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Casimir electromotive force in periodic configurations EvgenyG.Fateev∗ Institute of mechanics, Ural Branch of the RAS, Izhevsk 426067, Russia (Dated: January 28, 2016) The possibility in principle of the existence of Casimir electromotive force (EMF) is shown for nonparallel nanosized metal plates arranged in the form of a periodic structure. It is found that EMF does not appear in strictly periodic structures with parallel plates. However, when the strict periodicityisdisturbedinnonparallelplates,EMFisgenerated,anditsvalueisequaltothenumber ofpairsofplatesinaconfiguration. Moreover,therearesomeeffectiveparametersoftheconfigura- 6 tion(anglesbetweenplates,platelengthsandlengthtolengthratios),atwhichtheEMFgeneration 1 perunit of thelength of the periodic structureis maximal. 0 2 PACSnumbers: 03.70.+k,04.20.Cv,04.25.Gy,11.10.-z n a J INTRODUCTION a 4 Recently in work [1], the possibility in principle of the M M z 2 3 0 existence of Casimir electromotive force has been shown θ1 ] for a single open perfectly conducting nanosized struc- ϕ r θ h ture with nonparallel plates (wings). Naturally, in par- M1 2 p allel metal plates in the classical configuration studied - n by Casimir [2–6], electromotive force must not generate. R θ e However, at the ends of the plates, there can be certain b g P fluctuationsofelectricpotentialsduetoJohnson-Nyquist z . s thermalnoise[7]andinterferencecurrentsbecauseofra- ϕ c θ− i dio interference. P s y The possibility of Casimir EMF generation is associ- M2 Px h atedwithaneffectsimilartolight-inducedelectrondrag, x p which can appear in metals [8–10], graphite nano-films [ [11]andsemiconductors[12]. Inourcase,the drageffect FIG.1. Schematicviewoftheconfiguration with nonparallel 1 can arise in nonparallel and perfectly conducting wings metal plates with the surface length R, the particular cases v due to the total uncompensated action of virtual pho- of which are parallel plates for ϕ = 0 and a triangle at a = 6 0. The configuration with the width L in the y-direction tons upon electrons. Earlier it has been shown that in 6 (perpendicularto thefigure plane) is shown in the Cartesian addition to the EMF generationin its wings, the system 3 coordinates in the plane (x,z). The blue lines designate the 7 with nonparallel wings can also experience Casimir ex- virtual rays with the length b-outgoing from the point M1 0 pulsion force [13, 14] and other interesting effects [15]. underthe limit angles Θ1 and Θ2 toward theright plate and 1. The force shows up as a time-constant expulsion of non- finishing at the ends of the opposite wing at the points M2 0 parallel plates in the direction of the smallest angle be- and M3, respectively. 6 tween them. It should be noted that this force signifi- 1 cantlydiffersfromCasimirpressureandexpulsionwhich v: are capable of creating only effects of levitation above andtheseparationdistancebetweentheperiodsislarger i body-partners[16–20]. Theuncompensatedactionofthe than the minimal distance a between the wings. There X forces upon nonparallel structures is due to the nonuni- is an optimum of the d/a relation at which the maximal ar form action of Casimir forces upon the opposite ends of effect is obtained. the configuration asymmetrical along one of the coor- It is natural to ask if the EMF excitation is possible dinates. Optimal parameters have been found for the in periodic structures constructed on the basis of config- opening of the angles between the plates and for the urations having the effect of Casimir EMF. plate lengths, at which expulsion forces and EMF ob- THEORY tained should be maximal. Letusconsideraperiodicconfigurationwithnonparal- For obtaining large total uncompensated forces and lel nanosized metal plates for studying the possibility of EMF, it is important that the conditions for the exis- theCasimirEMFexistence. Theinnerandoutersurfaces tence of the discussed effects in periodic structures are of the plates should have the properties of