Chapter 4 SINGLE PARTICLE MOTIONS 4.1 Introduction

4.4 Inhomogeneous Fields 105r^F cθ^BR cFigure 4.9: The curvature drift arises due to the bending of lines of force. Againthis force depends on the sign of the charge.Combined grad B and curvature driftsConsider the ∇B drift that accompanies curvature in a cylindrical geometry:B = B θ =(B 0 /r)ˆθso∇B = ˆr ∂B 0/r= −ˆr(B 0 /r 2 )=−ˆrB θ /r = −r(B θ /r 2 )∂rwherewehaveused∇×B = 0 in vacuum andUsing Eq. (4.75) we have(∇×B) z = 1 r∂rB θ∂r⇒B θ ∼ 1 rv ∇B = ± 1 2 v B×∇B⊥r LB 2= ± 1 v⊥2 B×(−R c |B|)2 ω c B 2 Rc2= 1 mv⊥2 R c ×B(4.78)2 q RcB 2 2wherewehaveusedB/ω c = m/ |q|. Combining with the curvature drift we findv T = v ∇B + v R = m ( ) ( Rc ×Bv 2q RcB 2 2 ‖ + 1 )2 v2 ⊥ . (4.79)