Superluminal Near-field Dipole Electromagnetic Fields. 1 Introduction. 2 Analysis of electric dipole. 2.1 General solution
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1 Supeluinal Nea-field Diple Eleanei Fields Willia D. Wale KTH-Visby Caéaan SE-6 7 Visby, Sweden Eail: bill@visby.h.se Pesened a: Inenainal Wshp Lenz Gup, CPT and Neuins Zaaeas, Mexi, June -6, 999 Induin The pupse f his pape is pesen aheaial evidene ha eleanei nea-field waves and wave ups, eneaed by an sillain elei diple, ppaae uh fase han he speed f lih as hey ae eneaed nea he sue, and edue he speed f lih a abu ne wavelenh f he sue. The speed a whih wave ups ppaae (up speed) is shwn be he speed a whih bh dulaed wave infain and wave eney densiy ppaae. Beause f he siilaiy f he venin paial diffeenial equains, w he ysial syses (anei sillain diple, and aviainal adiain sillain ass) ae ned have siila esuls. Analysis f elei diple. Geneal sluin Nueus exbs pesen sluins f he eleanei fields eneaed by an sillain elei diple,. One siple and elean sluin slves he inheneus send de supepenial wave equain (pp. - 6). The eleanei fields an hen be deived f he Hez ve (Z). Fiue. Seial dinan syse used in pble: z x PDE (supepenial wave equain): E H E y Z () Sluin: Z R Cs( ) e i Z i Sin( ) e Z () Definin equains: C xz C R C C i Sin( ) i e () Field alulains: E xc B C whee: H B ()
2 Resulan eleial and anei field pnens f an sillain elei diple E H Cs( ) Sin( ) i e i i E Sin( ) i i e () i e (6) I shuld be ned ha his sluin is nly valid f disanes () uh eae han he diple lenh (d ). In he ein nex he sue ( ~ d ) an n be deled as a sinusid: Sin. Insead i us be deled as a sinusid inside a dia dela funin: d Sin. The sluin his hype-nea-field pble an be alulaed usin he Lienad-Wieha penials,, 8.. Lines f elei fe analysis Tadiinally he elei lines f fe an be deeined f he elain ha a line eleen (ds) ssed wih he elei field is ze. The esulin paial diffeenial equain an hen be slved yieldin he lassial esul. Resulan Equain: CSind CSind (7) Sluin: Sin Cs Tan Cns (8) A nu pl f his sluin (Eq. 8) eveals he lassial adiain sillain elei diple field paen (Fi. ). Caeful exainain f he paen eveals ha he wavelenh f he eneaed fields ae lae in he neafield (a) and edue a nsan wavelenh afe he fields have ppaaed abu ne wavelenh f he sue (b). The speed f he fields (ase speed = ) nea he sue an hen be nluded ppaae fase han he speed f lih f he elain ha wave speed ( ) is equal he wavelenh ( ) uliplied by he fequeny (f), whih is nsan. f if Fiue. Maheaia aniain nu pl f abve sluin a > b Radial Field (E ) Maheaia de used eneae aniain a final b a b = Tansvese Field (E )! " " #" " " " "! " " " " $% &!" #'" #(" #)" #" #" #" #" #)" #(" #'" *+
3 The Maheaia de (Ve..) shwn abve eneaes pls f he ppaain elei field a diffeen islaed ens in ie. Muse liin any f he faes in Maheaia aniaes he pl, evealin ha he vals f nsan elei field enlae as hey ppaae away f he sue (laed a = ). I is als ineesin ne ha as he elei field lines ae eneaed, se f he elei lines f fe vey nea he diple (~ / wavelenh) appea ppaae nly a sh disane and hen evese and ppaae ba in he sue. Fiue. Maheaia aniain f nu pl f elei lines f fe nea sue Analysis f ase speed and up speed. Phase alulain The eneal f f he eleanei fields (ef Eq.,6) eneaed by a diple is: i Field x iy e (9) If he sue is deled as Cs, he esulan eneaed field is: Field Ma Cs Ma Cs () whee: Ma x y I shuld be ned ha he fula desibin he ase is dependan n he quadan f he plex ve. y y Tan x Cs x () x y x. Definiin and alulain f wave ase speed Sinal pee ppaaes wih speed equal ase speed ( ) Phase speed an be defined as he speed a whih a Sinusidal sinal wave psed f ne fequeny ppaaes. The ase speed ( ) f an sillain field f he f Sin( ), in whih (, ), an be deeined by sein he ase pa f he field ze, diffeeniain he esulan equain, and slvin f.
