Charged particle motion in magnetic field

Transcription

1 Chaged paticle otion in agnetic field Paticle otion in cued agnetic fieldlines We diide the equation of otion into a elocity coponent along the agnetic field and pependicula to the agnetic field. Suppose that the fieldlines ae cicula. d d d db + b+ q b ϕ sinϕ x+ cosϕ y d : b coponent constant db dϕ dϕ cosϕ x+ sin ϕ y ρ dϕ db ρ ρ ρ

2 We diide the elocity into a peiodic pat and a dift pat ~ t t + V d The pependicula coponent of theequ. of otion db d + q + qvd d q + qvd + Taking thetie aeageof this equation gies ρ ρ qv d + ρ ρ

3 Soling fo the dift elocity we get the so called cuatue dift elocity V d q ρ ρ Genealising this expession to a cued field linewith cuatue we hae V q Fo a cued agnetic field line the agnetic field is non-unifo we ust then add the gad -dift to get the total dift elocity. Fo a acuu field we hae locally

4 The local total dift of a paticle in a cued iotational field then becoes finally V + V + q xaple: Paticle otion in dipole agnetic field Poble.8 in Chen Suppose the eath s agnetic field is at the equato and falls off as /, as fo a pefect dipole. Let thee be an isotopic population of ev potons and -kev electons, each with density n 7 - at 5 eath adii in the equatoial plane. a Copute the ion and electon gad dift elocities. b Does an electon dift eastwad o westwad? c How long does it take fo an electon to encicle the eath? d Copute the ing cuent in A/ 5 T

5 Poble.8. The gad dift is gien by 5 T eath adius q V Isotopic potons kineticenegy + κt κt pe degees of feedo Conention Chen ev 9 ev.6 J 6 κ T

6 Poble.8 s q V get we at ealuated q q q q V / The potons dift in the westwad diection

7 Poble.8 s e V get we at electons the Fo s q V get we at ealuated / /

8 Poble.8 c How long does it take fo an electon to encicle the eath? T 6 π 5 π sek V h 4.8h 6 d Copute the ing cuent in A/ j n ev ev 7 9 p, e, A/ 7

9 Paticle otion in agnetic ios Fo we hae + If is gien at and does not ay uch withthenwehaeappoxiately

10 Magnetic ios cont. The -coponent of the equation of otion can be witten L q q q d θ θ d q q d Fo which we get θ Ion with guiding cente at the -axis +

11 Magnetic ios cont. d W d d d W dw d d theefoe d d so field agnetic a in paticle fo a conseed is enegy kinetic The d W d d +

12 Magnetic ios cont. otion of constant appoxiate an is oent agnetic The constant W d W dw eaanging d W dw d d µ

13 The agnetic oent of a paticle + The agnetic dipole oent is defined as I S Fo a chaged paticle we can define the agnetic oent as W q Ω q µ IS S q S π T π π q

14 Tapped paticles in a io field Suppose we hae a paticle at and with initial elocity µ,,, µ the agnetic oent of the paticle is then which is a constant of otion, theefoe, We also hae enegy conseation so,, whee the pied elocity is the elocity at an abitay position. the coesponding agnetic field at this point. + +,,

15 Tapped paticles The conseation of agnetic oent and kinetic enegy gies µ, + +,, +,, We note that thee is a possibility to hae the paallell elocity eo at axiu if the following condition of the initial paallel elocity fulfills the condition,, These paticles ae theefoe tapped in the agnetic io

16 Tapped paticles Loss cone, We define a pitch-angle Θ as follows tan Θ,,,, The axiu paallel initial elocity fo the paticle to be tapped is ax / /,,ax,, whee ax is the so called io atio Using the definition of the pitch-angle this condition can be witten tan Θ Paticles that dont fulfill this condition ae lost

17 Paticles tapped in the eath s agnetic field

18 Adiabatic inaiants The agnetic oent µ is an exaple of an adiabatic inaiant. The definition of an adiabatic inaiant is ~ L µ t µ + Oexp / ε f t whee ε << L The eo is the second tewhich when Taylo-expanded fo sall ε gies exp / ε ε + ε + ε + ε This eans that the eo i salle than any powe of the sall paaete ε.

19 Adiabatic inaiants g ω t ε << l εt If the length of the pendelu is odified slowly then Wt ω t ω const << ω oe a peiod Adiabatic inaiant

20 Poble. A plasa with an isotopic elocity distibution is placed in a agnetic io tap with io atio 4. Thee ae no collisions, so the paticles in the loss cone siply escape and the est eains tapped. What faction is tapped? Isotopic elocity distibution f F F is constant on a sphee with adius y x Kinetic enegy is a constant of otion + +,,

21 Poble.,, + + The agnetic oent is a constant of otion ->, µ Cobining and we get +,, Paticles ae eflected at a point whee the paallel elocity becoes eo +,,

22 Poble. -> +,, The axiu paallel elocity at the oigin fo the paticle to be tapped is then,,,,,, tanθ Paticles that ae lost fo the io ae theefoe the paticles located in the two loss cones.

23 Poble. y x The nube of paticles pe olue in the two loss cones is gien by loss cones θ π θ x y sin θ θ ϕ 4 π [ cos θ ] F d d d F dd d F d 4 π Fd cos θ

24 Poble. The nube of paticles pe olue in the two loss cones is gien by loss cones θ π θ x y sin θ θ ϕ 4 π [ cos θ ] F d d d F dd d F d 4 π Fd cos θ The total nube of paticles pe olue is π π F dxdyd F sin θ d dθ dϕ 4 π F d All elocity space The faction of paticles lost and tapped paticles in the io is theefoe Losses cos θ Tapped cos θ

25 Poble. In the pesent poble the io atio is 4,, θ tanθ π 6 π Losses cos 6 Tapped faction

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