mirrors with investigated. In Ref. [3], for example, it is shown that the reflection coefficient ρ. The configuration can com- there are no uncompensated forces in strictly periodic pletely be merged into a material medium or be its part structures with parallel plates which are equally spaced. with the parameters of dielectric permeability different Theexpulsioneffectariseswhenthewingsareunparallel from those of physical vacuum. In a Cartesian coordi- 2 − z a d d>a + FIG. 2. Schematic view of the periodic configuration with pairs of nonparallel wings in the z-direction at the distance d from one another. nate system, a single nonparallelconfigurationlooks like r+asinϕ Θ =arccos , (5) two thin metal plates with the surface width L = 1 m 2 "− a2+r2+2rasinϕ# (oriented along the z-axis) and length R at the distance a from one another; the angle ϕ between them can be p . changed (at the same time and by the same value for sin(2ϕ Θ2)(a+rsinϕ) s= − (6) both wings) as it is shown in Fig.1. sin(ϕ Θ ) 2 − In the Cartesian coordinates, the periodic configura- Thus, here the scheme is presented for calculating the tionwithpairsofnonparallelwingslookslikeitisshown EMF generation due to virtual photons in nanosized in Fig.2. Each figure in the configuration period is com- metal configurations in optical approximation. When pletely similar to the single figure depicted in Fig.1. In such configurations are arrangedin a periodic structure, the period, between the ends of the figures there is the it is necessary to take into consideration the following. distance d. The periodic arrangementofs pairsofnonparallelwings In the first approximation, Casimir EMF for one non- withdistancedbetweenthemandagainstthex-axiswith parallel wing ∆E can be found in the form [1] k their wider sides leads to the formation of n 1 pairs of − 1 L rmax wings with oppositely directed sides (see Fig.2). In this ∆E = [1 ρ k] dy P(Θ,r,φ)dr. (1) k 2n e − − caseonewing ofeachconfigurationis atthe sametime a b Z0 Z0 wingoftheotherconfigurationthewideopeningofwhich Here n is volume density, e - electron charge, ρ - reflec- b isoppositelydirected. Thus,fornpairsofconfigurations tioncoefficient,k -photontransmission,P(Θ,r,φ)-local periodically arranged along the z-axis it is possible to specific pressure at each point r on the wing with the writethefollowingexpressionfortheirtotalEMF[21](if length r and width L max thewingsasthesourcesofEMFareconnectedinachain P(Θ,r,φ)= ~cπ2 Θ2dΘsin(Θ 2ϕ)4sinΘcosΘ in series and not in parallel) 240s4 Θ1 − (2) =−2R~4c0πs24A(ϕ,Θ1,Θ2), n ∆E = ∆E =n∆E(a) (n 1)∆E(d) (7) where total n − − n=1 A(ϕ,Θ ,Θ )= 1 [24Θ sin4ϕ 24Θ sin4ϕ X 1 2 96 1 − 2 +18cos2Θ 18cos2Θ Here ∆E(a) is the EMF for the shortest distance a be- 1 2 − +6cos(4ϕ 4Θ ) 6cos(4ϕ 4Θ ) (3) tween the pair of plates, and ∆E(d) is the EMF for the 2 1 − − − +3cos(8ϕ 2Θ ) 3cos(8ϕ 2Θ ) distance d in formulae (1-6). Naturally, if all the wings 2 1 − − − +cos(8ϕ 6Θ ) cos(8ϕ 6Θ )]. in the periodic configuration of plates are connected in 1 2 − − − parallel and not in series, it is necessary to calculate the In formula (2), ~ = h/2π. is reduced Planck constant, equivalent summed current in the system. c – speed of light. The functional expressions for limit CALCULATION RESULTS angles between the wings Θ , Θ and the s parameter 1 2 From formula (7) it follows that for d = a in periodic have the following form configurations Casimir EMF is ∆E = ∆E(a) at any total a. Itmeansthatintheperiodicstructure,evenford=a, Θ =arccos r+asinϕ−Rcos2ϕ , 1 −√(a+Rsinϕ+rsinϕ)2+(rcosϕ−Rcosϕ)2 at n EMF is at the same level as that for one pair (cid:20) (cid:21) →∞ (4) ( n = 1) of the configuration. However, it is clear that 3 E(arb. unit) Q (arb. unit ) E (arb. unit) Q (arb. unit) 10 10 ddd///aaa === 111...63125 a 10 b ddd///aaa === 111...63125 10 RRR///aaa === 416050 c 8 d RRR///aaa === 416050 = 1 deg 6 = 1 deg 5 5 5 4 = 1 deg 2 = 1 deg 100-1 100 101 102 103 104 100-1 100 101 102 103 104 010-1 100 101 102 103 010-1 100 101 102 103 R/a R/a d/a d/a FIG. 3. The