4 ( ) () Diffeeniain wih espe yields: () Cbinin hese esuls and usin yields: (). Definiin and alulain f wave up speed Pea f dulain pa f sinal ppaaes wih speed equal up speed ( ) In se ysial syses he wave ase speed Mdulaed sinusidal is a funin f fequeny. In hese syses when waves psed f diffeen fequenies ppaae, he wave up (wave envelpe) ppaaes a a diffeen speed (up speed) han he individual waves. The up speed is als nwn be he speed a whih wave eney and wave infain ppaae 6 (pp.68-69), (p.). The up speed f an sillain field f he f Sin( ), in whih (, ), an be alulaed by nsidein w Fuie pnens f a wave up: Sin( ) Sin( ) Sin( ) Sin( ) () in whih:,,, The up speed ( ) an hen be deeined by sein he ase pa f he dulain pnen f he field ze, diffeeniain he esulan equain, and slvin f : ( ) Diffeeniain wih espe yields: Cbinin hese esuls and usin he elain (6) (7) yields: li sall (8)
5 . Radial elei field (E ) Applyin he abve ase and up speed elains (Eq., 8) he adial eleial field (E ) pnen (Eq. ) yields he fllwin esuls: Fiue y x (9) Tan () ( ) () ( ) ( ) ( ) () Fiue Fiue 6 E E E 9 de. Tansvese elei field (E ) Applyin he abve ase and up speed elains (Eq., 8) he ansvese eleial field (E ) pnen (Eq. ) yields he fllwin esuls: y x () Cs () () ( ) ( ) 6( ) 7( ) ( ) ( ) (6) 6 8 Fiue 7 E 8 de Fiue 8 Fiue 9 E - E C vs E -
6 .6 Tansvese anei field (H ) Applyin he abve ase and up speed elains (Eq., 8) he ansvese anei field (H ) pnen (Eq. 6) yields he fllwin esuls: y x Cs ( ) ( ) (7) (8) ( ) ( ) ( ) (9) () Fiue H 8 de Fiue Fiue H H Gaial evidene f supeluinal ase and up speed. Supeluinal nea-field ase speed f adial elei field T densae he supeluinal nea-field ase veliy f he lniudinal elei field, he alulaed ase and apliude funins an be inseed in a sine sinal and he field apliude an hen be pled in he nea field as a funin f spae () a seveal islaed ens in ie (), (Fi. ). A field ppaain a he speed f lih (shwn as a dashed line) is als inluded in he pl f efeene. The fllwin paaees ae used in he subsequen pls: wavelenh (), GHz sinal fequeny (f),.ns sinal peid (T). The fllwin Maheaia de (Fi. ) is used eneae hese pls: Fiue. Maheaia de used eneae pls! " " " #" "!,"!+%,"," " " 6
7 Fiue. E vs Spae Csinusidal Sinal E T T T T The lniudinal field (shwn as a slid line in he pl abve) is bseved ppaae away f he sue, whih is laed a =. As i ppaaes away f he sue, he sillain apliude deays apidly (/ ) nea he sue ( < ), and deeases e slwly (/ ) in he fafield ( > ) (ef Eq. ). A field ppaain a he speed f lih (shwn as a dashed line in he pl abve) is als inluded in he pl f efeene. Bh sinals sa ehe in ase. The lniudinal field is seen ppaae fase han he lih sinal iniially when i is eneaed a he sue. Afe ppaain abu ne wavelenh he lniudinal elei field is bseved slw dwn he speed f lih, esulin in a final elaive ase diffeene f 9 deees. In de see he effe e lealy he sinals an be pled wih he apliude pa f he funin se uniy (Fi. ). 