Energy cnsieratins 99.0.8 DRFT. Energy cnsieratins The wrk reuire t assemble tw charges, an is fun by first bringing frm infinity t its esire psitin (which reuires n wrk) an then bringing frm infinity against the electric fiel f. The assciate wrk is then Φ, (.) where Φ is the ptential ue t at the lcatin f. Of curse, the same tw charges cul have been assemble in the ppsite rer, giving Φ (.) Bth f the abve meths must f assembling the system must give the same result. Hence, t eliminate bias in the rering, we may write -- ( Φ + Φ ) (.3) Nte that, by the way this wrk has been efine, it must represent the wrk that must be supplie by an external agent t assemble the system. Thus, if the charges are f the same sign, an external agent must supply psitive wrk t assemble the system. If, n the ther han, the charges are f ppsite sign, then the external wrk is negative the charges want t cme tgether. The wrk reuire t assemble a system f N charges {,,, N } is the similar well-knwn result * N N -- i Φ ji i j (.4) where Φ ji is the ptential at the lcatin ue t the presence f the charge j, an we efine Φ ji 0 if i j. The ttal ptential at the i th lcatin ue t all f ther charges is i th Φ i N j Φ ji (.5) Hence, the wrk t assemble the system can be written mst cmpactly as N -- i Φ i i * See, fr example, Engineering Electrmagnetics, by U.S. Inan an.s. Inan, 999. (.6)
DRFT 99.0.8 Energy cnsieratins Fr a cntinuus istributin f charge, this becmes --, (.7) ρ Φv where ρ is the vlume ensity f charge an v is the vlume. Since E. (.5) was erive taking accunt f all charge, it appears that ρ shul represent all charge, bth free an bun. The part f the wrk assciate with the free charge is f -- ρ Φv f (.8) where ρ f is the free charge ensity. The part f the wrk assciate with the bun (plarizatin) charge is b -- ρ b Φv (.9) The istinctin between f an b becmes imprtant when analyzing a system t fin the usable wrk available frm that system. If a charge cnfiguratin is isassemble, nly the wrk assciate with the liberatin f the free charge can be recvere. The wrk assciate with the bun charge ges back int the atmic internal energy an is therefre nt available fr usable wrk. Expressing wrk in terms f fiels By Gauss s law, ρ ---- Ẽ (.0) Furthermre, fr electrstatics, the ptential is efine such that Ẽ Φ (.) Substituting this int E. (.7) gives -- ---- ( Ẽ)Φv (.) pplying the vectr ientity that ( Ẽ)Φ ( ΦẼ) ( Φ) Ẽ an then applying the ivergence therem gives
Energy cnsieratins 99.0.8 DRFT ñ ( ΦẼ) S + Ẽ Ẽv S (.3) here we have als use the fact that Φ Ẽ. The integratin vlume merely has t enclse the entire charge system. If we take the integratin vlume t be infinitely large, it can be shwn that the surface integral ges t zer. Thus, the wrk t assemble the cnfiguratin can be written w v, where w Ẽ Ẽ (.4) Fr the free charge, recall that ρ f D (.5) Substituting this int E. (.8) gives f w f v, where w f -- Ẽ (.6) D Fr the bun (plarizatin) charge, recall that ρ b P (.7) an therefre, b w b v, where w b Ẽ (.8) --P Nting that Ẽ is typically psitive, we bserve that is typically negative, meaning P b that material plarizatin typically reuces the amunt f external wrk that must be supplie by external means in rer t assemble a particular cnfiguratin f charge. Frces reuire t hl a cnfiguratin f charge Fr a cnfiguratin f N iscrete charges, the wrk reuire t mve the i th charge by a tiny vectr isplacement i is δ i i. Nte that it is the ttal energy f the system that etermines δx frce. Likewise, F δx fr the case f a cntinuus charge istributin, it is the the ttal wrk, nt just the usable wrk that etermines the frces hling the system tgether. f 3
