1 Introduction. K. Morawetz*, M. Gilbert and A. Trupp Induced Voltage in an Open Wire

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1 Z. Ntufosh. 7; 7(7): K. Mowetz*, M. Gilbet nd A. Tupp Indued Voltge in n Open Wie DOI.55/zn-7-6 Reeived My, 7; epted My, 7; peviously published online June, 7 Abstt: A puzzle ising fom Fdy s lw hs been onsideed nd solved onening the question whih voltge will be ued in n open wie with time-vying homogeneous mgneti field. In ontst to losed wies whee the voltge is detemined by the time vine of the mgneti field nd the enlosed e, in n open wie we hve to integte the eleti field long the wie. It is found tht the longitudinl eleti field with espet to the wve veto ontibutes with / nd the tnsvese field with / to the ued voltge. In ode to f the eleti fields the soues of the mgneti fields e neessy to now. The epesenttion of sptilly homogeneous nd time-vying mgneti field implies unvoidbly etin symmety point o symmety line whih depend on the geomety of the soue. As onsequene the ued voltge of n open wie is found to be the e oveed with espet to this symmety line o point pependiul to the mgneti field. This in tun llows to f the symmety points of mgneti field soue by mesuing the voltge of n open wie pled with diffeent ngles in the mgneti field. We pesent etly solvble models of the Mwell equtions fo symmety point nd fo symmety line, espetively. The esults e pplible to open iuit poblems lie oosion nd fo stophysil pplitions. Keywods: Eletodynmis; Indution; Mwell Equtions. Intodution Fdy s lw is stndd tetboo nowledge. The ued voltge of losed ile in mgneti field *oesponding utho: K. Mowetz, Münste Univesity of Applied Sienes, Stegewldstsse 9, Steinfut, Gemny; Intentionl Institute of Physis (IIP) Fedel Univesity of Rio Gnde do Note, Av. Odilon Gomes de Lim 7, Ntl, Bzil; nd M-Pln-Institute fo the Physis of omple Systems, 87 Desden, Gemny, E-mil: mowetz@fh-muenste.de M. Gilbet: Münste Univesity of Applied Sienes, Stegewldstsse 9, Steinfut, Gemny, E-mil: gilbet@fh-muenste.de A. Tupp: Bndenbug Univesity of Applied Polie Sienes, Benue Stße 46, 655 Onienbug, Gemny, E-mil: tuppfhpolbb@ol.om is eithe used by the time-dependent hnge of the enlosed e o the time-dependent hnge of the mgneti field [, ]. Inteesting puzzles nd the ution in defomble iuits n be found in []. Fdy s ution epeiments now hve gined etin evivl when nnostutues e onsideed [4] nd ply uil ole in type II supeondutos [5, 6], see [7] fo efeenes. The mgneti field effets in uents e even used fo mesuing the speed of light [8]. In ft, Fdy s ution lw povides mny puzzles nd pdoes whih help the student to undestnd the beutiful physis beh. Hee we povide new one whih is stonishingly not teted in litetue. To solve this puzzle we povide genel fomuls fo ued voltges nd povide thee etly solvble models of Mwell equtions whih by itself might be inteesting fo univesity tehing. The messge of this tile is tht the ued voltge in n open wie n only be detemined if the geneting geomety of the even ssumed homogeneous nd time-vying mgneti field is nown. This evels deepe insight into eleti nd mgneti fields nd how they ontibute to ution. Though ution in losed wies nd the foes ting on wies in mgneti fields [9] e well undestood, the ution in open wies is ely studied, pobbly beuse the effets thee e espeilly puzzling. Insted of wies, open iuit poblems hve been studied intensively. As we n imgine n open iuit s supeposition of wies we thin tht this study hee might shed some light on open iuit poblems too. Mgneti effets due to open iuits hve been nown fo moe thn yes nd they emin of inteest with espet to ooding poblems in feomgneti eletodes. Fo n oveview of the epeimentl tivities nd thei