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# PLASMA PHYSICS I PHY 380L

UT

GPA 3.77

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This 20 page Class Notes was uploaded by Malvina Orn on Monday September 7, 2015. The Class Notes belongs to PHY 380L at University of Texas at Austin taught by Staff in Fall. Since its upload, it has received 50 views. For similar materials see /class/181832/phy-380l-university-of-texas-at-austin in Physics 2 at University of Texas at Austin.

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Date Created: 09/07/15

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621693 Mws f gt 912 kmpm Hana M 0 6 SySRM 0 NonQ cm ex 14 9gtm 3er 1 WLltOI no GKJPML 6 4ij l s V1 eo6 bl Qu l l39 89qu gy f m JQ LOhrms I jMWHaD Co 1an Oug L lCcquot f4 L 4Mquot39 9 CKQMAQ 9V cJltxC quot1 o Liam qf39t39 c U it jQ J Q 03qu P QV AE or Wu lL39l l U01 Ma aquot I4 Owe Ccm skew ww ifhf gt0 0M7 Cfowq e PQS 1W001390 Ej39 cwvH zlolt vow Hm fowl WM 7W Mm We 5y 3 RM rs Haw Maui m 04 as M 9 1me we Magnum fa La4 Pwohueg H19 P9 Hw qul 924233 y VVQ 0g t Rl n quot1903 Chile 10 440 NM 9 HVL syghml Hnwcue Vw 5 L Ly COWSCJtJ R WLAWXSJ Showhj SW VO 5 i cluem h QMOUJH 9 w qu M wwa The 495 ax Ed u s e C Ax wnire j WW 1 5 9 show E S QH CLzojoM ih M 4 7 W04 C f WC CL fLu In jammed P CHgt Cq hul w quotQ5 CWI gt3 9de q Ijt am OMJcselvag 9 op qgmq CM Hog VAN1 q COMJHLHU Law So n 0 no w 0 14 97 4 m vo M s 39f Hm wo lc mgi APPENDIX A SELFADJOINTNESS OF THE FORCE OPERATOR F The goal of Appendix A is to show that the force operator F is selfadjoint that is 39 a ned amer Al where E and 11 are two arbitrary vectors satisfying the boundary condition 11 E n o 1 0 on the surface This corresponds to the perfectly conducting wall boundary condition The integrand can be written as if 39 nFEn53VXBXQIVXQXB V 39Vp YpVE A2 with Q V X E X B The last term is integrated by parts yielding a n iwxn x micx Q XB VE39VP ypw XV 1 A3 Onenow writes E E Eub 1 ILL nub The term in the square bracket in Eq A3 has no parallel component speci cally 9sz a ElgBVXBXQ QJXB QVp 39 V 39 VP X E X 3 39V 39 E39 VFW A4 Yl x 39vp b B39V 39VPV39 39VPB WW W79 2 V Z x fo Clearly the parallel component cancels Consequently one nds 1 rs vpv vn 1 lat components of E and n where I is a function only of the perpendicu loam 39m 5307 x3 x Hiv x Q x3 vetVP A5 7 Iquot m39 7an WW 5 Appendices 466 Appendices The last term is now integrated by parts and the rst two terms rewritten using stande vector identities I miniown Bvo VBQl atvmvm A6 The three terms inside the square brackets in Eq A6 are expanded as follows 0 b m Q 39VB m 3 WET VB Er 39 VB39VB 3201 KV39EL 1h B VQB VTIL39QQ39B39V11L V3939LL39QBquotB Vgr gr39VB BV Er 39 B Vm A7 quotB 39 VEL 39 BfV11 EL VB B 39 V1 32ml Kw at quot mVBQV39BQmBQV39m BZV LXV 39 m El 39VBzZ 320 KV 39 m In the second and third terms full divergence contributions have been dropped since they integrate to zero Combining terms one nds ll nF Bv ivnn iBVErBVm YpVEVn 32 B2 Er39Vp 271 EErK Vm B2 moiorg R where OR 1h 39 39V J 39 VB gr 39 VB g 39 VB 39 B 39 V114 Q 0 C The middle line is simpli ed by noting that E Vp 322110 B2 JOKE K The quantity R can be rewritten by using the two identities V J EL VBB B 39 Vnr 39 E1 39 VB 39 W at naval VB m Baywn m gt VB VB m at VB VB m Brawn A10 5 U Appendices To within a divergence term which integrates to zero R is given by 1 B2 LL J o L V 39 R on E VB VB 1 E V p 2 10 The nal result is fnFadr dr iBVEi39BVmYPVEVn 32 7V EJ 2E 39 XXV 39 TLL 21h 39 A12 0 2 432 B 39M O39EJ39K1IJ39K11JE 3VV P Zuo which is clearly a selfadjoint form by inspection m b gsv ks 1 n gkkwg N L L v 8 34 N VJ P T i VKDJmf 7

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