total EMF in the periodic configuration at series connection of the wings in the chain depending on the relation R/a in thestructure (a) and theeffectiveness Q of thestructure(b) at different relations of d/a at ϕ=1 deg E (arb. unit) Q (arb. unit) E (arb. unit) Q (arb. unit) 180 ddd///aaa === 111...63125 a 180 ddd///aaa === 111...63125 b 10 ϕϕϕ === 000...100 20d 7ed gedgeg c 10 d ϕϕϕ === 000...100 20d 7ed gedgeg 6 6 R/a = 250 R/a = 250 R/a = 250 R/a = 250 4 4 5 5 2 2 010−5 10−3 10−1 101 10−5 10−3 10−1 101 010-1 100 101 102 103 010-1 100 101 102 103 (deg) (deg) d/a d/a FIG. 4. The total EMF in the periodic configuration at series connection in the chain of the wings depending on the angle ϕ in the structure (a), and the structure effectiveness Q(b) at different relations d/a and R/a = 250. The total EMF (c) for different angles ϕ dependingon the relation d/a at R/a=250. atd=a, the periodic configurationEMFwilldepend on Let us note, that analytically and in the result of the 6 thed/aratioinaccordancewithformula(7)fordifferent numerical calculations, the value of the optimal relation R/aratiosandanglesϕbetweenthewingsasitisshown d/a 1.62 at ϕ 0 and R/a has been found → → → ∞ in Fig.3a,c. When the number n of the pairs of wings for the periodic configurations. At the combined search grows, the behavior of the curve will be similar to those with the use of angles and relation d/a, the maximum showninFig.3,however,foranyanglesϕandparameters of the EMF generation shifts to the values of the order R/a, however, naturally, the their EMF level will grow d/a 1.8 at ϕ=1 deg. → linearly depending on n. CONCLUSIONS It is possible to determine the effectiveness Q of the Thus, in the present paper, the possibility in principle EMF generation in n pairs of wings as the relation of is shown for the existence of Casimir EMF in nanosized ∆Etotal tothetotalconfigurationlengthalongthez-axis nonparallel metal wings arranged in periodic structures. It is found that in the periodic structures with pairs of ∆E total Q= . (8) nonparallelwings,allCasimircurrentsarecompensated, n[a+d+2Rtan(ϕ)] and EMF is not generated. However, when the peri- The dependence of Q on the relations R/a and d/a is odicity of the pairs of nonparallel wings is disturbed, shown in Figs.3b.d. It can be seen, for example, that for uncompensated currents are generated in them in the any length R of the wings with different angles ϕ, there direction of the smallest angle between the wings, and is the maximal effectiveness Q of the generation of the consequently, EMF is generated. In this case, at any re- total EMF ∆E . lations of the configuration parameters (angles between total As it is known [1], there is the maximum of the EMF the wings and the wings lengths, the distance between generationfor eachpair ofwings depending onthe angle the wings, etc.) and at any number of the pairs of wings ϕ between the wings. The similar effect is also found for at their series connection in the chain, there is the max- the periodic configuration Fig.4a. The combined depen- imum of the effectiveness of the EMF generation in the dences of EMF on two parameters, i.e. the angle ϕ and period. The value of the total EMF in the nonparallel the relation d/a, and also R/a and d/a are displayed in pairs of wings connected in series linearly growsdepend- Fig.5a,c.. ThecorrespondingeffectivenessesQareshown ing on the number of elements in the chain. in Fig.5b,d. TheauthorisgratefultoT.Bakitskayaforhershelpful 4 )tinu.bra(E∆8642 1R0/-3a=1205-20 a 101102 )tinu .bra(Q861402-4 10-3 10-2 10-1 b R/a=250 101 102 1ϕ02=1deg c 10110202468∆E (arb.unit) 102 ϕ=1deg d 10110202468Q (arb. unit) 10-1 ϕ1 0(0deg)101 102 100 d /a ϕ (1d00eg)101 102 100 d/a 101 d /a 100 100 R/a 101 d/a 100 100 R/a FIG. 5. The total EMF in the periodic structure at series connection in the chain of the wings depending on the angle ϕ and the relation R/a in the structure (a), and the corresponding structure effectiveness Q (b). 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