Capacitance Calculations

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1 Cpctce Clcultos Atoo Clos M. de Queoz Abstct Ths documet descbes clculto methods fo dstbuted cpctces of objects wth sevel ptcul shpes, d methods fo the evluto of the electc felds d foces. It s fudmetlly collecto of fomuls, some ot vey esy to fd the ltetue. The lgothms wee mplemeted the Ic pogm, vlble t I. INTRODUCTION Most of the fomuls below e ow sce log tme, most dtg fom wos the XIX cetuy. Some ppe Mxwell s boo [], d some othe collectos of explct fomuls fo electomgetc poblems, s [], o othe ely wos s [3]-[5]. I most cses I hve just dpted the otto, but some devtos ot foud othe wos e peseted too. I most of the ely wos, cpctce s expessed uts of legth. Fo exmple, the cpctce of sphee of dus fee spce s lsted [] d [] s C. To covet ths ut to Fds, t s ecessy to multply the vlue by 4ε, whee ε s the pemssvty of vcuum, ε x -. ε c be clculted fom the speed of lght c d fom the mgetc pemeblty of vcuum, µ 4 x -7 ( defto), fom the elto: c () µ ε The cpctce of sphee of dus metes s the: C two sphees 8ε L() pf (6) A oblte spheod s the fgue geeted by the otto of ellpse oud ts mo xs. A polte spheod s geeted by the otto of ellpse oud ts mjo xs. The cpctces of these fgues e, cosdeg the mjo xs wth legth d the mo xs wth the legth b []: C C oblte polte 4ε s b b (7) b 4ε (8) + b l b Note the lmts whe b educg to (), d the educto to (3) whe b (7). Fo bodes embedded mtels wth othe pemssvtes, t s just questo of multplyg ε by the eltve pemssvty ε of the mtel. The cse whe dffeet delectcs e peset o the stuctue wll be ot dscussed hee. II. CAPACITANC OF A TOROID D C sphee 4ε.65 pf () Othe fgues tht hve smple expessos fo the feespce cpctce e: A d A th flt ds wth dus []: C ds 8ε pf (3) A ope hemsphee wth dus []: C ope hemsphee 4ε (/+/) pf (4) A closed (wth flt ds) hemsphee wth dus []: 8ε / pf (5) C closed hemsphee ( ) Fg.. Tood wth mjo dmete D, mo dmete d, cete dus A, d tube dus. Fom [] (the sme fomul ppes [3], tht s pobbly the og of ths fomul, but somewht dffeet otto) the cpctce of tood wth mjo dmete D d mo dmete d, d<d/, (fg. ) s: Two sphees wth dus cotct []: Ceted: 5//3. Lst updte: 6//

2 Q C ε A 6 σ P σ fo, fo > D d A d x A ( ( (9) whee P ) d Q ) e Legede fuctos, o ( x ( x ths cse, toodl fuctos. These fuctos c be evluted the followg wy: The fst two tems c be obted fom the eltos wth the complete ellptc tegls of fst d secod ds: Q ( K K Q ( K P ( K' ' P ( K' The modulus fo the ellptc tegls K d s: A + () () Ad fo the ellptc tegls K d (evluted the sme wy, wth modulus ): ' () Ths s eough fo the evluto of the two fst tems of the sees (eough fo th toods). The othe tems c be obted usg the ecuso fo Legede fuctos, detcl fo both fuctos: mxpm ( ( m ) Pm ( Pm + ( m + (3) mxqm ( ( m ) Qm ( Qm+ ( m + whee m-. All the tems c the be esly computed, sttg wth the sees (9), o m. The complete ellptc tegls e the educble fuctos: K F(, / ) F (, / ) ( ) dϕ s ϕ ( ) s ϕ dϕ (4) They c be qucly d pecsely evluted usg the thmetc-geometc me method, below mplemeted Pscl oute: { Complete ellptc tegls of fst d secod clsses - AGM method. Retus the globl vbles: (c) d FF(c) ( d K) Does t eque moe th 7 tetos fo c betwee d Refeece: P d the AGM, J. Bowe d