PLASMA PHYSICS. 1. Charge particle motion in fields of force 2. Plasma particles and their interaction 3. Characteristic propriety of plasma

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1 PLASMA PHYSICS 1. Charge particle otion in fields of force. Plasa particles and their interaction 3. Characteristic propriety of plasa 1

2 1. CHARG PARTICL MOTION IN FILDS OF FORC quations of otion: F = q( + ) d q = + ( ) q, : particle charge and ass;, : electric field and agnetic flux density field; : particle velocity. F = q (F Ε parallel to ) F L = q x (F perpendicular to and )

3 a. equal to zero ( 0, = 0): F = q q d = q q F // : if constant in space and tie, a uniforly accelerated straight otion in the direction of is induced. 3

4 b. = 0; 0, constant in tie and space F L = q d q = When is perpendicular to, lays in the x-y plane and is along the z axis, a z x centripetal force perpendicular to is obtained. This force induces a unifor circular otion. Fro the equality beteen centripetal force acting on the particle and centripetal acceleration it results: r = q ω = r = q r = q y y F x r. (ω - angular velocità velocity, angolare) gyration frequency). (r - gyration raggio di radius) girazione)

5 b. = 0; 0, constant in tie and space d α ω = q x G, y G r = = ω d x q dx q = y x = = cos( ωt + α) = d y q dy = x y = =! sin( ωt + α) A unifor circular otion is induced ith: phase angle defined by the initial conditions; angular velocity or gyration velocity; gyration centre; q Laror radius or gyration radius. y z y F x L x. x G, y G r. 5

6 b. = 0; 0, constant in tie and space is not perpendicular to : - in the plane perpendicular to a unifor circular otion is induced as described previously. If is the coponent of, perpendicular to : r = ω = q In the direction of the coponent of is //. The gyration centre oves along ith velocity //. 6

7 b. = 0; 0, constant in tie and space xaple: lectrons at T ~ 1000 K in = 1 T: = 10 5 /s (T ~ 1000 K if is due to theral energy) = 1 T q = - e = C e = kg ω e 11 1 = ωe = = s, 7 r = r = e e 7

8 1.1 DRIFT VLOCITY lectric Field Drift a. and different fro zero,, and constant in space and tie, The particle oves around the gyration centre in the plane x-y due to the Lorentz force induced by. At the sae tie it is accelerated by the Coulob force due to. Fro the cobination of the to forces a drift otion ith velocity D in the direction perpendicular to and, is obtained. 8

9 1.1.1 lectric Field Drift For q > 0 a clockise circular otion in the plain perpendicular to is induced by the Lorentz force. The Coulob force due to accelerates a positive charged particle hen it oves donard toard increasing x, and decelerates it hen it oves upard. The gyration radius, given by: r = /( q ), is increased by increases of the particle speed and it is decreased for decreasing. Therefore r reaches its axial value in the upper part of its trajectory and it reaches its inial value in the loer part of it. Therefore this results in a epicyclic trajectory ith a drift velocity D of the gyration centre perpendicular to and (electric field drift). For a negative charged particle the circular otion is counter clockise. The acceleration induced by is upards and the deceleration is donards. The gyration radius is big in the top and is sall in the botto of the trajectory resulting in a drift velocity D in the sae direction.

10 1.1.1 lectric Field Drift The electric field drift velocity due to an La electric velocità field di is: deriva di capo elettrico è: infatti A ne definito reference un syste nuovo is sistea defined, di here coordi - nate the particle per cui velocity la velocità is given della by: particella sia: d q = + F = q( + ) ( ) D, = ' = D, = ' + D, d' d q 1 + ( ) = [ + ' + ] ( ) = ( ) - - = - d' q = ' this is the equation of a unifor circular otion. Therefore in a reference syste oving ith a velocity given by the electric field electric field drift velocity D, the charged particle is oving ith a unifor circular otion around the gyration centre. 10

11 1.1.1 lectric Field Drift When ( coponent of perpendicular to ) and // ( // coponent of parallel to ) is different fro zero, the gyration centre oves along ith the velocity //. The coponent of the velocity // reains constant if is perpendicular to ( = ). The coponent of parallel to, //, is equal to zero ( // = 0). The plane on hich the particle oves and the electric field velocity lays, is oving in the direction of ith velocity given by //. // 11

12 1.1.1 lectric Field Drift b. and field different fro zero, and constant in space and tie, and not perpendicular to each other. D, = = D, // induces an acceleration of the gyration centre in the direction of. 1

13 1.1. Gravity Field Drift = 0 and g 0 (g gravity acceleration) g and constant in space and tie. g is the coponent of g perpendicular to = g D,g q = g ω As is a force per unit of charge, is replaced by g /q. The direction of D,g is perpendicular to g and. The orientation of D,g depends on the sign of particle charge q. g = 0 q > 0 g 0 D,g g q < 0 D,g 13

