5. Charge Particle Motion in Fields

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Marvin Harper
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charge and ass;, : electric field and agnetic flu densit field;

1 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 F (F parallel to ) F L (F perpendicular to and )

2 a eual to zero ( 0, 0): F d F // : if constant in space and tie, a uniforl accelerated straight otion in the direction of is induced 3 b 0; 0, constant in tie and space is perpendicular to : F L d When is perpendicular to, las in the - plane and is along the z ais, a centripetal force perpendicular to is obtained This force induces a unifor circular otion Fro the eualit beteen centripetal force acting on the particle and centripetal acceleration it results: r ω r r z F^ (ω ( - angular velocità velocit, angolare) gration freuenc) (r - gration raggio di radius) girazione) r

3 b 0; 0, constant in tie and space d α G, G r d d cos( t + a ) d - d! sin( t + a ) A unifor circular otion is induced ith: phase angle defined b the initial conditions; angular velocit or gration velocit; gration centre; Laror radius or gration radius z F L G, G r 5 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 previousl If is the coponent of, perpendicular to : r ω In the direction of the coponent of is // The gration centre oves along ith velocit // 6

4 b 0; 0, constant in tie and space aple: lectrons at T ~ 1000 K in 1 T: 10 5 /s (T ~ 1000 K if is due to theral energ) 1 T - e C e kg e 11-1 e s, e - 7 r r e 7 11 DRIFT VLOCITY 111 lectric Field Drift a and different fro zero,, and constant in space and tie, The particle oves around the gration centre in the plane - due to the Lorentz force induced b At the sae tie it is accelerated b the Coulob force due to Fro the cobination of the to forces a drift otion ith velocit D in the direction perpendicular to and, is obtained 8

5 111 lectric Field Drift For > 0 a clockise circular otion in the plain perpendicular to is induced b the Lorentz force The Coulob force due to accelerates a positive charged particle hen it oves donard toard increasing, and decelerates it hen it oves upard The gration radius, given b: r /( ), is increased b increases of the particle speed and it is decreased for decreasing Therefore r reaches its aial value in the upper part of its trajector and it reaches its inial value in the loer part of it Therefore this results in a epicclic trajector ith a drift velocit D of the gration centre perpendicular to and (electric field drift) For a negative charged particle the circular otion is counter clockise The acceleration induced b is upards and the deceleration is donards The gration radius is big in the top and is sall in the botto of the trajector resulting in a drift velocit D in the sae direction 111 lectric Field Drift The electric field drift velocit due to an La electric velocità field di is: deriva di capo elettrico è: infatti A ne definito reference un sste nuovo is sistea defined, di here coordi - nate the particle per cui velocit la velocità is given della b: particella sia: d F ( + ) ( + ) D, ' - D, ' + D, ( ) ( ) d' d æ ö 1 + ç ( ) [ + ' + ] è ø d' ' this is the euation of a unifor circular otion Therefore in a reference sste oving ith a velocit given b the electric field electric field drift velocit D, the charged particle is oving ith a unifor circular otion around the gration centre 10

6 111 lectric Field Drift When ( coponent of perpendicular to ) and // ( // coponent of parallel to ) are different fro zero, the gration centre oves along ith the velocit // The coponent of the velocit // reains constant if is perpendicular to ( ) The coponent of parallel to, //, is eual to zero ( // 0) The plane on hich the particle oves, is oving in the direction of ith velocit given b // // 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 gration centre in the direction of 1

7 11 Gravit Field Drift 0 and g 0 (g gravit acceleration) g and constant in space and tie g^ is the coponent of g perpendicular to g^ D,g g^ As ^ is a force per charge unit, ^ of D, is replaced b g^ / The direction of D,g is perpendicular to g^ and The orientation of D,g depends on the sign of particle charge g 0 > 0 g 0 D,g g < 0 D,g Curvature Drift 0, constant in tie ( is the gradient of scalar perpendicularl to ) For increasing values of, the radius of gration r /() decreases This causes a drift of the charged particle Fro a first order theor it is obtained: r I Ñ ^ r R I ^ ^ If the plasa current is eual to zero, it is 0 and / 1/R (R is the radius of curvature of ) 0 > 0 < 0 0

