Chapter 4 SINGLE PARTICLE MOTIONS 4.1 Introduction
Chapter 4 SINGLE PARTICLE MOTIONS 4.1 Introduction Chapter 4 SINGLE PARTICLE MOTIONS 4.1 Introduction
104Since the B-field is time invariant, we can average over a cyclotron period〈F y 〉 = 〈m ˙v y 〉 = ∓qv ⊥ r L∂B∂y 〈cos2 (ω c t)〉 (4.72)so that there is a residual y-force (but no x directed force - show this).resulting drift is given by Eq. (4.21)Thev ∇B = 1 F ×Bq B 2= 1 〈F y 〉Bîq B 2= ∓ v ⊥r L2BAlternatively, this can be expressed in vector formwhereB×∇B =∣∇B ≡∇|B|=î∂|B|∂xî ĵ ˆk0 0 B z0 ∂B∂y∂B î. (4.73)∂y0+ ĵ ∂ |B|∂y∣∂ |B|+ ˆk∂z .∇|B| often simplifies to ∇B z because B z ≫ B r ,B θ . The general result is(4.74)v ∇B = ± 1 2 v B×∇B⊥r L . (4.75)B 2The drift is in opposite directions for electrons and ions (see Fig. 4.8) but of thesame magnitude. The drift therefore results in a net current across B.Curvature driftIf the magnetic lines of force are curved, the charged particles feel a centrifugalforce proportional to the radius of curvature R c (see Fig. 4.9). The force felt isF c = mv2 ‖R cˆr = mv2 ‖ R cR 2 c(4.76)and the resulting drift can be written asv R = mv2 ‖qB 2 R c ×BR 2 c(4.77)
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)
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104Since the B-field is time invariant, we can average over a cyclotron period〈F y 〉 = 〈m ˙v y 〉 = ∓qv ⊥ r L∂B∂y 〈cos2 (ω c t)〉 (4.72)so that there is a residual y-force (but no x directed force - show this).resulting drift is given by Eq. (4.21)Thev ∇B = 1 F ×Bq B 2= 1 〈F y 〉Bîq B 2= ∓ v ⊥r L2BAlternatively, this can be expressed in vector formwhereB×∇B =∣∇B ≡∇|B|=î∂|B|∂xî ĵ ˆk0 0 B z0 ∂B∂y∂B î. (4.73)∂y0+ ĵ ∂ |B|∂y∣∂ |B|+ ˆk∂z .∇|B| often simplifies to ∇B z because B z ≫ B r ,B θ . The general result is(4.74)v ∇B = ± 1 2 v B×∇B⊥r L . (4.75)B 2The drift is in opposite directions for electrons and ions (see Fig. 4.8) but of thesame magnitude. The drift therefore results in a net current across B.Curvature driftIf the magnetic lines of force are curved, the charged particles feel a centrifugalforce proportional to the radius of curvature R c (see Fig. 4.9). The force felt isF c = mv2 ‖R cˆr = mv2 ‖ R cR 2 c(4.76)and the resulting drift can be written asv R = mv2 ‖qB 2 R c ×BR 2 c(4.77)
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