DMO / ΗΛΕΚΤΡΟΜΑΓΝΗΤΙΚΗ ΑΟΡΑΤΟΤΗΤΑ ΜΕ ΠΑΡΑΛΛΗΛΑ ΗΛΕΚΤΡΙΚΑ Κ ΜΑΓΝΗΤΙΚΑ ΠΕ ΙΑ Θ.. ΡΑΠΤΗΣ lecomagneic Sealh wih Paallel elecic and magneic Fields T.. RAPTΙS ΕΚΕΦΕ «ΗΜΟΚΡΙΤΟΣ» Τ. Θ. 68, 53 ΑΓΙΑ ΠΑΡΑΣΚΕΥΗ (Αθήνα) ΕΛΛΑΣ DMOKRITOS Naional Cene of Scienific Reseach P.O. BOX. 68, 53 AGIA PARASKVI (Ahens) GRC /8
ΗΛΕΚΤΡΟΜΑΓΝΗΤΙΚΗ ΑΟΡΑΤΟΤΗΤΑ ΜΕ ΠΑΡΑΛΛΗΛΑ ΗΛΕΚΤΡΙΚΑ ΚΑΙ ΜΑΓΝΗΤΙΚΑ ΠΕ ΙΑ Θ. Ε. Ράπτης ΕΚΕΦΕ «ΗΜΟΚΡΙΤΟΣ» ιεύθυνση Τεχνολογικών Εφαρµογών (heo@da.demokios.g) LCTROMAGNTIC STALTH WITH PARALLL LCTRIC AND MAGNTIC FILDS T.. Rapis NCSR DMOKRITOS Division of Applied Technology (heo@da.demokios.g) Aθήνα,, Ιουλίου, Ahens,, July, /8
DMO / lecomagneic Sealh wih Paallel lecic and Magneic Fields T.. Rapis Division of Applied Technologies, Naional Cene fo Science and Reseach Demokios, Paiachou Gigoiou & Neapoleos, Ahens, Geece -mail: heo@da.demokios.g Absac We analyze a heoeical example of paallel elecic and magneic fields in a hypoheical anisoopic medium wih vaying suscepibiliy. We deduce he polaizaion chaaceisics and we discuss he condiions unde which his could be uilized in elecomagneic invisibiliy.. Inoducion In a pevious wok [] we have pesened a geneal mehod fo solving he Belami poblem fo a spheically symmeic veco poenial. This class of fields has been peviously discussed in he lieaue [-7], wih espec o hei peculia popey of having boh he elecic and magneic componens in paallel. This siuaion caused an iniial conovesy which was seled down afe he ealizaion ha such special cases consiue vey special soluions of Maxwell equaions ha can only exis wihin caviies wih pescibed bounday condiions [7]. In fac, an expeimenal ealizaion of such a sae in a lase caviy wih opposie ciculaly polaized modes was shown in [5]. In he pesen wok we pesen an example which poves ha he applicaion of such peculia saes is fa fom exhaused. In paicula, in secion we conside a spheical egion whee a vaying chage densiy followed by he associaed polaizaion and cuen souces exiss and we analyze he geneal fom of soluions of Maxwell equaions unde he condiion ha he elecic field is an eigen-oaion field [9-] alhough he magneic field is no. This appoach diffes fom pevious sudies whee he magneic field was supposed o saisfy a Belami condiion which is a special class of eigen-oaion fields. In secion 3, we pesen a special mulipole soluion fo he eigen-oaion equaion fom which we deive he genealized suscepibiliy and pemiiviy of he enclosing space. In secion 4, we discuss he possibiliy ha such soluions could have poenial applicaions wih espec o elecomagneic invisibiliy (plasma sheah).. Paallel elecic and magneic fields We sa fom a polaizable medium which conains a hamonically ime vaying adial chage disibuion ρ(, ) = ρ(, θ, φ) exp( iω) whee ω sands fo a monochomaic fequency, inside a spheical egion of maximum adius R, an associaed cuen J and a polaizaion P. The need fo he addiion of a polaizaion em will become appaen in he nex secion. Ou aim will be o define he spaial 3/8
