On the Resistance of an Infinite Square Network of Identical Resistors (Theoretical and Experimental Comparison)

Transcription

1 On he esisance f an Infinie Square Newrk f Idenical esisrs (Thereical and Experimenal Cmparisn) J. H. Asad, A. Sakai,. S. Hiawi and J. M. Khalifeh Deparmen f Phsics, Universi f Jrdan, Amman-1194, Jrdan. Phsics Deparmen, Aman Universi, UAE. Phsics Deparmen, Muah Universi, Jrdan. Absrac A review f he hereical apprach fr calculaing he resisance beween w arbirar laice pins in an infinie square laice (perfec and perurbed cases) is carried u using he laice Green s funcin. We shw hw calculae he resisance beween he rigin and an her sie using he laice Green s funcin a he rigin, G (0,0), and is derivaives. Experimenal resuls are bained fr a finie square newrk cnsising f 30x30 idenical resisrs, and a cmparisn wih hse bained hereicall is presened. Ke wrds: Laice Green s Funcin, esisrs, Square laice.

2 1. Inrducin I is an exiing quesin find he resisance beween w adacen laice pins f an infinie square laice where all he edges represen idenical resisrs. This prblem was sudied well in man references 1 7, where fr finding he resisance beween w arbirar grid pins f an infinie square laice he used a mehd based n he principle f superpsiin f curren disribuins 3 5. One can find a full discussin fr he elecric circui in van der Pl and Bremmer 1. In he 1970 s Mngmer 8 inrduced a mehd fr measuring he elecrical resisivi f an isrpic maerial, where he prepared a recangular prism wih edges in principal crsal direcins wih elecrdes n he crners. Based n Mngmer paper, Lgan e. al 9 develped Mngmer s mehd, where he inrduced penial series fr cmpuing he curren flw in he recangular blck. ecenl Cseri 10 and Cseri e. al 11 sudied he prblem in which he used a alernaive mehd based n he Laice Green s Funcin (LGF) which enables us calculae he resisance beween an w arbirar sies in a perfec and perurbed infinie square laice. The LGF has man applicains in phsics such as describing he ineracin beween he elecrns which is mediaed b he Phnns 1, suding he effec f impuriies n he ranspr prperies f meals 13, suding he ranspr in inhmgeneus cnducrs 14, suding he phase ransiin in classical w-dimensinal laice culmb gases 15, and finall resisance calculain 10, 11. The LGF fr he w-dimensinal laice 1619 has been sudied well, and in hese references he reader can find useful papers. The LGF presened in his paper is relaed he LGF f he Tigh- Binding Hamilnian (TBH) 13. In he fllwing we shw hw calculae he resisance beween he rigin and an her sie using G (0,0) and is derivaives. The resisance beween he rigin and a laice sie ( l, m) in a cnsruced finie perfec square mesh (30x30 resisrs) is measured. Als, he resisance beween he rigin and a laice sie ( l, m) in he same cnsruced mesh, when ne f he resisrs is brken (i.e. perurbed) is measured. Finall, a cmparisn is carried u beween he measured resisances and hse calculaed b Cseri s mehd 10, 11. We believe ha invesigain f he resisance f square newrk f resisrs shuld be ineresed in he field f arras f Jsephsn uncins f high TC 0 supercnducing maerials. The invesigain f elecrphsical prperies f such ssems in he nrmal sae befre supercnducing

3 ransiin, and he cnen f his manuscrip is helpful fr elecric circui design and he mehd is insrucive.. Thereical esuls.1 Perfec Square Laice In an infinie square laice cnsising f idenical resisances, he resisance beween he rigin and an laice pin ( l, m) can be calculaed using 10 ( l, m) [ G (0,0) G ( l, m)] (1) where G (0,0) is he LGF f he infinie square laice a he rigin, and G ( l, m) is he LGF a he sie ( l, m). Firs f all, he resisance beween w adacen pins can easil be bained as (1,0) ( 1,0) [ G (0,0) G (1,0)]. () G can be expressed as (see Appendix A) 1 G ( 1,0) [ G (0,0) 1] ;. (3) Where is he energ. and = refers he energ f he infinie square laice a which he densi f saes (he imaginar pars f he LGF) is singular (Van Hve 3 5 singulariies). Thus, Eq. () becmes 1 (1,0) [ G (0,0) G (0,0) ]. (4) S, (1,0) (0,1). (due he smmer f he laice). The same resul was bained b Venzian 3, Akinsn e. al. 4, and Cseri 10. T calculae he resisance beween he rigin and he secnd neares neighbrs (i.e. (1, 1)) hen 3