P. Bowe, Joh Wley & Sos. } pocedue F(c:el) v,b,,b,,:el beg : b:sqt(-sq(c)) :-sq(c)/ : epet :(+b)/ b:sqt(*b) :-*sq((-b)/) :* : b:b utl bs(-b)<e-5 F:p/(*) :*F ed III. APPROXIMAT CALCULATIONS FOR PARTIAL TOROIDS A ptl tood c be descbed s sufce geeted by the evoluto of ptl ccle of dus ceteed t dstce A log the dl xs fom the evoluto xs z. The ccle lmts e defed by two gles θ d θ. See fg.. z A θ θ Fg. : A ptl toodl sufce s geeted by the otto of ptl ccle oud the vetcl xs. Wth ths fomulto sevel fgues c be geeted, s egul tood whe θ -θ d <A, sphee whe A, θ θ /, ope hemsphee, etc. ve ovelppg toods, wth >A, c be geeted. Ths sufce c be decomposed set of ftely th ccles wth xles t the z xs, postoed t heghts z, d wth d, ufomly spced t gles θ log the sufce: θ θ θ θ θ θ + + ( ) θ, A + cosθ z s θ... (5) ch of these gs hs ufom chge dstbuto, wth totl chge q. The potetl Ψ due to ech g t y gve posto, z s gve by: Ceted: 5//3. Lst updte: 6//

3 3 q Ψ (, z) 4 ε R R ( + ) + ( z Q z ) / q R ε K (6) The bsolute vlues llow coect tetmet of the cses whe some d e egtve. Cosdeg the the mutul flueces mog ll the gs, mtx P c be computed, tht llows the clculto of the potetls v t ech g, oce the chges q e ow []: v Pq P P Ψ (, z ) / q j j P Ψ (, z + R) / q j j (7) Fo the clculto of the self-potetls P, somethg must be ssumed bout the dus of the gs, R. The fomulto clcultes the the potetls t dstce R bove the gs. The mxmum physclly possble vlue of R would be whe djcet gs touch: θ R mx s (8) Ay esoble fcto of ths vlue c be used wth sml esults, but thee s oe tht poduces bette esults the ext clculto, tht ws foud (by tyg!) to, cuously, be: R s θ (9) Ths dus mes the e of the sufce of the g to be detcl to the flt e epeseted by t, t lest the cses whe θ / d smll θ (s the equto d poles of sphee splt my gs). The chge dstbuto fo ufom potetl V t ll the gs c be clculted by vetg the mtx P. The totl chge ech g s the obted fom sum of the coespodg les of the vese of P, C. The coeffcets of C e the fluece coeffcets j : q V j j C P () Ad the cpctce of the whole ssembly s smply the sum of ll the elemets of C: Sufce electc feld C totl j j () The electc feld t y pot of the sufce s oml to t d c be clculted by Guss lw s popotol to the chge desty t tht pot of the sufce: ρ () ε whee ρ s the sufce chge desty, ufom oud the g. Fo closed sufce, the electc feld s etely t the oute sufce. I ths cse, t c be clculted dectly fom the chge dstbuto loe. Assumg costt voltge t the sufce of the object, the chges t the gs c be clculted by (). The g hs legth d totl chge q. The g epesets th belt wth wdth equl to θ. The chge desty d the electc feld smll legth l e the: q q ρ l l θ θ q θε (3) A mpott pplcto of ths clculto s the detemto of the beout voltge of the object, the voltge tht cuses ozto of the oud t whe the electc feld eches bout 3 MV/m: V mx 3/Mx V (4) Fo tood, ths vlue occus t the mxmum dmete. I the cse of ope objects, t s ot possble to clculte the sufce electc feld ths wy, becuse t s splt uow wy betwee the two sdes of the sufce. The clculto s lso megless f the object hs edge, whee the electc feld s delly fte. A stge poblem wth (3) s tht t fls whe the gs e close to the cete of sphecl sufce. The lst g ppes to hve sgfctly less chge th t should hve (oud 9%). The clcultos fo cpctce, howeve, cotue to esult good vlues. IV. GNRAL TRUNCATD CONS Ay othe fgue wth ccul symmety c be lyzed by the sme method. A smple cse s the evoluto of stght le oud the cetl xle, tht geetes fgues gg fom flt ds wth possble cetl hole to coe o ope cylde. h z Fg. 3. A le tht ottes oud the vetcl xs. The coodtes o the gs e the: Ceted: 5//3. Lst updte: 6//

