Massachusetts Institute of Technology Introduction to Plasma Physics

Transcription

1 Massachustts Insttut of Tchnology Intoducton to Plasma Physcs NAME 6.65J,8.63J,.6J R. Pak Dcmb 5 Fnal Eam :3-4:3 PM NOTES: Th a 8 pags to th am, plus on fomula sht. Mak su that you copy s complt. Each poblm s woth %, although th poblms dff somwhat n complty. It s a good da to look though all poblms bfo dcdng n whch od to attack thm! Poblm. Th fgu shows fld lns n a cylndcally symmtc magntc mo plasma confnmnt dc. z Th fld s gn by B = B z l z ˆ l zˆ B a A patcl obts n th mo wth ts gudng cnt mong along th z-as. At z=, t has total ngy W = W W// wh th subscpts and // f to ppndcula and paalll to th fld. Calculat zt, th moton of th gudng cnt. b Calculat th gad B and cuatu componnts of th dft locty of th patcl whn ts gudng cnt s locatd at

2 z =, = l / c Fnd an quaton fo, th tunng pont of a patcl wth a gudng cnt that passs though th pont z =, = l / wth ngy W = W calculaton of th quaton of th fld lns. z t W//. Do not sol th quaton. Hnt: ths wll qu

3 Poblm. Th dal MHD quatons togth wth th appopat fom of Mawll s quatons a: ρ ρ = t d ρ = j B p dt γ d pρ dt E B = B = μ j, B E = t B = As w ha sn n class, th lnazd son of ths quatons suppots undampd sha and compssonal Alfén was n a homognous plasma mmsd n a unfom magntc fld. Fo ampl, wth B = zˆb th E -fld n th sha wa can b wttn E = E cos k Az ωt ˆ wh k = ω / and ca s th Alfén spd. A c A If th conductty of th plasma s consdd to b lag but fnt, th only modfcaton to th abo dal quatons s n th Ohm s law, whch taks th fom E B = η j As pctd, th sstty lads to dampng of th Alfén wa, as you a askd to show n ths poblm. a Assum th qulbum plasma s homognous, and wthout cunt o flow. Spcfcally assum ρ = ρ ρ, t B = B B, t j = j, t =, t E = E, t p = p p, t wh th quantts labld wth a subscpt a constants whl th quantts labld wth subscpt a small ptubatons. Wt down a complt lnazd st of quatons fo th ptubd quantts. b Assum that w a ntstd n was wth spac-tm dpndnc gn by p k ωt. In th absnc of sstty, th ptubd magntc fld s gn by k B B = ω In th psnc of sstty, th ptubd magntc fld can b wttn n th fom 3

4 k B B = ω Dtmn ω. c Agan n th absnc of sstty, th locty s dtmnd by soluton to th quaton [ k k V zˆ ] zˆ c kk V ω V = c, A wh w ha takn th qulbum magntc fld to b n th z-dcton. In th psnc of sstty, th sam quaton appls, but wth c A c and c S c. Dtmn and. A S S d By makng th substtuton suggstd n pat c, dtmn th dspson laton of th sha Alfén mod popagatng n th dcton of B wth V B, ncludng th ffct of sstty. Assum that ω s al cospondng to a souc of ths fquncy. Thn, fo th wa consdd n pat d, k = zˆ k k, wh k and k a th al and magnay pats of th wanumb. Dtmn k and k. c A c S 4

5 Poblm 3. Consd an asymmtc θ-pnch n whch th plasma gdly otats,.., V = θˆ Ω wh Ω s a constant. Th cntal bta KT aω = m p / B / s 5%, and th dg spd achs sonc lls,.., μ, wh T s th cntal tmpatu and a s th plasma adus. Thoughout ths poblm assum T = T and nglct scosty. Not: Th two pats of ths poblm a ndpndnt. a Fnd Ba-, th B-fld just nsd th dg of th plasma n tms of B, th fld at th cnt of th plasma. Assum that th plasma pssu anshs at = a. b Th plasma tmpatu as a functon of adus s gn by T = T / a whl th dnsty s gn by n = n wh n s a constant. Dtmn th confnmnt tm of th plasma dfnd by τ = W E P wh W s th thmal ngy stod p unt lngth and P s th pow lost by conducton p unt lngth. Nglct any spatal aaton of l nλ. Epss you answ n tms of: a, th plasma adus; τ, a collson tm aluatd at = ;, th Lamo adus of a thmal patcl also aluatd at = ; and Ba-/BB. B su to poply dntfy th collson tm and th Lamo adus n tms of typ and spcs. A pag of possbly usful fomulas poducd fom th Plasma Fomulay s found on th nt pag. ρ L 5

6 Poblm 4. Consd -D plasma oscllatons popotonal to p k t ω n a hot plasma wth a -D lcton dstbuton functon gn by ~ f =. Fo smplcty assum that k s al, but that ω could b compl. a Dtmn an algbac dspson laton fo lcton oscllatons, assumng that th ons a mmobl. b Sol th dspson laton obtand n a fo. ωk c Now assum that th ons ha a dstbuton functon gn by ~ f =, k << ω / whl th lcton dstbuton functon s th sam as n pat a. Assumng, dtmn fo on acoustc was. ωk Possbly usful ntgals: > = d < = d > = d < = d > = d < = d 6

7 Poblm 5. A plan wa s ncdnt on a plasma slab, as ndcatd schmatcally n th fgu blow. y n k z Th plasma dnsty s unfom an th y- and z-dctons, but as n th -dcton and s gn by n = n wh l, th gadnt scal lngth, s long compad to th walngth..., kl >>. Th s a magntc fld n th z-dcton, B = zˆb wh s constant. Th paamts a such that B l ω n = n = ω. p Thoughout ths poblm, consd only th lcton spons to th wa fld. c 7

8 a On th lft gaph blow, ndcat th gons of popagaton up to / l = 4 fo th odnay mod. b On th ght gaph blow, ndcat th gons of popagaton up to / l = 4 fo th taodnay mod. c What s th mamum dpth of popagaton of th taodnay mod fo fquncs lss than th lcton cycloton fquncy? ω/ω c 4 3 ω/ω c / l 3 4 / l 8