4 ( 1,1) [ G (0,0) G (1,1)]. (5) G (1,1) can be expressed in erms f G (0,0) and G (0,0) as (see Appendix A) G (1,1) ( 1) G (0,0) (4 ) G (0,0). (6) G (0,0) K( ) ( E( ) and G ( 0,0) K( ). (7) ( ) ( Where K ) and E ) are he ellipic inegrals f he firs kind and 1 secnd kind respecivel. Subsiuing he las w expressins in Eq. (4), ne bains ( 1,1). (8) Again ur resul is he same as Cseri 10 and Venezain 3. Finall, find he resisance beween he rigin and an laice sie ( l, m) ne can use he abve mehd, r we ma use he recurrence frmulae presened b Cseri 10 (i.e. Eq. 3). S, using Eq. (3) in Cseri 10 and he knwn values f ( 0,0) 0, (1,0) and ( 1,1) we calculae exacl he resisance fr arbirar sies. The same resul was bained b Akinsn e. al 4, and belw are sme calculaed values: (,0) 0.767, (3,0) , and (4,0) Fr large values f l r/and m he resisance beween he rigin and he sie ( l, m) is given as 10 ( l, m) ( Ln Ln8 l m ) (9) 4

5 where is he Euler-Mascherni cnsan 6. Venezain bained he same resul 3. Finall, as l r m ges infini hen he resisance in a perfec infinie square laice divergence.. Perurbed Square Laice (a bnd is brken) The resisance beween he sies i and f he perurbed infinie square laice where he bnd beween he sies i and is brken can be calculaed using 11 ( i, ) [ ( i, ) ( i, ) (, i ) 4[ ( i ( i, i ), )] (,. (10) where i ( i x, i ), J ( x, ), i ( ix, i ) and ( x, ) The resisance beween he ends f he remved bnd (i.e. ( i, )) is equal 11. )]. T calculae he resisance in he perurbed newrk, ne has specif clearl he ends f he remved bnd. Fr example, when he brken bnd is aken be beween he sies i ( 0,0 ) and (1,0) we fund using Eq. (10) ha: (1,0) , (,0) , (3,0) , (4,0) and Nw, if he brken bnd is shifed and aken be beween he sies i (1,0 ) and (,0) we fund again using Eq. (10) ha: (1,0) , (,0) , (3,0) , Fr large separain beween he w sies, hen ( i, ) Ln ( i, ) i i i i i 4[ i ( i, )] i (4,0) and i. (11) Nw, as i r/and ges infini hen ( i, ) ( i, ). (1) ha is, he perurbed resisance beween arbirar sies ges he perfec resisance as he separain beween he w sies ges infini. 5

6 . 3. Experimenal esuls T sud he resisance f a finie square laice experimenall we cnsruced a finie square newrk f idenical ( ) carbn resisrs, each have a value f (1 K ) and a lerance f (1%). 3.1 Perfec Case Using he cnsruced newrk, he resisance beween he rigin and he sie (l,m) is measured (using w-pin prbe). Belw are sme measured values: (1,0) , (,0) (3,0) 0.783, 0.864, and (4,0) The abve measured values are ver clse hse calculaed in secin Perurbed Case In his secin he bnd beween he sies i (0,0) and (1,0) is remved and he resisance beween he sies i (0,0) and ( x, ) is measured using he same newrk. Belw are sme measured values: (1,0) 1.000, (,0) , (3,0) , (4,0) and Nw, he brken bnd is shifed be beween he sies i (1,0) and (,0). Again we measure he resisance beween he sies i (0,0) ( x, ) (1,0) 0.537,, and belw are sme measured values: (,0) , (3,0) , (4,0) and As shwn, ne can see ha here is an excellen agreemen beween he measured and he calculaed values. 4. esuls and Discussin Frm he Figures shwn he resisance in an infinie square laice is smmeric under he ransfrmain ( l, m) ( l, m) due he inversin smmer f he laice. Hwever, he resisance in he perurbed infinie and 6