histoy see []. As most of the epeiments e pefomed with espet to the question of ooding mteils [ ], it is ovelid by the poblem of hemil etions. Then nonequilibium situtions hve to be onsideed suh tht Loentz nd gdient foes beome impotnt on stem in nodi dissolution of miostutues [4]. These mgneti field effets e uil fo pttening of eletodeposits [5]. Also eddy uents, mesued fo emple with ontt-less methods [6], e still mjo poblem. In [, ] the oienttion of the eletode in the mgneti field evels opposite esponses when oiented pllel o pependiul to the field. This will be eplined by ou ppoh. Beeitgestellt von MPI fue Physi omplee Systeme Heuntegelden m.7.7 :59

2 68 K. Mowetz et l.: Indued Voltge in n Open Wie + E Figue : Indution in two identil wies (solid) diffeing only by two losing pths (dshed) oveing the sme e but leding to n opposite sign of the ued voltge. Despite these viety of pplitions the simple question wht voltge is ued in n open wie o iuit when pled in homogeneous nd time-vying mgneti field is not teted to the best of ou nowledge. Hee we wnt to esolve this puzzle poviding unique epession of the voltge ued t the ends of n open wie within time-dependent nd homogeneous mgneti fields nd its dependene on the dietion of the mgneti field. A fist pdo ises sing the simple question whih voltge might be ued in n open uved wie eposed to time-vying but homogeneous mgneti field. A gednen epeiment seems to onvine us tht this voltge is undetemined. Dependent on whethe one loses the wie lowise o ntilowise, diffeent sign of the ued voltges is obtined t its ends s shown in Figue. The pth used to tun theses wies into losed loop detemines whethe the eleti field is followed fom left to ight o fom ight to left nd onsequently the sign of the ued voltge. Mny diffeent setups n be onstuted whih show the sme ontdition. The solution of this pdo is enlightening the ingeniousness of Fdy s lw. In ode to mesue the voltge one hs to lose the open wie in some wy whih povides losed e evey time. It is oet tht the bove setup yields two diffeent signs dependent on how one loses the iles by mesuement. The eplntion so f seems to led to the onlusion tht the voltge in n open wie by itself emins undetemined until we lose the loop nd n pply the Fdy lw. This is fotuntely not the se. We hve to speify wht we men with ued voltge. As pointed out by Rome [7] the voltge wht voltmete is mesuing is the pth integl long the wie of the voltmete. We will onside ou wie s the pth long whih we mesue the voltge. As we e onsideing sptilly homogeneous mgneti field without bodes we hve simply E + onneted e with nonvnishing mgneti nd eleti fields. In this sense n open wie possesses n ued voltge t its ends if it ests in time-vying homogeneous mgneti field. This is due to the ft tht homogeneous mgneti field n be only elized in n symptoti limit of finite geomety. This implies etin symmety points o lines fiing n oigin of the oodinte system. We will show tht this unvoidbly leds to n ued voltge whih is the e spnned by the open wie with espet to this symmety point o line of the mgnetifield-eting setup. We wnt to suppot the gumenttion fist by genel epliit lultion fom Mwell equtions nd then we will illustte it with the help of two etly solvble models whee the homogeneous nd time-vying mgneti field is elized in n symptoti wy. Fist we deive the genel fomul fo the ued voltge showing tht the tnsvese nd longitudinl pts ontibute with / nd / espetively. Then in the thid setion we pesent some etly solvble models whih illustte the neessity to onside the geomety of the soue of the mgneti field poviding unique ued voltge. The summy ontins suggestion fo detemining these geometies nd the ppendi pesents fou diffeent wys to lulte used integl. Indution in Open Wies Fist we onside the genel fomuls fo the ution. Without doubt we n f the ued voltge in wie if we integte the pesent eleti fields long the wie () (, )d. U t = E t The question is only whih eleti fields e pesent. Theefoe we ty to f n nswe by solving the Mwell equtions. The fist eqution we onside, E (, t) = B (, t), () is best Fouie tnsfomed by to te the fom E t = E t i () d e (, ) E = ib. () () (4) Beeitgestellt von MPI fue Physi omplee Systeme Heuntegelden m.7.7 :59