4 4 h z z z + ( ) z, ( ),... (5) The dus to use the clculto of the selfpotetls would be, stll usg the mxmum dvded by : R ( ) + h (6) Wth ths dus, the sufce chge desty d the sufce electc feld (fo closed object) c be clculted cosdeg tht the sufce e of the g s detcl to the belt e epeseted by t, wht s ppoxmtely vld lso fo the cse o cuves, usg R gve by (9): q ρ l l q 4 Rε q R 4 R V. LCTRIC FILD FROM A RING (7) The electc feld ywhee c be clculted by ddg the electc felds due to the gs. Fom (6), the dl d xl compoets of the electc feld c be clculted by dffeetto, esultg : dl dψ d ( ) K + q sg 3 ' K ε + R ( ' xl 3 εr ' ) (8) dψ q ( z z ) (9) dz dl xl totl + (3) whee the devtve of the ellptc tegl K elto to the modulus ws used (the devtve of s lsted below too fo efeece, but ws ot ecessy): dk ' K d ' ' d d K (3) VI. GNRAL CAS WITH AXIAL SYMMTRY The cpctce mtx d the potetl d electc feld oud sees of objects wth xl symmety decomposed th gs c the be esly clculted. The objects e decomposed sees of ptl toods cocl sheets, d othe shpes (s ellpses) d these pts e decomposed gs. To obt the cpctce mtx, t s just questo of ddg the tems of the totl cpctce mtx tht coespod to the gs tht belog to the objects, sted of ddg them ll to obt the cpctce of the ete object. The chges ll the gs c be obted fom the complete equto qcv, wth the ssged voltges the objects ged V coespodece wth the gs tht belog to the objects. The potetl ywhee oud the objects s obted by ddg (6) fo ll the gs, d the electc feld by ddg (8) d (9) d usg (3). The tems t the dgol of the cpctce mtx coespod to the cpctces of the objects to goud whe ll the othe objects e gouded too. The fluece coeffcets out of the dgol mesue the elto betwee the chge duced oe object d the voltge othe, whe ll the othe objects e gouded. Fom the cpctce mtx, model of the ccut usg lumped cpctos c be deved, by obsevg the equvlece: C C + C C C + C C C + C C (3) C, C,, C e dect cpctces betwee the elemets d the goud, d the othe elemets e the egtve of the flotg cpctces betwee the objects. The dect cpctce to goud fo the object s just the sum of the elemets the le, o colum, of C. VII. MAXIMUM LCTRIC FILD BTWN TWO SPHRS A good test fo these feld clcultos s the ow fomul fo the mxmum electc feld betwee two dffeet sphees [4]. The expesso comes dectly fom the method of mges developed by Lod Kelv []. Fo two sphees of d d b, <b, wth dstce betwee cetes c, t potetls v d v, the mxmum electc feld t the sufce of the smlle sphee (ssumed s beg whee the sufce of the smlle sphee tecepts the le betwee the cetes of the sphees) s gve by: mx 4 ξ ξα ξα v +α +α + 4 ( +ξ) ( +ξ) ( +ξα ) ( +ξα ) ( ξ) 3 ηα ηα ηα v η +αη +α η + bα b + α ξ η c c α oot < of 3 5 ( +ηα) ( + ηα ) ( +ηα ) ( α + b)( bα + ) c α (33) 5 + Ths fomul coveges slowly whe the sphees e t smll dstce, but the speed s cceptble. [4] Ceted: 5//3. Lst updte: 6//