7 Fiue. E (Nalised) vs Spae Csinusidal Sinal E ( ) T ( ) T ( ) 7 T ( T ) I is als insuive pl he sinals as a funin f ie () f seveal psiins () away f he sue (Fi 6). A he sue ( = ) bh sinals ae bseved be in ase. Fuhe away f he sue he lniudinal field sinal is bseved shif 9 deees, indiain ha i aives ealie in ie. The pls shwn belw ae nalized f laiy, bu i shuld be ned ha he sinals have he sae f even if he apliude pa f he funin wee inluded. The nly diffeene is he veial salin f he pl. F hese pls i an als be seen ha he lniudinal field ppaaes uh fase han he speed f lih nea he sue ( < ), and edues he speed f lih a abu ne wavelenh f he sue ( ), esulin in a final elaive ase diffeene f 9 deees beween he lniudinal field (shwn as a slid line), and he field ppaain a he speed f lih (shwn as dashed line). 7
8 Fiue 6. E (Nalised) vs Tie Csinusidal Sinal E. Supeluinal nea-field up speed f adial elei field T densae he supeluinal nea-field up ppaain speed f he lniudinal field, he alulaed ase and apliude funins an be inseed in he speal pnens f an apliude dulaed sine sinal, and he field apliude an hen be pled as a funin f spae () a seveal islaed ens in ie (), (Fi. 8). T densae his ehnique he up ppaain (shwn as a slid line) is paed he ase speed ppaain (shwn as a dashed line) f waves f he f: Cs. Ne ha he ase pnen () is independen f fequeny. This esul is nwn pdue up waves and ase waves ha bh ppaae a he speed f lih. This an be seen by usin (Eq. ): sine. The fllwin paaees ae used in. Usin (Eq. 8): he fllwin pls: Caie pa f sinal - wavelenh ( ), MHz sinal fequeny (f),.ns sinal peid (T), Mdulain pa f he sinal - wavelenh, MHz sinal fequeny (f),.ns sinal peid (T). Ne ha he ase elain f bh sinals ae he sae a diffeen islaed ens in ie and ha bh sinals ppaae away f he sue a he sae speed. The fllwin Maheaia de (Fi. 7) is used eneae hese pls: Fiue 7. Maheaia de was used eneae hese pls * * * - %+$ - -. *.. *.. *....!-". " " " " " " Fiue 8. Lih Phase (Csine Wave) and Gup (AM Sinal) vs Spae T T T 7 T 8
9 Plin he sinals as a funin f ie f seveal spaial psiins f he sue als shws ha up and ase sinals avel a he sae speed and eain in ase as hey ppaae. Fiue 9. Lih Phase (Csine Wave) and Gup (AM Sinal) vs Tie The supeluinal nea-field up ppaain speed f he lniudinal elei field an als be densaed in he sae way as in he abve exaple. The alulaed ase funin f he field an be inseed in in he speal pnens f an apliude dulaed sine sinal and he field apliude (shwn as a slid line in he pl belw) an hen be pled as a funin f spae () a seveal islaed ens in ie (), (Fi., ). An apliude dulaed field ppaain a he speed f lih (shwn as a dashed line) is als inluded in he pl f efeene (envelpe ppaaes a speed f lih). Ne ha f his efeene sinal bh he ase speed and he up speed ae equal he speed f lih (ef Fi. 8, 9). The fllwin aheaia de (Fi. ) is used eneae he pls belw. The sae sinal paaees used in he pevius exaple ae used in he alulain. Fiue. Maheaia de used eneae pls -.. * * * * * * - %+$ - -. *.. *.. * -. *.. *..*....!-" -" " " "!," " " Fiue. E (Nalized) vs Spae AM Sinal E T T T 7 T 9
