Effects of an Inhomogeneous Magnetic Field (E =0)

Transcription

1 Effects of an Inhoogeneous Magnetic Field (E =0 For soe purposes the otion of the guiding centers can be taken as a good approxiation of that of the particles. ut it ust be recognized that during the particle s Laror gyrations, it saples regions where or E ay be different fro what they are at the guiding center (GC. For the agnetic field, the effects are iportant, and they are to be studied here. We decopose forally the particle s velocity v into a Guiding Center part v GC and a Laror part v L. In turn, the GC velocity has a coponent v along, plus a coponent (a drift perpendicular to : The Laror velocity is defined by, v GC v = v GC + v L ( = v + v D dv L = qv L GC ( v L = 0 where has been explicitly taken to be at the particle s Guiding Center. We next assue the plasa is agnetized with respect to the particles in question, so that the Laror radius is sall : R L << Distance over which changes appreciably (3 We also assue (to be verified later that the drift velocities are sall copared to those involved in Laror otion: v D << v L (4 ecause of (3, (r (r being the instantaneous location of the particle, easured fro the GC can be expanded to first order about GC : The full equation of otion is then, d(v + v D + v L (r = GC +(r (5 = q(v + v D + v L ( CG +(r (6 Of the ters on the right, clearly v GC = 0 (parallel vectors. The ter v D (r is of second order (product of sall ters, and will be neglected. The ter q(v L GC cancels the dv L / on the left, according to (. This leaves, dv + dv D = q [ ] v (r + v D GC + v L (r (7

2 n If the line through the GC has a radius of curvature, then, GC dv = dv b b + v n (8 where b and n are unit vectors tangent to and along the inner noral to the line, respectively. We can now separate out the coponents of Eq.(7 along ( and perpendicular to ( Along : Perpendicular to : v n + dv D dv = q [ ] v L (r = q [ ] v (r + v D GC +[v L (r ] (0 (9 The next step is to average these equations over each quasi-orbit about the GC ( quasi, because, in an inhoogeneous field the Laror orbits ay not quite close on theselves - we ignore that. During each such orbit, v, v D and GC are regarded as constant, but r and v L vary cyclically. ecause of this, any ter where r and v L appear linearly will average to zero, but care ust be taken when they appear in pairs. In particular, v (r = 0 ( Laror Note: If we had retained E in the forulation (even with soe inhoogeneity, this averaging step would have eliinated any first order E (no E drift ters exist. x n b y To see what the averaging results are for the other ters, we now re-write (9 and (0 in a local Cartesian coordinate frae z as shown in the figure. The y direction copletes a righthanded set. unit vector Writing out Eq. (9: dv z = q ( v Lx x y x The corresponding unit vector is the bi-noral bn = b n ( y + y y ( v Ly x x x + y x y Laror (3 Projecting Eq. (0 along the noral (x direction, { vz + dv D x = q ( v Dy z + v Ly x z x z + y y Laror } (4

3 Projecting Eq. (0 along the bi-noral (y, { ( dv Dy q = v Dx z v Lx x x z z + y dy Laror } (5 y r The Laror otion is shown in the figure for a positive q. If the Laror radius is r L,wehave x = r L cos ω c t y = r L sin ω c t x ω c t v Lx = ω c r L sin ω c t v = ω r cos ω t Ly c L c (6 (7 where ωc = q is the gyro frequency. Therefore, when doing v L the averaging, xv Ly = ω c r L cos ω c t = ω L L c rl yvly = ω c rl sin ω c t cos ω c t L L = 0 xv Lx L = ωcr L sin ω c t cos ω c t L = 0 yv Lx L = ω c r L sin ω c t = ω L c rl (8 Notice the occurrence of the group q ω c r L. Fro q = ωc and ω cr L = v, this group is, q ω crl = v = μ (9 where μ is the agnetic oent μ = v /. We now have fro (3, (4, and (5, dv Dz dv Dx = v z v + ω c v D y z (a Rc x dv Dy v = ω c v Dx z (b (0 y ( = v x x + y v = z (c y z Notice we have used = 0 in the last equation. Also, since = + + x y z and x = y = 0 due to the choice of axes, = z. Furtherore, taking gradients, = x x + y y + z z = z and so z =. Hence the subscript z will be oitted fro z fro here on. 3