5 5 develops bette expesso fo the cse of sphees t smll dstce too. The choce of α ppes to wo coectly too whe the othe oot s used. The fomul wos fo y choce of v d v, but lwys clcultes the electc feld t the pot of the sphee wth dus closest to the othe sphee, eve whe ths s ot the pot of mxmum electc feld (s whe v d v hve the sme sg). Note tht t tht pot the clculto usg gs, s fomulted, clcultes electc feld slghtly smlle th the coect vlue. VIII. CAPACITANCS OF TWO SPHRS Sml fomuls, due to Kchhoff, led to the cpctce mtx of two sphees [5]. Fo two sphees wth d d b d dstce betwee cetes c: ξ λ ξ αξ α ξ 8ε λ ξ + α ξ + α ξ η αη α η 8ε λ η + α η + α η 3 α α α 8ε λ α + α + α λ λ + cξ α α η b ξ ( c + + b)( c b)( c + b)( c + b) c (34) These fomuls lso covege slowly whe the sphees e t smll dstce. [5] shows bette fomuls fo smll dstces. [][] hve the exct soluto whe the sphees e touchg, eq. (49). The coeffcets of the cpctce mtx epeset the to betwee the duced chges d the voltges. d epeset cpctces to goud fom sphee wth the othe sphee gouded, d s the flotg cpctce betwee the sphees. The dffeetl cpctce betwee the sphees s obted by ssumg opposte chges ±q o them: v v C dff q + + (35) The cpctces to goud wth the othe sphee flotg c lso be clculted, by ssumg zeo chge the flotg sphee: C C (36) The cpctce of both sphees to goud s smply the sum of the fou coeffcets, becuse both e t the sme potetl d the totl chge ppes the to C Q/V. Ths cpctce c be used to fd the elto betwee totl chge d voltge pth-bll electoscope. Fo equl sphees t ves betwee (6) whe the sphees e touchg d two tmes () whe they e f pt: C p + + (37) IX. POTNTIAL AND LCTRIC FILD AROUND A TOROID The soluto of ths poblem c be tced to [3]. The fomul fo the potetl lso ppes []. The potetl oud solted tood fee spce, wth cetl dus A d tube dus, t dl dstce d xl dstce z fom the cete, s foud s: V (, β) ( cosh β cos α) σ P ( cosh β) Ψ α σ fo,fo > A x c A z + ( + c) β l z + ( c) z c s α sh β cos α cosh β sh β s α cz α t t cos α + z c Q P ( ( / cos α (38) The sufce electc feld c be foud by the dffeetto of (38). The mxmum occus whe cosh β x A/ (tood sufce) d α (mjo dmete). The esult, hted [3] but ot developed, s the sees: 3/ 4 V ( x cos α) cos α σ (39) d ( x ) P ( The del exct bedow voltge c the be obted s (4). Ths sees coveges somewht moe slowly th (9) but stll c cheve hgh pecso. The sees (38) my lose pecso due to eos the evluto of Q +/ ( by the ecuso (3). X. OTHR CAPACITANCS Othe cpctces, of shpes wthout ccul symmety, e ow fom umecl lyss. Clsscl cses e the cpctces of the sque d tgul flt pltes [6], the cube [7] d the tethedo [6], ll wth sde : C sque 4ε pf (4) C tgle 4ε pf (4) C cube 4ε pf (4) C tethedo 4ε pf (43) XI. FORCS BTWN RINGS The foce betwee th coxl flmets c be clculted by tegto of the Coulomb foce betwee them. Fo two gs wth d A d d septo b, cotg chges q d q, the foce s gve by the double tegl (44). The tem cosθ pojects the foce log the xs: Ceted: 5//3. Lst updte: 6//