dependence of he quaniies ρ, J and P unde he condiion of paallelizaion of he and B fields and find he chaaceisics of he medium. Denoing wih ε and µ he especive pemiiviy and pemeabiliy of he vacuum, Maxwell equaions in he SI sysem ake he fom D= ρ, B= = B H= D+ J (a) wih he consiuive elaions D= ε + P B (b) H= µ The las of (a) can be ewien in ems of a polaizaion cuen souce em aking ino accoun he consiuive elaions as D= ρ, B= = B B= ε µ + µ P+ J () We assume a hamonic ime-dependence of he fom exp( i ω) and a linea Ohm s law such ha J = σ so ha we can ewie he las wo as = iωb B= iε µω+ iµωp+ σ (3) The cenal hypohesis of his secion is ha we can find soluions fo he souces such ha //B. We expess his hough he condiion B Λ(, ω) = (4) whee Λ is an abiay scala funcion. Subsiuing his ino he fis of (3) we deive he eigen-oaion equaion = iω Λ (5) This is simila wih he equaion descibing a Belami flow in hydodynamics bu wihou he addiional condiion fo o be a puely solenoidal field [9-]. The em i ωλ is hen he field s eigen-voiciy. Applying he divegence opeao o he lhs leads naually o he condiion ( Λ) = which esuls in he elaion 4/8
Λ = Λ (6) This also auomaically saisfies B =. We hen subsiue (4) ino he second of (3) which esuls in he equaion Λ +Λ ) = iε µω+ iµωp+ σ ( (7) Fom he above and using (5) we can find he polaizaion cuen in ems of he elecic field as i P = χ, ω) Λ µω ( (8a) Λ χ(, ω) = ε + σ + (8b) µ We now see ha in he absence of any polaizaion he cuen would have o obey a songly non-linea fom wih unnaual chaaceisics like negaive esisance. Fom (8a) we see ha D can now be expessed as Λ i D = ε + P= Λ µ µω (9) Accodingly, he chage disibuion can now be found fom he fis of () as i µω Λ Λ µ µ ρ = D= ( Λ ) Λ () Fom sandad veco ideniies we also have ha ( ) = iωλ Λ ( Λ ) = Λ () Using boh () and (6) in (9) finally yields Λ ρ = Λ () µ 3. Soluions of he eigen-oaion equaion Complee soluion of he poblem of souces fo he peviously pescibed elecic and magneic fields equies a soluion of he eigen-oaion equaion (5). We have povided a geneic semi-analyical echnique fo solving such a poblem elsewhee []. Hee we will appoach he poblem fom a diffeen viewpoin ha allows finding a special analyical soluion ha can be uilized fo compuaions. 5/8
Specifically, we avoid he use of a veco poenial and we ecouse o a wo sep echnique. We will have o inoduce a se of veco spheical hamonics as Υ lm =Υ Υ lmˆ, Ψ lm = Υlm, Φ lm = lm whee Υ lm ae he usual scala spheical hamonics. Nex, we poceed in wo seps. We seek fo a se of dual fields wih espec o he oaion opeao ha ansfom as = λ = λ (3) A soluion of he he eigen-oaion equaion can be found fom a linea combinaion of a se of special soluions of he above ha allow a common faco. Fom sandad ideniies of veco spheical hamonics as pesened in [] we have f ( ) Υ = f ( ) Φ l( l+ ) df f ( ) Φ= f ( ) Υ d + f Ψ (4) We see ha he second pa of he hs of (4) cancels ou if we choose which leads o he choice f ( ) = k / k k = Υ lm, = Φ lm (5) fo which (3) is saisfied wih λ = /, λ = λ /, λ = l( l+ ). In ode o find a symmeic linea combinaion of hese fields we assume a se of coefficiens ha depend on λ and wie he oal field in he fom = λ +λ (6) µ ν whee µ and ν ae unknown exponens. Then we obseve ha he acion of he oaion opeao is o inechange he coefficiens of (6) in he fom ν µ =λ ( λ + λ ) + (7) In ode fo he final combinaion o be wien in ems of he oiginal we ewie he above as ν µ + µ µ ν ν = λ λ λ + λ λ ) (8) ( In ode o ge a common faco he following condiion mus be saisfied ν µ + µ ν ( ν µ ) + λ = λ λ (9) = 6/8