4 These averaged equations contain rich inforation. Let us first consider separately (0a and (0b, i.e. the part. Following the procedure used before, we define a coplex v D vector, v D = v Dx + iv Dy ( and cobine the equations as (a + i(b: d (v Dx + v Dy dv D + iω cv D = v + ω c (v Dy iv Dx v = v v ( where = i + j. The hoogeneous solution of ( is the Laror otion, which we x y disregard once again, since we are after the drifts. These drifts are given by the particular solution, v D = v D,part = i + i ω c ω c v v v D = i v q + i v q (3 In vector ters, v D = v q bn + v q b (4 Of these, the first ter is the Inertia Drift due to the centrifugal force on a particle which is sliding along a curved line. One echanistic way to view this is the following: The centrifugal force is v n. This can be countered by the agnetic force due to a drift along bn, chosen such that, v n = qv DI R = qv DI ( bn b c and, fro (, ( bn b = n, leaving v DI = v q, as in (4. b b The second ter in (4 is the so-called - Drift (Notice that b = b. This will be later shown to be also related to - line curvature, since, due to = 0, the agnitude of can only vary if the lines are curved. This - drift is seen to be perpendicular to b and the and directed to the left of. A physical picture is in shown in the figure. 4

5 To show how is related to line curvature, we start fro Frenet s first forula (illustrated in the figure: n n b ds dθ b + d b = b +( b bds and so, (b b = (b b (5 and use the vector identity ( = ( Dividing this by, and reebering = b, (b = b ( ( =( b (. ( =, = [ b ( ] b bn which is also n, according to (5. ties b, to fro b n = ( bn : b = b ( b ( }{{} We now cross-ultiply or b b ( = ( b bn = ( and substituting this into the expression for (v D (fro (4, ( v bn (v D = q b = v ( q ( In the absence of appreciable net current to, one would have = 0, and (v D would be strictly proportional to -line curvature (/Rc, and directed along the binoral vector just as the inertia drift. We should therefore group these ters together. Starting at Eq. (4, ( v + v v D = v bn ( (7 q q Parallel Drifts (6 To coplete the picture, we now return to Eq. (0c and investigate the parallel drift effects. We had, dv = v (8 z 5

6 dv In steady state, dv = v = d( v. Since the total kinetic energy is conserved (no dz dz E forces, and forces perpendicular to v, ( d v = d ( v dz dz Substituting in (8, or ( d v dz d dz ( = v d dz ( v = 0 (9 The quantity μ = v (the agnetic oent, μ, according to our previous definition is seen to be invariant for a particle which oves along a -line while executing Laror rotations perpendicular to. This is soeties called the second adiabatic invariant ( adiabatic refers to the absence of energy input, as fro a tie-varying field. There are now two constants of the otion: and E = v + v (30 μ = Eliinating the perpendicular energy v v = μ, we obtain (3 v = E μ (3 Showing that the parallel energy decreases as the particle oves into a region with a stronger field. At the sae tie, of course, the perpendicular energy increases: the varying field has the effect of transferring energy fro ( to( or viceversa. If increases enough, a particle of given energy and agnetic oent can be stopped in its parallel otion, and reflected. Suppose the particle has a parallel velocity v 0 when the field is 0. It will be reflected at turning point T if v 0 + μ 0 = μ T : or T = 0 + T 0 v 0 μ ( v = + 6 = 0 + v 0 0 v 0 v 0 (33

7 Defining the pitch angle θ = atan ( v /v, T = + cotθ0 = 0 sin θ 0 (34 0 ax v v θ 0 0 v 0 v Any particle for which sin θ 0 > 0 / ax will be reflected at soe T given by (34. ut those with sall enough pitch angle (sin θ 0 < 0 / ax will not, and will escape through the agnetic bottleneck. If a Magnetic ottle is used to confine plasa between solenoids, this leakage of low-pitch particles creates a peculiar distribution, with particles within a loss-cone in pitch being ostly absent. 7

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