Chapter 4 SINGLE PARTICLE MOTIONS 4.1 Introduction

4.4 Inhomogeneous Fields 103where Φ is the magnetic flux linked by the particle orbit. Thus, if the magneticfield increases (decreases) slowly compared with ω c , the orbit radius decreases(increases) in such a way that the particle always encircles the same number ofmagnetic “lines of force”.4.4 Inhomogeneous Fields4.4.1 Nonuniform magnetic fieldGrad B driftB∆|Β|-+yxzBFigure 4.8: The grad B drift is caused by the spatial inhomogeneity of B. It isin opposite directions for electrons and ions but of same magnitude.In this case we consider E = 0. As alluded in the introduction, we Taylorexpand the variation of B, B = bˆk, assuming that the variation in B across aLarmor orbit is small. This obtainsB = B 0 + y ∂B∂y+ ... (4.70)wherewehaveassumedthatB varies only in the y-direction and that the firstorder term is small. Since we consider variation in y of order the Larmor radiusr L ,werequirey < ∼ r L ≪ B/( ∂B∂y ) ∼ Lwhere L is the scale length for variation of B. Substituting into Eq. (4.3) andusing Eq. (4.10) we obtainm ˙v y = −qv x B[= −qv ⊥ cos (ω c t) B 0 ± r L cos (ω c t) ∂B ]. (4.71)∂y