8 113 Curvature Drift If // 0, the particle oves along flu tubes of, as e ill see later, hich has a curvature radius R On the particle acts a centrifugal force given b // /R and a force per unit of charge // /(R): II // R // Rω // R T Δ '+" i! I + 1 R II ( / ) D, C ^ + // Displaceent Drift constant in space and tie, variable in tie, ( along, along z) z D, D,S (t) (t) i æ ç 0 + è t ö ø i (ipotesi : (assuing that! cost) d d d ( + ) t æ ç è ö ø d The solution of the associated hoogeneous euation is the usual rotational otion æ ç è 0 + ö t ø 16

9 114 Displaceent Drift A particular solution leads to a drift velocit: ' ' 1 æ - ç 0 è + ö t ø (t) ' - ' D, this che in D,S d + 0 z D, D,S D,S t 17 DRIFT VLOCITY ^ D, D, g g ^ g^ I II + D, C D,S t I II r Ñ ^ ^ // // R R D, C 18

10 1 MAGNTIC MOMNT The agnetic oent associated ith a circuit here a current i flos, is: n is the unitar vector perpendicular to n µ isn S, surface the border of hich is given b the circuit The orientation of n is i S given b the right hand rule For a charged particle, rotating around, the current i is given b: i, S p r, n - p µ r 1 - p - - p p p r ^ Module of µ æ W^ ç 1 µ p r F ç here dove W^ ^ ; p p ç è Magnetic oent associated ith the circular otion of ith r / /( ) F ö p r ø 11 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 ef is: d ò dl - ò ò n ds S The associated electrical poer is given b the ef ties i ω/(π): Pl p r d µ d p For the conservation of energ the variation of the kinetic energ W^ has to be eual to the poer P l (in the direction parallel to there is no influence of on and dw // / 0) 0

11 11 Magnetic Moent Invariance It follos: P l ;< Fro the definition of µ: µ <? Therefore: ;< ;A 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: ;(A?) a μ Φ : μ const Φ const Therefore the particle is o ving on flu tube of b μ W / const, W W + W // const For increasing, W increases and W // decreases and fro the epression of P l : P l µ ;? µ ;? + ;A ;< ;(A?) and ;< µ ;? ;A 0 As 0 it follos that 0 and that µ costant in t 1 13 MAGNTIC MIRRORS As μ W / is constant and as the kinetic energ W + // is constant the charged particle tends to be reflected fro regions of increasing agnetics fields

12 13 MAGNTIC MIRRORS + + / /a a a a b b / /b b W is constant for a otion of in fields Invariance of μ / /b + / /a a 1 b 6 The parallel coponent of the velocit becoes eual to zero for // D a 0: / /a b 1 a a 3 13 MAGNTIC MIRRORS The Si definisce irror coe ratio rapporto is defined di as: specchio: R a in The le particelle reflected cariche charged riflesse particles sono are those uelle for per hich: cui: / /a R - 1 ^ a That od anche: is: a ^ a R 4

13 13 MAGNTIC MIRRORS The escape cone and the escape angle θ are defined b: tan -1 θ 5 // M 5 M liit R 1 The particle is reflected if it oves ith a velocit ith a direction θ > θ For velocities directed ithin the escape cone ith θ > θ the particle leaves the agnetic sste θ 5 13 MAGNTIC MIRRORS 6

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

PLASMA PHYSICS. 1. Charge particle motion in fields of force 2. Plasma particles and their interaction 3. Characteristic propriety of plasma PLASMA PHYSICS 1. Charge particle otion in fields of force. Plasa particles and their interaction 3. Characteristic propriety of plasa 1 1. CHARG PARTICL MOTION IN FILDS OF FORC quations of otion: F =