DRFT 99.0.8 Energy cnsieratins EXMPLE: linear ielectric in a parallel plate capacitr 5 metal + + + + + + + + + + + + + + + + Φ 4 h 3 unifrmly plarize linear ielectric z metal Φ0 Figure.. Parallel plate capacitr with a ielectric an vi in between the metallic plates (f infinite lateral imensin). Prblem statement. Fr the iealize parallel plate capacitr in Fig.., we will assume that the system is in cmplete (electrical an mechanical) euilibrium. Befre the applicatin f the vltage, the gemetry may permissibly have been ifferent. In ther wrs, the values fr h an shwn in the figure are permitte t be ifferent frm their initial values h an. Hwever, fr the system t be in euilibrium, we assume that an external agent is applying sufficient surface an (if necessary) by frces t the system s that the cnfiguratin will remain statinary. The ielectric is assume t satisfy the electrical cnstitutive relatin D Ẽ + Π (.9) where is the permittivity an is the mechanical (innate) plarizatin, which shul be taken cnstant an Π in the z-irectin fr this prblem. The ielectric is assume t satisfy an istrpic linear-elastic mechanical cnstitutive euatin. Hence, fr this plane strain prblem, σ H, where σ entes the stress an entes the strain. Here, H is the elastic cnstraine mulus ( H K + 4G 3, where K is the bulk mulus an G is the shear mulus). If we use lgarithmic strain, ln( ). The actual strain measure selecte has little cnseuence in terms f its impact n the general cnclusins f the upcming analysis. 4
Energy cnsieratins 99.0.8 DRFT The metallic cnuctrs are cnnecte by a battery such that the bunary cnitins n the electric ptential are Φ( 0) 0 an Φ( h) (.0) e seek the electric ptential Φ( z), the electric fiel E, electric isplacement D, the plarizatin P, an the external frces reuire t prevent the system frm mving. Furthermre, we wish t test whether the electrical tractins t can be btaine frm the electrstatics jump euatin, t ñ [[ ] ], where ẼẼ (.) T T -- ( Ẽ Ẽ )Ĩ Gverning euatins. ithin cntinuusly varying regins, the gverning euatins are ρf, (.) D v ρb (.3) P v (efinitin f electric isplacement) (.4) D Ẽ + P Ẽ + (the electric cnstitutive law) (.5) D Π Ẽ Φ (.6) ρ v f where is the free charge per unit vlume an ρb v is the bun (plarizatin) charge per unit vlume. crss a iscntinuity surface having a unit nrmal, the gverning euatins are nˆ [[ ]] ρf, (.7) nˆ D s [[ ]] ρb, (.8) nˆ P s [[ Φ] ] 0 (.9) where ρ f s is the free surface charge, an, fr any uantity, ζ, the uble bracket peratin [[ ζ] ] euals the value f ζ n the sie f the jump surface int which pints, minus the value f ζ n the ther sie. nˆ Fr this prblem, all vectrs are in the z -irectin, s we may write these gverning euatins as D ρ f z v P ρb z v D E+ P D E + Π lcally everywhere (.30) lcally everywhere (.3) lcally everywhere (.3) lcally within the ielectric (.33) 5
DRFT σ H ln ----- Φ E z [[ D] ] ρf s [[ P] ] ρb s [[ Φ] ] 0 99.0.8 Energy cnsieratins lcally within the ielectric (.34) lcally within the ielectric (.35) lcally everywhere (.36) acrss jump surfaces. (.37) acrss jump surfaces. (.38) Fr this prblem, the tractin euatin (.) that we wish t test becmes (.39) ( ) T ( j ), where T () i ---- [ E (.40) () i ] t( ij) T j t( ij) Here, entes the electrical tractin exerte n the surface between regins (i) an (j). The sign f t( ij) is efine such that the tractin is psitive if it pints int regin (j). Slutin fr fiels. In the regin between the plates, there is n free charge. That is, ρ f v 0 an ρ f s 0 fr 0 < z< h (.4) Therefre, applying Es. (.30) an (.37) D cnstant fr 0< z< h (.4) Regins () an (5) are metallic cnuctrs. Cnseuently, D () P ( ) D ( 5) P( 5) 0 an E ( ) E ( 5) 0 (.43) T () i pplying the efinitin f frm E. (.40), we nte that T ( ) T ( 5) 0 (.44) Regins () an (4) are vi, s P ( ) P ( 4) 0 (.45) Nte that [[ P] ] 0 acrss the interface f regins () an (). Cnseuently, E. (.38) implies that there is n plarizatin charge ensity there. Likewise, there is n plarizatin charge ensity between regins (4) an (5). Thus the free surface charge must eual the ttal surface charge at the cnuctr-vi interfaces: 6
Energy cnsieratins 99.0.8 DRFT f f at z0. (.46) + at zh. (.47) The cnuctr charge is unknwn, but it will be relate t the applie vltage nce all f the fiels are knwn in terms f. Fr the lwer cnuctrvi interface, E. (.37) gives D () D ( ). Similarly, fr the upper cnuctr-vi interface, D ( 5) D( 4) +. Recalling E. (.4), D () D ( 3) D 4 ( ) (.48) In regin (), the plarizatin is zer an the permittivity is that f free space. Thus, slving E. (.3) fr the electric fiel gives E ( ) ρ s, (.49) T () i pplying efinitin f frm E. (.40), T ( ) (.50) Substituting E. (.48) int the cnstitutive law f E. (.5) gives ρ ( ) s + Π - E 3 (.5) Hence, T ( 3) ---- ( + Π) (.5) Regin (4) is nt plarize, s E ( 4) ρ s (.53) an T ( 4) (.54) The plarizatin is relate t D an E via E. (.3). Hence 7
