Chapter 4 SINGLE PARTICLE MOTIONS 4.1 Introduction
Chapter 4 SINGLE PARTICLE MOTIONS 4.1 Introduction Chapter 4 SINGLE PARTICLE MOTIONS 4.1 Introduction
106Figure 4.10: The grad B drift for a cylindrical field.Note that the two contributions add with similar magnitude because 〈v 2 ‖ 〉∼k B T/m and 1 2 〈v2 ⊥ 〉∼k BT/m.Magnetic mirrors — ∇B ‖ BWe have looked at particle drifts when ∇B is at an angle to B. What happenswhen the gradient is aligned with B? This situation is encountered in magneticmirrors where the magnetic field strength increases along the direction of thelines of force as shown in Fig. 4.11.Figure 4.11: Schematic diagram showing lines of force in a magnetic mirror device.We shall show that a charged particle inside such a magnetic topology canbe trapped under certain circumstances. Let’s describe mathematically the fieldstructure. The field must be divergence free (no sources or sinks): ∇.B =0. In
4.4 Inhomogeneous Fields 107cylindrical geometry this gives1 ∂rB rr ∂r+ ∂B z∂zProvided ∂B z /∂z does not vary much with r we have∫ rrB r = −0=0. (4.80)r ∂B z ∂B zdr ≈−r2∂z 2 ∂z(4.81)orB r = − r ∂B z2 ∂z . (4.82)Any radial inhomogeneity of B r gives an azimuthal drift B z ˆk×∇Brˆr about theaxis of symmetry [see Fig. 4.11] but there is no radial drift (why?).What is the effect of the Lorentz force in the cylindrical field?F = qv×B =∣∣ˆr ˆθ ẑ ∣∣∣∣∣∣v r v θ v zB r 0 B z= ˆr(qv θ B z ) − ˆθq(v r B z − v z B r ) − ẑ(qv θ B r ). (4.83)For simplicity, consider a particle spiralling along the axis (r = r L )sothatwecan ignore grad B drifts. The logitudinal (axial) force isq ∂B zF z = v θ r L2 ∂z= ∓v ⊥ r Lq2∂B z∂z= ∓ qv2 ⊥ ∂B z2ω c ∂zions and electrons= − mv2 ⊥2B ∇ ‖Bwhere v ⊥ is the cyclotron speed. Expressed in terms of the magnetic moment,F ‖ = −µ∇ ‖ B. (4.84)This force is away from increasing B and is equal for particles of equal energy(independent of charge).A particle moving from a weak field region to a strong field sees a time changingmagnetic field. However, the magnetic moment stays constant during thismotion provided the rate of change is slow. Since µ is a constant of the motion,thenv 2 ⊥0B 0= v2 ⊥mB m(4.85)
- Page 9: 4.3 Time Varying Fields 95analogous
- Page 12: Materials, Trainings & Technical As
- Page 15 and 16: 4.3 Time Varying Fields 101withD =
- Page 17 and 18: 4.4 Inhomogeneous Fields 103where
- Page 19: 4.4 Inhomogeneous Fields 105r^F cθ
- Page 23 and 24: 4.4 Inhomogeneous Fields 109Figure
- Page 25 and 26: 4.4 Inhomogeneous Fields 111Figure
- Page 27 and 28: 4.4 Inhomogeneous Fields 113Passing
- Page 29 and 30: 4.4 Inhomogeneous Fields 115Compari
- Page 31 and 32: 4.4 Inhomogeneous Fields 117Problem
4.4 Inhomogeneous Fields 107cylindrical geometry this gives1 ∂rB rr ∂r+ ∂B z∂zProvided ∂B z /∂z does not vary much with r we have∫ rrB r = −0=0. (4.80)r ∂B z ∂B zdr ≈−r2∂z 2 ∂z(4.81)orB r = − r ∂B z2 ∂z . (4.82)Any radial inhomogeneity of B r gives an azimuthal drift B z ˆk×∇Brˆr about theaxis of symmetry [see Fig. <strong>4.1</strong>1] but there is no radial drift (why?).What is the effect of the Lorentz force in the cylindrical field?F = qv×B =∣∣ˆr ˆθ ẑ ∣∣∣∣∣∣v r v θ v zB r 0 B z= ˆr(qv θ B z ) − ˆθq(v r B z − v z B r ) − ẑ(qv θ B r ). (4.83)For simplicity, consider a particle spiralling along the axis (r = r L )sothatwecan ignore grad B drifts. The logitudinal (axial) force isq ∂B zF z = v θ r L2 ∂z= ∓v ⊥ r Lq2∂B z∂z= ∓ qv2 ⊥ ∂B z2ω c ∂zions and electrons= − mv2 ⊥2B ∇ ‖Bwhere v ⊥ is the cyclotron speed. Expressed in terms of the magnetic moment,F ‖ = −µ∇ ‖ B. (4.84)This force is away from increasing B and is equal for particles of equal energy(independent of charge).A particle moving from a weak field region to a strong field sees a time changingmagnetic field. However, the magnetic moment stays constant during thismotion provided the rate of change is slow. Since µ is a constant of the motion,thenv 2 ⊥0B 0= v2 ⊥mB m(4.85)
Change language