Chapter 4 SINGLE PARTICLE MOTIONS 4.1 Introduction

114This shift is shown in Fig. 4.17=( ) vthq(r)ω c= q(r)r L . (4.97)Trapped particlesLet’s now assume that (v ⊥ /v ‖ ) 2 > 1/ɛ so that particles are trapped and bouncein the mirrors produced by the 1/R variation of the toroidal field.We wish to find a relationship between the radial motion of trapped particlesand their parallel velocity. Geometry tells that the radial component of thevertical drift velocity isv r = v z sin θ= mv2 ⊥2qB 0 R sin θ (for v2 ‖ ≪ v⊥). 2 (4.98)The parallel force felt by the particle due to the increasing toroidal field ismv ˙ ‖ = −µ ∂B∂l(4.99)where l ≈ Rφ is the distance coordinate along the magnetic field line. Thecoordinate l is related to the poloidal angle θ through the safety factor q:l ≈ Rφ = rB 0θB θorθ = κl with κ ≡ B θrB 0where r is the radius of the field line with respect to the magnetic axis. The rightside of Eq. (4.99) can now be evaluated as∂B∂l= ∂ ∂l [B 0(1 − ɛ cos κl)]= B 0 ɛκ sin κl=r B θB 0 sin θR 0 rB 0= B θsin θR 0(4.100)where we have used Eq. (4.92). Equation (4.99) now givesv ˙ ‖ = − mv2 ⊥2B 0= − v2 ⊥2B θsin θmR 0B θsin θ. (4.101)R 0 B 0