7 square laice is n smmeric due he brken bnd, excep alng he [01] direcin since here is n brken bnd alng his direcin. Als, ne can see ha he resisance in he perurbed infinie square laice is alwas larger han ha in a perfec laice and his is due he psiive secnd erm in Eq. (10). Bu as he separain beween he sies increases he perurbed resisance ges ha f a perfec laice. The cnsruced mesh gives accurael he bulk resisance shwn in Figs , and his means ha a crsal cnsising f (30x30) ams enables ne sud he bulk prperies f he crsal in a gd wa. Bu, as we apprach he edge hen he measured resisance exceeds he calculaed ne and his is due he edge effec. Als, ne can see frm he figures ha he measured resisance is smmeric in he perfec mesh, which is expeced. Fig. 3.4 and Fig. 3.6 shw ha he measured resisance alng he [01] direcin is nearl smmeric wihin experimenal errr, which is expeced due he fac ha here is n brken bnd alng his direcin, and his is in agreemen wih he hereical resul. Finall, ur values are in gd agreemen wih he bulk values calculaed b Cseri s mehd 10, 11. Derivain f he resisance f a finie square laice is under invesigain in rder cmpare wih he realisic experimenal resuls. eferences 1- B. Van der Pl and H. Bremmer, Operainal Calculus Based n he Tw-Sided Laplace inegral (Cambridge Universi Press, England, 1955) nd ed., p P. G. Dle and J. L. Snell, andm walks and Elecric Newrks, (The Carus Mahemaical Mngraph, series, The Mahemaical Assciain f America, USA, 1984) pp Venezian, G Am. J. Phs. 6, Akinsn, D. and Van Seenwik, F. J Am. J. Phs. 67, E. Aichisn Am. J. Phs. 3, F. J. Baris Am. J. Phs. 35, Mnwhea Jeng Am. J. Phs. 68(1), H. C. Mngmer J. Appl. Phs. 4(7): B. F. Lgan, S. O. ice, and. F. Wick J. Appl. Phs. 4(7): Cseri, J Am.J.Phs.68, Cseri J, Gula, D. and Aila P. 00. Am. J. Phs, 70, ickazen, G. Green s Funcins and Cndensed Maer Academic Press, Lndn. 13- Ecnmu, E. N. Green s Funcin in Quanum Phsics Spriger-Verlag, 7

8 Berlin. 14- Kirparick, S ev. Md. Phs. 45, Lee, J. and Teiel, S Phs. ev. B. 46, Hiawi,. 00. PhD Thesis, Universi f Jrdan( unpublished). 17- Mria, T. and Hrigucih, T J. f Mah. Phs. 1(6), Mria, T J. f Mah. Phs.1(8), Kasura, S. and Inawashir, S J. f Mah. Phs. 1(8),16. 0-G. Alzea, E. Arimnd,. M. Celli and F. Fus J. Phs. III France H.S.J.Van der Zan, F.C.Frisch, W.J.Elin,L.J.Geerligs and J.E.Mi.199. Phs. ev. Le. 69, D. Kimhi, F. Levraz, and D. Arisa Phs. ev. B 9, J. M. Ziman, Principles f The Ther f Slid (Cambridge U.P.,Cambridge, 197). 4-N.W.Ashcrf and N.D.Mermin,Slid Sae Phsics (Sanders Cllege Publishing, Philadelphia, PA, 1976). 5-C. Kile, Inrducin Slid Sae Phsics (Wile, New Yrk, 1986), 6 h ed. 6- Arfken, G. B. and Weber, H. J Mahemaical Mehds fr Phsiss. Academic Press, San Dieg, CA. 4 h. Ed. 7- T. Hriguchi, and T. Mria. " Ne n he Laice Green's Funcin fr he Simple Cubic Laice J. Ph. C (8) :L3(1975). 8-. S. Hiawi, and J. M. Khalifeh. "emarks n he laice Green's Funcin, he General Glasser Case". J. The. Phs., 41, 1769 (00). 9-. S. Hiawii, J. H. Asad, A. Sakai, and J. M. Khalifeh. " Laice Green's Funcin fr he face Cenered cubic Laice". In. J. The. Phs., (43) 11: I.S. Gradshen and I. M. zhik, Tables f Inegrals, Series, and Prducs (Academic, New Yrk, 1965). 31- Baeman Manuscrip Prec, Higher Transcendenal Funcins, Vl. I,edied b A. Erdeli e al.(mcgraw-hill, New Yrk, 1963). 3- P. M. Mrse and H. Feshbach, Mehds f Thereical Phsics (McGraw-Hill, New Yrk, 1953). 8