6 6 F 4ε cos θ b / q dq dϕ q dq' dϕ' F A + qq 3 6 ε cos θ dq dq' + b A cos ( ϕ ϕ' ) bdϕdϕ' 3 ( A + + b A cos( ϕ ϕ' )) (44) Ths tegl c be exctly solved tems of the complete ellptc tegl of the thd d (45), whch s lso esy to evlute. The code below computes few tetos the thee complete ellptc tegls (), F(), d Π(,c): / d ( c s ϕ ) ϕ Π(, c) (45) s ϕ { Complete ellptc tegls of fst, secod, d thd ds - AGM Retus the globl vbles (), FF(), d IIcII(,c) Refeece: Gett, Joul of Appled Physcs, 34, 9, 963, p. 57 } pocedue FII(,c:el) v,b,d,e,f,,b,d,e,f,s,:el beg : b:sqt(-sq()) d:(-sq(c))/b e:sq(c)/(-sq(c)) f: :/ S:*sq(-b) epet :(+b)/ b:sqt(*b) :* S:S+*sq(-b) d:b/(4*)*(+d+/d) e:(d*e+f)/(+d) f:(e+f)/ : b:b d:d e:e f:f utl (bs(-b)<e-5) d (bs(d-)<e-5) F:p/(*) :F-F*(sq()+S)/ IIc:F*f+F ed The soluto fo the foce s: F q q b c 3 ε A Π ( A + ) + b (, c) (46) The fomul educes to the well ow cse of chged g d pot chge f oe g hs zeo dus. I ths cse c d Π(,)/: F q q b q q b (47) ε ( ) 3 / ( ) 3 / A + b 4ε A + b To clculte the foces coductos decomposed gs, the chges the gs e fst clculted usg qcv d the the foces ech g c be obted by (46), by ddg the vlues obted betwee gve g d ll the othes. Flly, the foces ll the gs belogg to ech coducto e dded. The clcultos fo gs belogg to sgle coducto c be omtted, becuse they dd to zeo (d ths s good test of the lgothm). The sme esult c be obted by computg the foces multplyg the totl xl electc feld see by g by ts chge, usg (9): F 3 ε A q q b ( ) ( A + ) + b XII. XAMPLS ( ) (48) Some toods lyzed by the methods bove. V mx ws obted fom (39) d (4), except fo the holeless tood, whee (3) d (4) wee used. All the cpctces ( ths d the othe exmples) pf: D x d C exct gs gs V mx (V).x x x x Ope hemsphees (D dmete): D C exct gs gs Flt dss (D dmete): D C exct gs gs Hollow cyldes (D dmete, h heght): D h gs gs Ceted: 5//3. Lst updte: 6//

7 7 Hollow coes (D dmete, h heght): D h gs gs I the lst two cses o explct fomuls wee foud the ltetue, lthough vey pobbly they e ow. The geel lgothm fo objects wth xl symmety ws mplemeted the Ic pogm d used to geete the ext exmples: A closed hemsphee c be geeted by the combto of ope hemsphee d flt ds (hlf of the gs fo ech elemet, D dmete): D C exct gs gs Two sphees cotct (D dmete, hlf of the gs fo ech sphee): D C exct gs gs Two dffeet sphees cotct. Some cses e lsted [][]. Fo sphees of d d b, whe b C 4 ε L 3, d whe 3b C 4 ε 3/ 4L 4., b xct 4 gs 4 gs., , , The lst cse, 4b, s moe complcted, C 4ε ( / L /L( + 5) / ). The geel cse c be computed by the fomul [] ( dffeet fom) d []: b b C 4ε ψ ψ γ (49) + b + b + b whee ψ( s the dgmm fucto ψ( Γ (/Γ(, devtve of the logthm of the gmm fucto, d γ s ule s costt, γ A tood wth the cetl hole closed by th ds. Note the smll dffeece to egul tood. A tood whee the closue of the cetl hole doubles the cpctce would hve spect to of bout.4. Hlf of the gs fo ech elemet, D mjo dmete, d dmete of the tube: D x d gs gs 4 gs.3x x x Mxmum electc feld betwee sphees wth opposte voltges. Hlf of the gs to ech sphee. Dmesos s (33). Felds V/m/V:, b, c xct 4 gs 4 gs.,., ,., ,.3, Cpctce mtx fo two sphees. Hlf of the gs to ech sphee. Dmesos s (33): (dus ), b, c xct 4 gs 4 gs.,., ,., ,.3, (dus b), b, c xct (pf) 4 gs 4 gs.,., ,., ,.3, , b, c xct (pf) 4 gs 4 gs.,., ,., ,.3, Foce betwee the two hlves of chged sphee. The exct vlue s smply F ε v /, depedet of the dus of the sphee. The ppoxmte clculto s lso depedet of t. Vlues fo V t the sphee. Rdus (m) xct (pn) 4 gs 4 gs Foce betwee two equl sphees t the sme potetl, of