3 K. Mowetz et l.: Indued Voltge in n Open Wie 69 Obseving the seond Mwell eqution tht thee e no mgneti monopoles, B =, o Fouie tnsfomed B =, the (4) is solved by veto lgeb s tns long E = E + E whee the tnsvese field with espet to the wveveto eds tns i E = B. The longitudinl field with espet to the wve veto is given by the divegene of the hge s Guß lw o thid Mwell eqution (, t) div E (, t) = εε with the vuum pemittivity ε nd the dieleti onstnt ε hteizing the medium whih is ssumed to be onstnt hee. In ontst to (4) whih detemines only the tnsvese eleti field unmbiguously, the Guß lw detemines only the longitudinl field s =. The Guß lw eds fte Fouie tnsfom E long i = εε The hge density dops out of the finl fomul lte but it is equied to show tht this longitudinl field ontibutes with one thid to the ued voltge. The ued voltge is the line integl long the wie uve unning fom to. Let us emphsize gin tht the Mwell eqution (4) lone does not detemine the eleti field. One needs dditionl boundy onditions o the seond Mwell eqution (6) to detemine the omplete eleti field. We onside now the tnsvese nd longitudinl pts with espet to the wve veto septely. The tnsvese field is obtined fom (5) U tns. (5) (6) (7) tns = d E d i i = id e d e B(, t) ( π) = d B (, t) d 4 π (8) whee we hve used the invese Fouie tnsfom of the oulomb potentil d i i e = ( π) 4π (9) nd tivil ottion of the veto podut. In the futhe steps we ssume homogeneous but time-dependent mgneti field suh tht the integl in (8) 4π d = () n be pefomed (see Appendi A) with the finl esult U = B d = BA. tns () The wie oves n e A with espet to some oigin given by the used oodinte system. This will be found to be fied due to the soue of the mgneti field. We obtin just / of the epeted esult of Fdy s lw. In othe wods / must be ontibuted by the longitudinl field. Indeed, we n lulte the ued voltge of the longitudinl field (7) s long U = long d E d i i = id e d e (, t) ( π) εε = d d ( E (, t)) 4 π = d d E (, t) j i 4 π j i ij = d d E (, t) ( j i ) = d E (, t ) () whee we used (9) fom seond to thid line s well s (6). Then we used tht the divegene of longitudinl eleti field is the sme s the divegene of the totl field. A oesponding ptil integtion is pefomed when going fom the thid to the fouth line. As d E is supposed to be the totl ued voltge, the longitudinl pt () povides obviously only / of the totl ued voltge. We obtin the esult tht the tnsvese pt of the eleti field whih is the nononseving pt, ontibutes with / nd the onseving longitudinl pt with / to the totl ued voltge. As epeted, the longitudinl pt is onseving in the sense tht it is just the sl potentil diffeene seen fom the thid line of () U () t = d d (, t) long 4 π εε () = Φ(, t) Φ(, t) Beeitgestellt von MPI fue Physi omplee Systeme Heuntegelden m.7.7 :59

