq. 5.59a), in which we substitute according to the geometry of Fig. 5.10

i = jxk = -^x5i (5.60)

R Bo

and reintroduce cog = lqlB0/m to derive the curvature drift velocity as

which is thus again dependent upon the sign of the charge. Fig. 5.11 illustrates this drift of an ion spiraling around a curved magnetic field line.

A curved В-field is shown below to be accompanied by a gradient B-field, thus both specific drift velocity’s components add to yield the resultant drift. We remark that, although the expressions for the different drift velocities derived
here are correct, the assumptions and approximations used in the derivation are inconsistent in the sense that the suggested nonuniform В-fields do not satisfy Maxwell’s equations

V xB = 0 (5.62)

and V-B = 0. (5.63)

In order that these fundamental equations are met, it is seen that an inhomogeneous magnetic field has always to be associated with some curvature and vice versa. Hence grad-B drift and curvature drift will occur simultaneously.