10 Fiue. Z f E (Nalized) vs Spae AM Sinal E T T The abve pls shw an apliude dulaed lniudinal field up pae (shwn as a slid line) ppaain away f he sue, a = (up axia aed by veial aw). A ppaain speed f lih up wave (shwn as a dashed line) is als pvided f efeene. The up axia f he apliude dulaed lniudinal field is bseved ppaae he ih side f he pl befe he up axia f he speed f lih wave. These seies f pls lealy densae ha he lniudinal up wave ppaaes uh fase han he speed f lih nea he sue ( < ). Afe ppaain abu ne aie wavelenh f he sue ( ) he dulain pa f lniudinal field (envelpe f slid line) edues he speed f lih, esulin in a final elaive ase diffeene f 9 deees (elaive he aie sinal) beween he lniudinal field, and he field ppaain a he speed f lih. I is als vey insuive pl he field apliude f he apliude dulaed lniudinal wave (shwn as a slid line in pl belw) as a funin f ie (), f seveal psiins away f he sue (), (Fi., ). As befe, an apliude dulaed wave avelin wih lih speed is als pled f efeene (shwn as a dashed line). A he sue ( = ) bh sinals ae bseved be in ase. Fuhe away f he sue he dulaed lniudinal field sinal is bseved shif he lef, indiain ha he dulain pa f lniudinal field (envelpe) aives ealie in ie. The pls shwn belw ae nalized f laiy, bu i shuld be ned ha he sinals have he sae f even if he apliude pa f he funin wee inluded. The nly diffeene is he veial salin f he pl. F hese pls i an be seen ha he dulain pa f lniudinal field (envelpe f slid line) ppaaes uh fase han he dulaed lih speed sinal (envelpe f dashed line ppaaes a speed f lih) nea he sue ( < ). Afe ppaain abu ne aie wavelenh f he sue ( ) he dulain pa f lniudinal field (envelpe f slid line) edues he speed f lih, esulin in a final elaive ase diffeene f 9 deees (elaive he aie sinal) beween he lniudinal field, and he field ppaain a he speed f lih. Fiue. E (Nalized) vs Tie AM Sinal T 7 T E 7
11 Fiue. Z f E (Nalized) vs Tie AM Sinal Z E (Nalized) vs Tie Z E (Nalized) vs Tie Z E (Nalized) vs Tie Z E (Nalized) vs Tie E (x -8 s) (x -8 s) (x -8 s) (x -8 s) 7 The fis fae f he pl shws he w wave ups sain in ase a he sue ( = ). The fllwin faes shw ha as he w waves ppaae away f he sue, he up axia (aed by a veial aw) f he apliude dulaed lniudinal aives ealie in ie han he up axia f a lih speed apliude dulaed sinal, hus densain ha he up speed f an apliude dulaed lniudinal eleial field is uh fase han he speed f lih nea he sue ( < ), and edues he speed f lih afe i has ppaaed abu ne wavelenh f he sue ( ), esulin in a final ase diffeene f 9 deees. 7 Relain beween up speed and ppaain speed f infain and eney Seveal auhs have indiaed ha wave up speed (ppaain speed f wave envelpe) is als he speed a whih wave eney and wave infain ppaaes 6 (pp.68-69), (p.). One inuiive way undesand his is aheaially apliude dulae and dedulae a ppaain wave, heeby ansiin and deein infain. An apliude dulaed wave an be aheaially deled as fllws: AM Si Cs Cs () whee ( ) is he dulain fequeny, ( ) is he aie fequeny, and () is he index f dulain. Usin inei ideniies he AM sinal an be shwn be: AM Si Cs[ ] Cs[ ] Cs[ ] () Wave ppaain an hen be delled by insein he alulaed ase elain f he wave in he speal pnens f he dulaed sinal. AM Si Cs[ ] Cs[ ] Cs[ ] () If he lniudinal eleial field is used ansi he sinal, hen he nea-field ase ( < ), elains f he speal pnens ae (ef Eq. ): whee ()
