Constructing solutions using auxiliary vector potentials
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1 58 uilir Vtor Pottil Costrutig solutios usig uilir tor pottils Th obti o M thor is to id th possibl M ild oigurtios (ods or gi boudr lu probl iolig w propgtio rditio or sttrig. This b do b idig th ltri d gti ilds ( d or qull obtiig th uilir tor pottils ( d I dditio to uilir tor pottils d thr r othr possibl st. or pl rt tor pottils ( Π d Π. r w ol otrt o d Th pth or solig M ild oigurtio is th s ollows h Sours J M Itgrtio pth- ilds Itgrtio Pth- Vtor pottils or Π Π h Dirtitio Pth- Dpdig o probl t hd pth- b sir th pth- Trditioll d B r r iwd s phsil ild qutitis whrs tor pottil ( d its slr outr prt ( φ r osidrd s thtil ostruts. owr thr r dirgig iws o this poit!!! It is itrstig to ot tht Mwll hisl drid o his rsults b usig th opt o tor pottil ( whih h lld ltrogti otu. owr this pproh ws ltr ritiid b othr prtitiors suh s rt d isid.
2 59 Th qustio o th propgtio o ot rl th ltri pottil Ψ but th tor pottil... wh brought orwrd pro to b o o tphsil tur... th ltri or d th gti or... tull rprst th stt o th diu rwhr. isid Philosophil Mgi 889. r is wht rt ss bout Mwll s pproh: I tio th prdoi o th tor pottil i [Mwll s] udtl qutios. I th ostrutio o w thor th pottil srd s soldig... it dos ot ppr to tht... dtg is ttid b th itrodutio o th tor pottil i th udtl qutios. C.. Md Collti ltrodis. r is dirt (or odr poit o iw:... th tor pottil whih pprs i qutu his i pliit or produs lssil or whih dpds ol o its dritis. I qutu his wht ttrs is th itrr btw rb pths; it lws turs out tht th ts dpd ol o how uh th ild hgs ro poit to poit d thror ol o th dritis o d ot o th lu itsl. Nrthlss th tor pottil (togthr with th slr pottil φ tht gos with it pprs to gi th ost dirt dsriptio o th phsis. This bos or d or pprt th or dpl w go ito th qutu thor. I th grl thor o qutu ltrodis o tks th tor d slr pottils s th udtl qutitis i st o qutios tht rpl th Mwll qutios: r d B r r slowl dispprig ro th odr prssio o phsil lws th r big rpld b d φ. Lighto d Sds Lturs o Phsis Vol. II 984.
3 6 qutios gorig tor pottil Si B ( B d ( (3 Subsript is to rid us tht B r d r r du to tor pottil or M (o gti sour (rd s Lw (4 Us (3 i (4 ( (5 Si url o grdit o slr is ro i.. ( φ h φ φ Slr pottil Vtor pottil th ro (5 w φ whr (6 qutios (6 d (3 r th prssio or d i trs o d φ W ot tht ro or hoogous diu w writ r ( (7 Usig pr s Lw J i (7 w h J (8 ( Priousl w oud th prssio or (8 w h [ ] J ( φ to b φ. Usig this i (9 ll tht th (9 b writt s φ ( J W h did th url o s B w r t librt to di th.