9 Figure Capins Fig. 3.1 The resisance beween i (0,0) and,0) f he perfec square laice as a funcin f x ; calculaed (squares) and measured (circles) alng he [10] direcin. ( x Fig. 3. The resisance beween i (0,0) and x, ) ( f he perfec square laice as a funcin f x and ; calculaed (squares) and measured (circles) alng he [11] direcin. Fig. 3.3 The resisance beween i (0,0) and,0) funcin f ( x f he perurbed square laice as a x ; calculaed (squares) and measured (circles) alng he [10] direcin. The ends f he remved bnd are (0,0) i and (1,0). Fig. 3.4 The resisance beween i (0,0) and 0, ) ( f he perurbed square laice as a funcin f ; calculaed (squares) and measured (circles) alng he [01] direcin. The ends f he remved bnd are (0,0) i and (1,0). Fig. 3.5 The resisance beween i (0,0) and,0) funcin f ( x f he perurbed square laice as a x ; calculaed (squares) and measured (circles) alng he [10] direcin. The ends f he remved bnd are (1,0) i and (,0). Fig. 3.6 The resisance beween i (0,0) and 0, ) ( f he perurbed square laice as a funcin f ; calculaed (squares) and measured (circles) alng he [01] direcin. The ends f he remved bnd are (1,0) i and (,0). 9

10 (l,m)/ The Sie Fig. 3.1 (l,m)/ The Sie Fig

11 (l,m)/ The Sie Fig (l,m)/ The Sie Fig

12 (l,m)/ The Sie Fig (l,m)/ The Sie Fig

13 APPENDIX A The LGF fr w-dimensinal laice is defined b 11 G(m, n, ) Csmx Csn - (Csx Cs) dx d, (A1) Where (m, n) are inegers and is a parameer. B execuing a parial inegrain wih respec x in Eq. (A1), we bained he fllwing recurrence relain 0 : G ( m 1, n) G( m 1, n) mg( m, n), (A) where G / (m,n) expresses he firs derivaive f G(m,n) wih respec. Taking derivaives f Eq. (A) wih respec, we bained recurrence relains invlving higher derivaives f he GF. Puing (m,n)=(1,0), (1,1), and (,0) in Eq. (A), respecivel we bained he fllwing relains: G (,0) G(0,0) G(1,0), (A3) G (,1) G(1,0) G(1,1), (A4) G ( 3,0) G(1,0) 4G(,0). (A5) Fr m=0 we bain 10,1- G(0, n) 0 G(1, n) G(0, n 1) G(0, n 1) 0 (A6) n Inser n=0 in Eq. (A6) we find he well-knwn relain 1 ( 1,0) [ G(0,0) 1], G (A7) 13

14 fr m 0 we have G ( m 1, n) G( m, n) G( m 1, n) G( m, n 1) G( m, n 1) 0, (A8) Subsiuing (m,n)=(1,0), (1,1), and (,0) in Eq. (A8), respecivel we bained he fllwing relains: G( 1,1) 1 1 G(1,0) G(0,0) G(,0), G(,1) ( 1) G(1,0) G(0,0) G(,0), G(3,0) ( ) G(0,0) 3G(,0) ( ), (A9) (A10) (A11) Nw, b aking he derivaive f bh sides f Eq. (A11) wih respec, and using Eqs. (A3), (A4) and (A5), we bained he fllwing expressins: 3 G(,0) (4 ) G(0,0) G(0,0), (A1) G(1,1) ( 1) G(0,0) (4 ) G(0,0), (A13) 3 1 G (,1) ( 3) G(0,0) (4 ) G(0,0), (A14) 1 4 G(3,0) (9 ) G(0,0) 3 (4 ) G(0,0) ( ), (A15) Again, aking he derivaive f bh side f Eq. (A1) wih respec, and using Eqs. (A3) and (A7), we bained he fllwing differenial equain fr G(0,0): (4 ) G (0,0) (4 3 ) G(0,0) G(0,0) 0. (A16) Where G // (0,0) is he secnd derivaive f G(0,0). B using he fllwing ransfrmains G(0,0) = Y(x)/ and x=4/ we 3 5 bain he fllwing differenial equain : 14

15 d Y ( x) dy ( x) 1 1 x) (1 x) Y ( x) 0, dx dx 4 x (A17) ( This is called he hpergemeric differenial equain (Gauss s differenial equain). S, he sluin is 5 Y(x)= 1 F (1/,1/;1;x) = (/π) K(/) Then, G(0,0, ) K( ) (A18) B using Eq. (A18) we can express G / (0,0) and G // (0,0) in erms f he cmplee ellipic inegrals f he firs and secnd kind. E( ) G( 0,0, ), 4 G (0,0, ) [ E( )[3 ( 4) 4] K( )]. (A19) (A0) K(/) and E(/) are he cmplee ellipic inegrals f he firs and secnd kind, respecivel. S ha, he w-dimensinal LGF a an arbirar sie is bained in clsed frm, which cnains a sum f he cmplee ellipic inegrals f he firs and secnd kind. 15