4 6 K. Mowetz et l.: Indued Voltge in n Open Wie s the potentil of hge distibution itself is given by (, t) Φ t = 4 π εε (, ) d. (4) In othe wods the longitudinl field ues voltge whih is given by the diffeene of the potentils t the ends of the wie but epesents only / of the totl ued voltge. The totl ued voltge is, theefoe, eveytime lge thn the eletomotive foe whih is only the pt of eleti fields due to hge seption [] nd epesented by the potentil diffeene () nd not due to nononseving tnsvesl fields. Summizing we obtin the ued voltge of n open wie in homogeneous time-vying mgneti field s U = B A (5) with the e spnned by the wie A = d. (6) The fomul (5) povides the ued uent of uved wie in homogeneous time-dependent mgneti field. The mzing fetue of (5) is now tht fo n open wie its vlue depends on the point of oigin O fom whih we ount the uve s shown in Figue. Only in losed uves this point of oodinte oigin is dopping out of the integl. In othe wods the vlue fo the ued voltge in open wies is given by the oigin of the oodinte system of the geomety whih elizes the homogeneous mgneti field. One should note tht (5) nd the ptition of / fo longitudinl nd / fo tnsvesl field is guge ependent nd solely onsequene of Mwell equtions nd epesents the Helmholtz theoem whih sttes tht the soue of the field is the ul nd the d O O Figue : Ae spnned by the open wie nd by n infinitesiml d with espet to two diffeent oigins O nd O of oodintes oding to (6). div of the field. In [8] it ws hosen speil guge suh tht A= B() t nd Φ = onst nd one hs shotly U = E d = B( t) d (7) = Bt () d = Bt ( ) A. Hee in ou tetment we hve shown epliitly tht the / nd / shing is not mtte of speil hoie o due to the Fouie tnsfom but onsequene of Mwell equtions by et mnipultions. The ouing eleti fields n lso be mde visible by the lssoom demonsttion of ued nononsevtive eleti fields [9]. Etly Solvble Models fo Asymptoti Geomety. Genel Fomuls fo Indued Voltge in Open Wies In ode to onvine ouselves bout the bove sttement tht the ued voltge of n open wie is dependent on the symmety point of the eting mgneti field, we onstut time-vying homogeneous mgneti field by the epliit solution of the Mwell equtions. This solution is onveniently given by the Liénd Wiehet potentils []. The divegene-fee mgneti field is epesented by the veto potentil B = A nd the Mwell eqution B = E leds to the eltion ( A+ E) = whih mens tht the eleti field is given in tems of the veto potentil nd sl potentil Φ E = A Φ. (8) With the Guß lw it leds to = E= A Φ. (9) εε Using the othe Mwell eqution µµ ( j + εε E) = B= ( A) A one hs A A j A. () = µµ + Φ + hoosing the Loenz-guge A, Φ + = fom (9) nd () the symmeti wve equtions fo the veto nd sl potentil esult s Beeitgestellt von MPI fue Physi omplee Systeme Heuntegelden m.7.7 :59

5 K. Mowetz et l.: Indued Voltge in n Open Wie 6 A A µµ j = Φ Φ=. εε () These inhomogeneous wve equtions e solved by etded potentils j, t µµ A (, t) d 4 π, t Φ(, t) d 4 π εε = = () with the uent density j(, t) nd the hge density (, t). Employing the etded potentils we ensue the onsisteny with ll Mwell eqution. Now it is dvntgeous to epess the ued voltge in tems of the veto potentil. The ued voltge is given s line integl long the wie ove the eleti field pesent t the oesponding point. Insted of lulting the ued voltge long uve : = q ( ) with q q q, it is moe onvenient in the following to lulte thei time deivtive with the help of the Mwell eqution U = E d = µµ j d + B d. () The seond integl beomes with the help of ( A) = ( A) A nd epesenting the uve of the wie s = ( q) q d d q ( A) Ad dq q = A da + µµ d j q q (4) whee we used the wve eqution () fo the veto potentil in the lst line. One sees tht the lst pt of (4) nels just the fist pt of () nd we obtin q U = d A+ A. (5) Due to the Loenz guge the seond pt is nothing but the diffeene of the potentil t the ends of the wie. This pt vnishes fo losed loop q = q nd one obtins etly the time deivtive of the Fdy lw U = d A= df A= B df q (6) whee df is the sufe e element. Fo n open loop the fomul (5) is onvenient to use beuse the veto potentil