This is equivalen o he equaion ( µ ) + = (7) we find he eigen-voiciy as ν o = ν + /, ν =,,,... / λ ' = λλ = µ. Then fom l ( l+ ) of special soluions of he eigen-oaion equaion in he fom. We can hus wie a family ( ν ) k ν + ν = ( L Y lm + L Φ lm ), l > () whee we inoduced he abbeviaion L = l( l+). Fom (5) we now find ha he Λ L faco has he fom Λ (, ω) = i fom which we deduce he suscepibiliy funcion ω as L χ(, ω) = ε + σ + µ ω () We may now sepaae he oveall polaizaion ino an isoopic and an anisoopic pa as P= χ+ PA ν + () P A i L = Λ = µω µ ω ( ˆ Φ lm) We noice ha boh pas divege as o ω. In fac, boh ems divege as /( ω ) aking ino accoun ha Υ, Φ lm lm ae funcions of he spheical angles and ˆ is a uni veco. Fo he chage disibuion we also have ha ν + 3 L ρ = 3 µ ω Υ lm (3) Assuming ha we can se up appopiae bounday condiions in an ineio and exeio spheical suface R < < R, we may suppess he divegence o a scale less han R by aking he poduc Rω in which case he divegence can be educed o a vey low scale by inceasing he fequency. Thus, fo a fequency of he ode of GHz he divegence egion fo boh pas can be educed o a adius of less han nm. 4. Conclusion The pevious secions ae devoed o an absac eamen of a hypoheical polaizable medium capable of susaining paallel elecic and magneic fields. In secion 3 an exac analyical fom of he polaizabiliy and he suscepibiliy of such a medium wee deived. The impoan popey of he pescibed soluions of Maxwell equaions esumes ino hei abiliy o hold a lage amoun of elecomagneic enegy due o adiaion cancellaion ( B= ). 7/8
The las decade has seen an inceasing effo owads opical invisibiliy cloaks using special meamaeials wih negaive efacive index which wee fis inoduced wih he pioneeing wok of J. Pendy and ohes [3-4]. In ou eamen, a diffeen possibiliy aises due o he adiaion cancellaion condiion. While in odinay invisibiliy he effo is owads ligh bending aound an objec simila o gaviaional lensing, in ou case i appeas ha in pinciple adiaion could ge apped inside a spheical egion. This in pacice would epesen an alenaive mehod of shielding an aea agains any passive deecion mehod. In he pesen epo we found ha a soluion wih non-adiaing paallel elecic and magneic fields is possible inside an anisoopic polaizable medium wih a vaying chage concenaion and polaizaion cuen. Refeences [] C. D. Papageogiou, T.. Rapis, Chaoic Sysems: Theoy and Applicaions, C. H. Skiadas and I. Dimoikalis, Wold Sci. (). [] C. Chu, T. Okhawa, Phys. Rev. Le. 48, 837 (98) [3] H. P. Zaghloul, H. A. Buckmase, Am, J. Phys. 56 (9), 8 (987) [4] K. Shimoda e al., Am. J. Phys. 58 (4), 394 (99) [5] F. Beenake, A. Le Floch, Phys. Rev. A, 43 (7), 374 (99) [6] N. A. Salingaos, Phys. Rev. A, 45 (), 88 (99) [7] J.. Gay, J. Phys.: Mah. Gen. 5, 5373 (99) [8] U. Leonhad, T. G. Philbin, New J. Phys. 8, 47 (6) [9] A. Lakhakia, Belami Fields in Chial Media, Wold Sci. Seies in Conempoay Chem. Phys. V (994) [] G. Mash, Foce-Fee Magneic Fields, Wold Sci. (996) [] Ray D Inveno, Inoducing insein s Relaiviy, Claendon Pess (993) [] R.G. Baea e al., u. J. Phys. 6, 87-94 (985) [3] U. Leonhad, Science V. 3 No 578, 777 78 (6) [4] J. B. Pendy e al., Science V.3 No 578, 78 78 (6) 8/8