4 6 I light o ( φ J lt us di th dirg o r to b φ ( ( Usig ( i ( w h J d (3 φ (4 ill our prssios or d i th lst pg [qs. (6 d (3] b writt s φ ( (5 (6 Now qutios (5 d (6 r prssios or subt to Lort gug. d i trs o ol qutios gorig th tor pottil Cosidr rgio o sp r o hrgs i.. q th D D Subsript is to rid us D is du to tor pottil ll tht pr s Lw with J is gi b ( ( (3 (4 Us (4 i (3 d w h (5 (
5 Copr ( φ φ with ull idtit ( φ th it is lr tht ( 6 or hoogous di ro ( w h ( ro rd s Lw w h M (3 th substitut (3 i ( M ( (4 But w lrd oud prssio or i (. Us ( i (4 d w h M ( φ (5 Whr gi O gi url o is did b dirg o. Lt φ φ Usig (6 (5 sipliis to M D. W r t librt to hoos th ill ot tht [q. (] d [q. (3 o lst pg] b writt i trs o ordig to φ ( (6 (7 (8
6 63 Sur. id ro J. id ro M 3. id ro ( ( (3 4. id ro ( or 5. id ro 6. id ro ( or 7. Th totl is gi b ( or 8. Th totl is gi b ( or (4 (5 (6 (7 (8 (9 (
7 64 Solutios or d ll tht gorig dirtil qutios or d r J ( M ( or sour lotd t ( d obsrtio poit dist ro th sour th solutios to ( d ( r gi b ( ( d J 4 r (3 ( ( d M 4 r (4 whr J d M h disios proportiol to / or s J d s M disios proportiol to / w h ( ( s s s d J 4 r r (5 ( ( s s s d M 4 r (6 or ltri d gti urrt dsitis I r [pr] d Ir [olt] w h ( ( l d I 4 r (7 ( ( l d I 4 r (8 r r r ( (
8 65 TM T d TM ods Th trsrs ltrogti ild oigurtio is od or whih ltri d gti ild opots r trsrs to gi dirtio. This dirtio ot but ot lws is th pth tht th w is trlig. or T od th ltri ild is trsrs to gi dirtio d or TM od th gti ild is trsrs to gi dirtio. gi or T d TM ods th ortiod dirtio is ot but ot lws th dirtio o propgtio. Th oditios o uilir tor pottils d or TM T d TM ods ll tht d i trs o d wr gi b ( ( ( ( Lt ( ( ( (3 ( ( ( (4 Us (3 d (4 i ( d (. W gt
9 66 or w h ( ro prssio or d i trs o d w s thr r t lst 3 ws or whih w obti TM od with rspt to -dirtio i.. TM (W or pl i ll th oditio listd blow r stisid w h TM od d d d d d ( ( th } } o ( } } ( ( [ ] (3 Not tht ro ( d (3. W urthr lult th d to b } ( (
10 67 } ( ( ( d it b show (W (3 (W (4 Whr prssio or wr gi priousl (.g. ( d ( (5 Trsrs gti w WT -dirtio (TM To sur tht w is trsrs gti (TM ild WT -dirtio it is suiit to sur th uilir tor pottil hs ol -opot d. or TM ( d (6 Th ild opots r th gi b (7 (8 (9 ( ( ( ll th ild opots o th TM od lso b prssd i trs o
11 68 Trsrs ltri ild WT -dirtio (T To h T w rquir to h ol -opot d i.. ( d ( Th ild opots r gi b ll th ild opots o th T od lso b prssd i trs o tgulr tlli w guid tgulr tlli wguids r routil usd t d irow rquis. Thir stud is ot ol otitd b thir us s /irow opots but will hlp us bttr udrstd th opt o od d guidd w propgtio I studig th guidd w struturs w r usull itrstd i prtrs suh s: ild oigurtios (ods tht r supportd b th strutur th strutur uto rqu guidd wlgth w ipd phs ostt ttutio ostt t. or tlli rtgulr wguid it b show tht lthough TM ild oigurtio is th lowst ordr od it dos ot stis th boudr oditios d s suh th wguid dos ot support TM ods owr th T d TM ods stis th rquird boudr oditios d s suh r supportd b th strutur