is given in tems of the mgneti-field-eting uents whih povide unique esult if we integte ove the pmete nge q < q < q desibing the wie.. Simple Pmeteiztion One my simply use the epesenttion of the homogeneous mgneti field A= B()( t z,, ); B= B( t)(,, ) (7) As A = we hve fom () j A= = B ( z,, ) εε (8) nd Φ = o = whih identifies the uent s soue of this homogeneous nd time-vying mgneti field. Fom (5) one gets now fo the time deivtive of the ued voltge = d ( ) U B z (9) whih mens tht it is detemined by the e fomed by the wie with the -is pependiul to the mgneti field dietion whih is the y-is. In othe wods, we hve symmety line, the -is, with espet to whih the e hs to be lulted. Now we will see how homogeneous mgneti field is symptotilly elized by finite setup of geomety. A list of diffeent wys of integtion pths with diffeent esults in geement with the outlines model hee n be found in [].. Infinite ylil oil Let us onside n infinite ylil oil with hight h nd n inne nd oute dius of / = R ± / s shown in Figue with uent I(t) = I osωt unning though n infinitesiml thin wll. We tnsfom = in () oding to Figue nd obtin t + ( z z ) osω µµ A= d d dz e 4 πh + ( z z ) h π + I φ h φ () Beeitgestellt von MPI fue Physi omplee Systeme Heuntegelden m.7.7 :59

6 6 K. Mowetz et l.: Indued Voltge in n Open Wie R I h The epnsion o(z/r) mens to inese the dius fste thn the height of the yle. This is only speil hoie of eting homogeneous mgneti field nd shows the poblem of eting homogeneous mgneti field. The integtion z ove the length h hs beome tivil nd in the lst line of () we hve pefomed the limit. Futhe we see tht the time is delyed by R t = t whih is the time the field needs to oveome the distne R fom the soue. The lst ngul integtion n be pefomed with the finl esult y sin φ µµ I ω A= J φ os sin ωt () y ϕ α ϕ ϕ nd the Bessel funtion J ( ) = [ J J ( )]. The mgneti field beomes µµ I ω = = sinω ω J B A t (4) Figue : onsideed geomety of n infinite oil with the uent density j = I(t)/h unning though the shded e (bove) nd the hosen integtion pth (below). with e φ = ( sin φ, os φ, ). The os theoem leds to the lowe nd uppe integtion limits of = ( R± /) sin φ os φ nd the sin theoem ± leds to sinφ = sin α with α = φ φ. In the following we will wo in the limit of lge R. Sine R+ R () + one fs = + o(/r) nd φ = φ + φ + o(/r). We substitute p= + ( z z ) in () whih llows to pefom this integtion with the uppe nd lowe limits p = R os φ ± + o( / R) + oz ( / R) nd obtin ± ω sin µµ ω A= t + e π I R d os os φ ω φ φ πω sin( φ+ φ ) π µµ I ω = dφ os ωt osφ os( φ φ ). 4π + + () whih is dieted long the symmety is. We see tht homogeneous mgneti field is only elizble fo distnes = /ω fom the symmety is beuse then J (). Within this limit we hve J () / nd intoduing () into (5) we obtin sin φ φ I U µµ ω = sinωt d osφ = B d φ( φ) z 4 (5) φ whee we hve epesented the wie pmetilly by (φ). We obtin just the time deivtive of the ued voltge (5) with the setol e (6) spnned by the wie with espet to the z-is. The ltte is the symmety is povided by the setup of the symptoti homogeneous mgneti field. Plese note tht A = is etly vlid fo (), i.e. thee e no sl fields. We see tht the ued voltge in n open wie is just the seto e spnned by the wie with espet to the oigin pependiul to the mgneti field. This oigin is given by the z-is s the symmety is of the mgnetifield-eting setup..4 Infinite ylil Pltes In seond emple we wnt to onside sitution whee we do not hve symmety point but symmety line. We Beeitgestellt von MPI fue Physi omplee Systeme Heuntegelden m.7.7 :59