12 The speed a whih he wave up (envelpe) ppaaes an be deeined by squain he esulan dulaed sinal (Eq. ) and nin he ase shif ( ) f he dulain pnen ( ) f he esulan sinal. Pefin his puain n he lniudinal eleial field and usin inei ideniies yields: AM SiDe Cs[ ] ] Cs[ ] Cs[ = +O[ ] () The esulaan ase shif f he dulain pnen is:. Subsiuin he ase elains (ef Eq. ) yields: d The speed a whih he dulain envelpe (infain) ppaaes an hen be alulaed usin (Eq. ). Pefin his puain n he abve esul yields he sae answe as when alulaed usin he up speed (ef Eq. ): d Ne ha his del an als be used shw ha he nea-field eney densiy eneaed by an elei diple ppaaes a he up speed, sine he eney densiy (w) f an eleanei field is nwn be equal he su f he squaes f eah field pnen (p.7): w E E B (8) (6) (7) 6 Ppsed expeiens easue supeluinal nea-field ase and up speed f he lniudinal elei field The pevius aheaial auens have indiaed ha he up speed f a ppaain neafield lniudinal elei field is uh fase han he speed f lih f ppaain disanes less han ne wavelenh ( < ). The appxiae f f he up speed, f he nea-field lniudinal elei field (ef Eq. ) is: (9) The fllwin expeien is ppsed easue he nea-field up speed f a ppaain nea-field lniudinal elei field. I is suesed ha an apliude dulaed sinal be injeed in ne end f a paallel plae apai and deeed n he he side f he apai by an apliude dedula. The ase diffeene beween he esulan dedulaed sinal
13 and he iinal dulain sinal shuld hen be easued. The ap disane f he apai plaes shuld hen be ineased and he ase diffeene shuld hen be easued aain. The up speed an hen be deeined by enein he value f he dulain fequeny (), he easued hane in ase (), and he hane in apai plae ap disane (d ) in he fllwin elain: d () Beause he ase speed f he lniudinal elei field is uh fase han he speed f lih, he expeed ase hane ay n be easy easue. In he neafield, a disanes less han ne enh wavelenh (elaive he aie fequeny), he up speed f he lniudinal elei field is appxiaely (ef Eq. ): () d d d () Ne ha usin a hih dulain fequeny and an even hihe aie fequeny an inease he bseved ase hane. If a MHz dulain fequeny and a MHz aie fequeny wee used eneae he apliude dulain sinal hen a hane in apai plae disane wuld eneae a x - de ase hane. Ne ha hese fequenies espnd a 6 fa-field dulain eleial wavelenh and a.6 aie wavelenh. This ase hane wuld be vey diffiul easue bu i ay be pssible usin a hih ase sensiiviy l-in ehnique develped by he auh. 7 Supeluinal wave ppaain in he ysial syses 7. Eleanei fields eneaed by a anei diple. The eleanei fields eneaed by an sillain uen lp has been shwn by seveal auhs have a siila f he elei diple (p6) (pp. 6-6, 6). The nly diffeene beween he sluins is ha elei and anei fields ae evesed: Resulan eleial and anei fields f an sillain anei diple H Cs( ) i e i H Sin( ) i i e () E Sin( ) i i e ()
14 Cnsequenly, all he analysis pefed n he elei diple in he pevius paes an als be applied his syse. Speifially, he nea-field ase and up speeds f his syse ae als nluded be uh fase han he speed f lih. 7. Gaviainal fields eneaed by an sillain ass Maheaial analysis f an he aviainal fields eneaed by a vibain ass eveals ha f wea aviainal fields, aln he axis f vibain, he sillain aviainal diple is deled wih he a paial diffeenial equain siila ha f he sillain elei diple. One sinifian diffeene beween he w syses is ha in de nseve enu, a vibain ass us be apanied wih anhe sillain