12 69 Trsrs ltri ild T Cosidr th tlli wguid o si th -dirtio b s show. Th wguid is iiit i ro our prious disussio w h s tht T ods r obtid i d ˆ ( whih iplid b ust stis th tor dirtil qutio ( ( Not tht ( ribls ( ( g( h( is slr utio tht b writt s (usig sprtio o lso rll tht solutios to r ithr stdig ws (siusoidl or trlig ws (potil with opl rgut Th prtiulr or (stdig w or trlig w is hos bsd o th boudr oditios to b stisid I th s o our tlli wguid solutios i d ust b stdig ws d solutio i -dirtio (guid is iiit i th -dirtio ust b trlig w
13 7 ( ( g( h( C os( D si( with [ ] [ C os( D ( ] [ B ] si t ll tht or ti dpd is positil trlig w (w trls i positi -dirtio d is gtil trlig w (w trls i gti -dirtio I sour is lotd suh tht ol positil trlig w is prst th B B 3 3 I sour is lotd suh tht ol gtil trlig w is prst th I both positil d gtil trlig ws r prst ( wguid tritd o lod tht is ot thd th both 3 d B3 ust b iludd r or sipliit w ssu tht ol positi trlig w ist B 3 b Th is th gi b ( [ C os( D si( ] [ C os( D si( ] 3 ( W ipos th boudr oditios o th top botto lt d right wlls o th tlli wguid ssuig prt ltri odutor (PC boudr oditio i.. d tgtil r ro o th wlls Th boudr oditios r: ( ( b Botto d top wlls or ( ( ( b Botto d top wlls or (3 ( b ( b Lt d right wlls or (4 ( b ( b Lt d right wlls or (5 Not tht th boudr oditios (3 d (5 r ot idpdt d th rprst th s boudr oditios s ( d (4.
14 7 Th ssr d suiit oditios r to stis ithr ( or (3 ( d t botto d top wlls d ithr (4 or (5 ( d t th lt d right wlls urthror ot tht or T ods b diitio is ro. This s tht th ssr d suiit B.C. s or T r ( d (4 o th lst pg ( t th botto d th top d t th lt d right wlls ( ( b ( b ( b ll tht th tor pottil or T ws gi s [q. ( lst pg] ( [ C os( D si( ] [ C os( D si( ] 3. W th h 3 [ C os( D si( ] C si( D os( 3 [ ] ro ( D si b ro ( b ( b 3 or qull 3 b is sotis rrrd to s iglu b I w us our wl oud rsults w h ( C os( D si( C os 3 b [ ] Th b oud ro b [ C si( D os( ] C os 3 Boudr oditios or t lt wll is b D ( Boudr oditio or t right wll is si ( b (
15 7 3or qull 3 Puttig it ll togthr th tor pottil is gi b C os C os 3 os os b 3 with but 3 is ostt C C 3 ( b Propgtio ostt (w ubrs d wlgths i th d dirtio ro our prious disussio it is lr tht propgtio ostt (or w ubr b writt s log ( d log ( ; ( b ; d with b ( whr w h lso did th wlgth log to b d log to b ll tht or qull wll: (tril isid th guid ro ( d ( ot tht whr s whr is th wlgth i th diu with d d is otiuous prtr. r disrt (o s th r qutid Not tht i priipl thr r iiit ubrs o possibl d (iglus h thr r iiit ubr o T ods tht stis th w qutio d th gi boudr oditio. ro d d w s tht b
16 73 b b whr b diitio: uto propgtio ostt or uto w ubr Not tht b. th uto rqu ( is gi b ( i b 3 3 ro th prssio w s wh is lld th uto propgtio ostt. or this w ubr d th w o logr trls log th - dirtio. Th bo b or lrl s ro ( b. Clrl or Th ild opots or T r ow gi b ( ( si os ( ( os si ( ( os si ( ( Y si os ( ( si os To pprit th iport o th uto oditios osidr th ollowig: b
17 74 ( ± ± ± ± or > > ( or ( ± ± ± ± or > > I w ol osidr th positil trlig w w ust hoos th sig i rot o th squr root ppropritl i.. ( or < < ( or ( or > > Si ltri d gti ilds r proportiol to t th t rprsts propgtig w or < t t t rprsts stdig w or t t rprsts ttutd (st w or >
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