7 K. Mowetz et l.: Indued Voltge in n Open Wie 6 onstut the time-vying homogeneous mgneti field by two time-vying uents whih un in -dietion in n uppe plte t (,, ) nd in opposite dietion in plte t (,, ). We ssume tht the pltes e yles with the thiness in z-dietion nd dius R in nd y-dietion s shown in Figue 4. We will onside the limit of smll nd lge R whih should podue homogeneous nd time-vying mgneti field in y-dietion ne the ente fo smll z. The uent uns though the e R. With the uent pe length I/R = j os ωt we n define the e uent density fo the uppe nd lowe plte s It () j os ωt j =± e =± e. ± R (6) We onside the line uent density with espet to the line R seen by uent I. As we will wo in the limit R the diffeene between ltel midpoint nd endpoints does not mtte. The veto potentil follows the uent to be in -dietion A= (,, A A ) nd eds in yle oodintes fom Figue 4 t + ( z z ) π m + / µµ j os A d d dz = φ 4 π / + ( z z ) (7) fo the pt fom the uppe plte. The pt of the lowe plte hs to be subtted with the eplement. Fo the uppe limit one gets os φ. With the limit of smll + / d zf ( z ) = f( ) + o( ), / = R sin φ m (8) z + / / y R j nd epling p z = + ( ), the integtion yields µµ A = 4 π j z dφ sinω t πω + ( z) m sin ωt. (9) + / / j We n now lulte ll quntities epliitly nd use the limit R whih povides + ( z) = R os φ + o ( / R) nd the seond sin m tem in (9) is subtted when the ontibution of both pltes e dded. One obtins y µµ j ωz A = A A = os ω t sin ω (4) y nd the mgneti field beomes ωz B =, µµ j osω t os,. (4) φ φ Figue 4: onsideed geomety of two uents unning ntipllel though ylil pltes t the distnes ± fom the -is (bove) nd the integtion vibles fo pependiul vetos = (below). We see gin the ppene of time dely t = t whih is the time the field needs to oveome the distne fom the soue uent. A homogeneous mgneti field is only symptotilly possible fo z = /ω. In this limit o(wz/) we obtin fo the time deivtive of the ued voltge (5) q d q ( ) U = µµ j ω osωt d q zq ( ) = B dz dq (4) y q Beeitgestellt von MPI fue Physi omplee Systeme Heuntegelden m.7.7 :59

8 64 K. Mowetz et l.: Indued Voltge in n Open Wie whih is (9) nd hs the genel fom of (5). The e, howeve, is now the one whih the wie foms with the -is pependiul to the mgneti field dietion whih is the y-is. Fo losed wie the lst integl gives gin the Fdy lw fo the enlosed e in, y dietion whih is penetted by the time-dependent mgneti field. Fo n open wie we see tht the e is now given by symmety line, the -is, nd not the symmety point s in the lst emple. Plese not gin tht A = fo (4) whih mens tht no sl potentils e pesent. 4 Summy Iespetive of the tul poedue to elize homogeneous nd time-vying mgneti field one n integte the time deivtive of the eleti field long the line to obtin the time deivtive of the ued voltge. The longitudinl eleti field povides only / of the ued voltge nd is given by the potentil diffeene between the ends of the wie. The tnsvese nd nononseving field ontibutes with / to the ued voltge. A homogeneous mgneti field n be only elized in n symptoti limit of fied geometil setup. The ltte one defines etin symmety is o symmety points nd the oigin of the oodinte system. We f fom the genel solution of the Mwell eqution tht the ued voltge of n open wie is given by the e spnned with espet to this symmety point o line. oespondingly we obtin seto fomul fo the e if ylil field is pesent nd n e integl with espet to line in pln symmety. We hve illustted these two ses by two etly solvble models fo the Mwell equtions elizing the homogeneous nd time-vying mgneti field. The dependene of the open iuit voltge on the dietion of the mgneti field hs been mesued in []. Thee it hs been found tht if the sufe of the eletode is oiented pllel o pependiul to the mgneti field, the open iuit potentil moves in opposite dietions with the lgest hnges ouing when the