ass sillain wih he sae fequeny bu wih ppsie ase. The effe f he send vibain ass is anel he ppaain aviainal fields in he fa field. Hweve, nea he sue he send vibain ass des n nibue sinifianly he esulan ppaain aviainal fields, and an be neleed in he dellin 8. Usin he Einsein elain: 8G G T () Aln he axis f vibain, f sall asses and lw veliies, he Einsein equain edues : G T (6) The nly nn-vanishin e in he eney enu ens de is: d T dsin( ) O in whih: (7) Slvin he paial diffeenial equain yields 7 : K O l l Sin Cs (8) in whih: K = - G, d l, l d Sin (9) The aviainal field () an hen be alulaed usin he elain G d d : hh.. O O Sin ()
15 Sine d O, slvin his elain f he ase speed f he lniudinal aviainal field yields: d O () Theefe i is nluded ha nea he sue, aln he axis f vibain, he lniudinal aviainal ase speed is uh fase han he speed f lih. The up veliy has been shwn be (ef Eq. 8). Insein he elain f he ase speed (ef Eq. ) yields:. Insein he de esiae f he ase speed (Eq. ) yields: d O () F he abve esuls (Eq., ) i is nluded ha nea he sue bh he ase speed and he up speed f he lniudinal aviainal field, aln he axis f vibain f he ass, ae uh fase han he speed f lih. 8 Cnlusin This pape has pvided aheaial evidene ha eleanei nea-field waves and wave ups, eneaed by an sillain elei diple, ppaae uh fase han he speed f lih as hey ae eneaed nea he sue, and edue he speed f lih a abu ne wavelenh f he sue. The speed a whih wave ups ppaae (up speed) has been shwn be he speed a whih bh he wave eney densiy and dulaed wave infain ppaae. Beause f he siilaiy f he venin paial diffeenial equains, w he ysial syses (anei sillain diple, and aviainal adiain sillain ass) have been shwn have siila nea-field supeluinal esuls.
16 9 Refeenes W. Panfsy, M. Philips, Classial eleiiy and aneis, Addisn-Wesley Pb. C., (96) J. D. Jasn, Classial Eledynais, Jhn Wiley & Sns, (97) P. Lain, D. Csn, Eleanei fields and waves, W. H. Feean and Cpany, (97) R. Feynan, R. Leihn, M. Sands, Feynan leues n ysis, Vl., Ch., Addisn-Wesley Pb. C., (96) D. Twne, Wave enena, Dve Pub., New Y (967) 6 F. Cawfd, Waves Beeley ysis use, Vl., MGaw-Hill Pub. C. (968) 7 W. Ele, M. Heald, Physis f waves, Dve bs, New Y, (969) 8 M. Heald, J. Main, Classial Eleanei adiain, Saundes Pb. C. (99) 9 K. Gaff, Wave in in elasi slids, Dve pub., New Y, (97) W. Guh, J. Rihads, R. Willias, Vibains and waves, Penie Hall pub., (996) I. Main, Vibains and waves in ysis, Cabide Univ. Pess, (99) H. Pain, The ysis f vibains and waves, Jhn Wiley & Sns, (99) W. Wale, Supeluinal ppaain speed f lniudinally sillain eleial fields, Cnfeene n ausaliy and laliy in den ysis, Kluwe Aad pub., (998) W. Wale, J. Dual, Phase speed f lniudinally sillain aviainal fields, E Aaldi nfeene n aviainal waves, (997) L. Billuin, wave ppaain and up veliy, Aadei pess, (96) 6 W. Wale J. Dual, Expeien easue he ppaain speed f aviainal ineain, Vi Cnfeene n aviainal waves sues and dees, Wld Sienifi, (997) 7 N. Sauann, Geneal Relaiviy and elaivisi asysis, exs and nas in ysis, Spine Vela, Belin, Heidellbe, pp. 8-9 (99) 8 W. Wale, Gaviainal ineain sudies, Disseain ETH N. 89, Zuih, Swizeland, (997) 6
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