14 1.1.3 Curvature Drift 0, constant in tie. ( is the gradient of scalar perpendicularly to ) For increasing values of, the radius of gyration r = /(q) decreases. This causes a drift of the charged particle. Fro a first order theory it is obtained: I D,C I D,C r = r R = If the plasa current is equal to zero,it is = 0 and / = 1/R (R is the radius of curvature of ) Β = 0 q > 0 Β 0 D,C Β q < 0 D,C

15 1.1.3 Curvature Drift If // 0, the particle q oves along flux tubes of, as e ill see later, hich has a curvature radius R. On the particle acts a centrifugal force given by // /R and a force per unit of charge // /(qr): // R T Δ. ='+". i II D,C = // Rq = // Rω! I = + D,C D,C 1 ωr II D,C ( / ) D, C = + // 15

16 1.1.4 Displaceent Drift constant in space and tie, variable in tie, ( along x, along z). z x y. D, D,S (t) = (t) i = x x0 + x t i (assuing that (ipotesi :! = cost.) x d q q d d = x y q ( + ) + y x = = 0 x0 + x t q d The solution of the associated hoogeneous equation is the usual rotational otion. y + y = 1 x0 + x t 16

17 1.1.4 Displaceent Drift A particular solution leads to a drift velocity: ' ' y x = = 1 q ' ' x0 x + x t x (t) y = x = x q = = D, this che in D,S q z x d y y + x = 0. D, D,S D,S = q t x 17

18 DRIFT VLOCITY D, = D, g = g g = = q ω I II + D, C D,C D,S = q t D,C I D,C r = // Rq II = = D, C // Rω 18

19 1. MAGNTIC MOMNT The agnetic oent associated ith a circuit here a current i flos, is: i µ = isn qω =, S = πr, n = π n is the unitary vector perpendicular to S, surface the border of hich is given by the circuit. The orientation of n is given by the right hand rule. For a charged particle q, rotating around, the current i is given by: µ r ω 1 q = π q = = π π πr i S n Magnetic oent associated ith the circular otion of q ith r = /ω = /( q ). Module of µ ω W q 1 µ = πr q = = Φ here dove W = ; π π Φ = πr

1..1 Magnetic Moent Invariance Invariance of µ: for slo variations of in tie and space, the agnetic oent, associated ith the otion of a charged particle, reains constant in tie and in space.

20 1..1 Magnetic Moent Invariance Invariance of µ: for slo variations of in tie and space, the agnetic oent, associated ith the otion of a charged particle, reains constant in tie and in space. For slo variations in tie of, the induced e..f. is: d dl = n ds S The associated electrical poer is given by the e..f. ties i = qω/(π): Pl = π r d qω π = µ d For the conservation of energy the variation of the kinetic energy W has to be equal to the poer P l (in the direction parallel to there is no influence of on q and dw // / = 0). 0

21 1..1 Magnetic Moent Invariance It follos: P l = Fro the definition of µ: µ = 7 8 : Therefore: < 69 Ø It is possible to see also that for slo variations of in space, µ = const. Therefore µ is an invariance of the particle otion in a field. It follos that: = 6(<:) 69 and fro the expression of P l : P l = µ 6: = µ 6: < 69 = 0. As 0 it follos that 6< = 6(<:) 69 and a. µ Φ Β : µ = const. Φ Β = const. Therefore the particle q is o ving on flux tube of = µ 6: =0 and that µ = costant in t. b. µ = W / = const., W = W + W // = const. For increasing Β, W increases and W // decreases. 1

22 1.3 MAGNTIC MIRRORS As µ = W / is constant and as the kinetic energy W = + // is constant the charged particle tends to be reflected fro regions of increasing agnetics fields.

23 1.3 MAGNTIC MIRRORS + = + / /a a a = a b b / /b b W is constant for a otion of q in fields Invariance of µ = / /b + / /a a 1 The parallel coponent of the velocity becoes equal to zero for //? a = 0: / /a = b 1 a a 3

24 1.3 MAGNTIC MIRRORS The irror ratio is defined as: Si definisce coe rapporto di specchio: R = ax in The le particelle reflected cariche charged riflesse particles sono are those quelle for per hich: cui: / /a R 1 a That od anche: is: a a R 4

25 1.3 MAGNTIC MIRRORS The escape cone and the escape angle θ are defined by: tan -1 θ = I // J I J liit = R 1 The particle is reflected if it oves ith a velocity ith a direction θ > θ. For velocities directed ithin the escape cone ith θ > θ the particle leaves the agnetic syste. θ 5

26 1.3 MAGNTIC MIRRORS 6

5. Charge Particle Motion in Fields

5. Charge Particle Motion in Fields 5 Charge Particle Motion in Fields 1 1 CHARG PARTICL MOTION IN FILDS OF FORCS uations of otion: F ( + ) d ( + ), : particle charge and ass;, : electric field and agnetic flu densit field; : particle velocit