eletode sufe is pllel to the mgneti field. This obsevtion is eplined by ou fings. If the wie osses the symmety line o point we n estblish simple mio ule by onsideing the et mio imge of the wie in pependiul plne to the mgneti field. The wie nd its imge should hve the sme ued voltge if the wie osses the symmety line. Theefoe if we lose the open wie with its mio imge we should hve twie the ued voltge of the open wie. The now losed e obeys Fdy s lw. Theefoe we n suggest the ule tht the ued voltge in this se is hlf the one whih is ued by the e oveed by the wie nd its mio imge. Altentively we might onnet the uved wie with stight line nd use this e fo the ution lw. This mio ule is only pplible if the symmety is o point osses the wie. We n suggest n epeimentl setup to detemine the symmety point o symmety line of given mgneti field. Mesuing the ued voltge of n open stight line wie in diffeent dietions would yield zeo if the wie is ligned pependiul to symmety is. If this line wie is now otted fom to 9 one n ett fom the inesing voltge the geomety whethe we hve seto fomul s in ou fist emple o n e integl with espet to symmety is. Fom this one n onlude bout the oigin of the symptotilly homogeneous mgneti field. This might hve stophysil pplitions in detemining the symmety of soues of mgneti fields. Anowledgments: The uthos e gteful to Mtin Poppe bout questioning the pdo. Appendi A: Fou wys to lulte n integl We e going to lulte the integl 4π I = d = (A) in fou diffeent wys. () By Guß-Ostogtzy: The integl n be dietly tnsfomed into sufe integl by the integl theoem of Guß-Ostogtzy (one wites Guß theoem thee times fo eh oodinte nd ombines it s veto) d g= dag (A) nd pefoming the zimuthl ngle integtion leving the ltitudes, one gets I = da = πlim d + > 4π 4π = lim = < (A) (b) Diet integtion: Pefoming the zimuthl ngle integtion dietly Beeitgestellt von MPI fue Physi omplee Systeme Heuntegelden m.7.7 :59

9 K. Mowetz et l.: Indued Voltge in n Open Wie 65 I = d = dyy d 4π = y π / ( + y y) (A4) () By limit of nown integls: The nown men oulomb enegy in s-wve stte leds to the integl / e R / R / d = π / R( e ) e R (A5) whih we n use to pply nd pefoming the R limits leds etly to (A). (d) By veto ti: using = we n use ptil integtion fo the i-th omponent I = d ( ) i i 4π = d d ( ) j i j ij j = 4π = i (A6) s ll nondigonl ombintions e zeo due to ngul integtions nd we hve used = 4 π( ). Refeenes [] I. Glili, D. Kpln, nd Y. Lehvi, Am. J. Phys. 74, 7 (6). [] D. J. Giffiths, Intodution to Eletodynmis, Addison- Wesley, Boston. [] G.. Sogie, Eu. J. Phys. 6, 6 (995). [4] H. K. Kim, J. S. Hwng, S.-W. Hwng, nd D. Ahn, Nnotehnol IEEE Tns. 7, (8). [5] S. M. Khnn nd M. A. R. LeBln, J. Appl. Phys. 4, 565 (97). [6] Y. Levin nd F. B. Rizzto, Phys. Rev. E 74, 6665 (6). [7] P. Lipvsý, J. Koláče, K. Mowetz, E. H. Bndt, nd T. J. Yng, Benoulli Potentil in Supeondutos, Spinge, Belin 7, Letue Notes in Physis 7, ISBN [8] G. Spviei, Eu. Phys. J. D 66, 76 (). [9] J. A. Redinz, Am. J. Phys. 79, 774 (). [] A. Dss, J. A. ounsil, X. Go, nd N. Leventis, J. Phys. hem. B 9, 65 (5). [] M. Wss nd Y. I. Khts, J. Eletonl. hem. 5, 5 (). [] N. S. Peov, P. M. Shevedyev, nd M. Inoue, J. Appl. Phys. 9, 8557 (). [] R. Sueptitz, K. Tshuli, M. Uhlemnn, A. Gebet, nd L. Shultz, Eletohim. At 55, 5 (). [4] A. Bund nd H. H. Kuehnlein, J. Phys. hem. B 9, 9845 (5). [5] P. Dunne nd J. M. D. oey, Phys. Rev. B 85, 44 (). [6] Y. Kftmhe, Am. J. Phys. 68, 75 (). [7] R. H. Rome, Am. J. Phys. 5, 89 (98). [8] R. Bielein, Am. J. Phys. 6, 8 (995). [9] A. P. J. vn Deusen, Am. J. Phys. 7, 99 (5). [] J. D. Json, lssil Eletodynmis, John Wiley, New Yo 999. [] Beeitgestellt von MPI fue Physi omplee Systeme Heuntegelden m.7.7 :59