V. The exct vlues wee computed by the ppoxmtos [8] d [9]. The exct vlue fo sphees cotct s F 4ε v ( L / 4) / 6 []., b, c xct (pn) 4 gs 4 gs.,., ,.,.5 ~ ,.,. ~ Foce betwee the two hlves of hoed tood t V. Ths cse ppetly hs o ow exct soluto. The foce does t deped o the sze of the devce d s popotol to the sque of the potetl, s hppes ll cses of objects cotct. Foces pn D (m) 4 gs gs 4 gs Foce betwee two stced hoed toods cotct, t V. The exct soluto s uow, but ths cse the Ceted: 5//3. Lst updte: 6//

8 8 umecl lyss s expected to be pecse. Foces pn. D (m) 4 gs gs 4 gs The foce ceses s the spect to of the toods cese. It doubles fo.6874 tood. The poblem wth ths ppoch s tht s the umbe of gs ceses t becomes moe d moe dffcult to vet the mtx P wth pecso d esoble tme. The exmples show tht the method s ot vey pecse fo objects wth edges. The pecso c be ehced by ddg moe gs to the egos close to edges. Fo exmple, the exmple of foce betwee the two hlves of sphee, f the ± degees oud the equto of the sphee e modeled wth 3 gs, wth gs fo the emg sufce, the obted foce s of pn, wth 4 coect dgts. Fo the hoed tood, the sme dstbuto poduces pn, o 4ε.4583 N, possbly wth sml pecso. I the lst pge s tble of exct tood cpctces clculted by (9). Note tht t would be eough to hve sgle colum wth omlzed spect tos, sce fo fxed spect to the cpctce s dectly popotol to the mjo (o mo) dmete. Acowledgmets: Ths to Pul Ncholso fo dscussos d vefctos, d to Godfey Loude fo sevel ppes d the devto of (39). [8] A. Russel, The mutul ttctos d epulsos of two electfed sphecl coductos, Joul of the I, Vol. 48,, pp , 9. [9] A. Russel, The electosttc poblem of two coductg sphees, Joul of the I, Vol. 65, 365, pp , My 97. [] W. Thomsom (Lod Kelv), O the mutul ttcto d epulso betwee two electfed sphecl coductos, Phlosophcl Mgze, Vol. 5, 3, pp , 853. [] A. Russell, "The coeffcets of cpcty d the mutul ttctos o epulsos of two electfed sphecl coductos whe close togethe," Poceedgs of the Royl Socety of Lodo. Sees A, Vol. 8, No. 557, pp , July 99, [] A. Russell, The electosttc cpcty of two sphees whe touchg oe othe, Poc. Phys. Soc. Lodo 37 pp. 8-86, 94. CHANGS 9//: Coected eq. 9 d smll coectos the text. 6//: Coected eq //: Added secto IX. //: Added eq //: Added secto bout foces. 4//: Added fomul fo the cpctce of two dffeet sphees cotct. 5//: Added moe exmples of foces, smll coectos. Ths documet s ot publshed ppe. RFRNCS [] Jmes Cle Mxwell, A Tetse o lectcty d Mgetsm, Dove Publctos Ic, New Yo, 954 (ept fom the ogl fom 873). [] Cheste Sow, Fomuls fo Computg Cpctce d Iductce, Ntol Bueu of Stdds Ccul #544. [3] W. M. Hcs, O toodl fuctos, Poceedgs of the Royl Socety of Lodo,, 3, Mch 88. [4] Alexde Russel, The mxmum vlue of the electcl stess betwee two uequl sphecl electodes, Poceedgs of the Physcl Socety of Lodo, Novembe 9, pp. -9. [5] Alexde Russel, The cpcty coeffcets of sphecl electodes, Poceedgs of the Physcl Socety of Lodo, Jue 9, pp [6] H.J. Wtle, The cpctce of the egul tethedo d equltel tgle, Joul of lectosttcs, Volume 6, Issue, August 99, pp. 5. [7] Ch-O Hwg d Mchel Mscg, lectcl cpctce of the ut cube, J. Appl. Phys. 95, Ceted: 5//3. Lst updte: 6//

9 xct tood cpctces (dmetes metes, cpctces pf) Mo d Mjo d Ceted: 5//3. Lst updte: 6// 9