DRFT P ( 3) D 3 99.0.8 Energy cnsieratins ( ) E 3 ( ) Substituting in the previusly erive values fr ( ) an E ( 3) gives D 3 (.55) ( ) ρ s ---- + Π ---- P 3 (.56) t this pint, we have the electric fiel everywhere, s we may integrate E. (.36) frm z0 t zh t relate the capacitr charge t the vltage: 0 Φ h 0 Ez (.57) r E ( ) h ----- E ( 3) ( ) E ( 4) h ----- + + ρ ---- s + Π ( h ) + - (.58) Slving fr the charge ensity gives ρ Π s ( h ) + (.59) Tractins. Nte that the plarizatin is piecewise cnstant an therefre has a zer graient everywhere. Cnseuently, an external by frce is nt reuire t maintain euilibrium. Let t ( ij) ente the tractin exerte by the surface charge between regins (i) an (j), with psitive tractin crrespning t a frce that pints in the psitive z -irectin. The surface charge at the - interface cannt exert a frce n itself. The part f the fiel that exerts a frce n the surface charge at the - interface is E( ) avg -- [ E ( ) + E ( ) ] (.60) The tractin (frce per area) that the surface charge at the - interface exerts n the system will be ente t( ) euals the charge times the average electric fiel: 8
Energy cnsieratins 99.0.8 DRFT ( )E( ) [ avg ( ) E( ) + E( ) ] --- t( ) (.6) Substituting Es. (.43) an (.59) gives t( ) ρ s T (.6) ( ) T ( ) This tractin is psitive, inicating that the charge n the bttm cnuctr tens t pull the cnuctr up in the psitive z -irectin. T prevent mtin f the capacitr plates, an external agent must apply a tractin f eual magnitue an ppsite irectin. Thus, t ( ) t external ( ) (.63) Similar analysis reveals that the tractins n the ther cnuctr are t ( 45 ) t ( ) an t( 45) external t( ) external (.64) The tractin exerte by the plarizatin charge n the interface between regins () an (3) must eual the ttal surface charge ensity (which, in this case, is the plarizatin charge ensity) times the average electric fiel. t the interface between vi an a plarize material, the plarizatin surface charge ensity euals the plarizatin vectr tte int the utwar unit nrmal t the ielectric. Thus, the plarizatin charge ensity euals P ( 3) at the interface between regins () an (3). This result cul have been btaine irectly by simply applying E. (.38). The average electric fiel at this interface euals E( 3) avg -- [ E ( ) + E ( 3) ] -- ρ s Π + ----- (.65) Multiplying this by the plarizatin charge ensity P ( 3) frm E. (.56) gives the tractin exerte n the plarizatin charge at the interface between regins () an (3): t( 3) ρ s ---- Π + ---- ρ s ---- + + Π ---- T ( 3 ) T ( ) (.66) Recall that the current thickness f the ielectric might nt be eual t its initial thickness in ther wrs, it might be uner strain. ccring t the mechanical cnstitutive Es. (.34) an (.35), this strain is achieve by the applicatin f a ttal stress given by: 9
DRFT σ H ln ----- 99.0.8 Energy cnsieratins Stress is efine t be psitive in tensin. Hence, n the bttm sie f the ielectric, the tractin is negative f the stress: (.67) t( 3) elastic H ln ----- (.68) This tractin is achieve, in part, by the electrical tractin t( 3). The remaining part must be supplie by external means. Hence, t prevent mtin f the system, t( 3) external t( 3) elastic t( 3) (.69) If the externally applie tractin is knwn, then this euatin may be slve (numerically) fr the euilibrium thickness f the ielectric slab,. Similar analysis reveals that the tractins n the ppsite sie f the ielectric are t( 34) t( 3) ( 34) t( 3) elastic ( 34) t( 3) external, t elastic an t external (.70) Nte that all f the interface tractins have nw been emnstrate t satisfy E. (.). Energy. The electrstatic energy assciate with assembing the free charge cnfiguratin is given by f -- D Ẽ (.7) The internal energy assciate with the bun plarizatin charge is b -- P Ẽ (.7) Fr this prblem z, s h f -- DEz --- ( D ( ) E ( ) ( h ) + D ( 3) E ( 3) ) 0 (.73) r 0
Energy cnsieratins 99.0.8 DRFT f h ---- Π + (.74) In the labratry, nly free charge can be easily manipulate, s f may be regare as the useful ptential energy assciate with the capacitr. In aitin t this energy, the bun charges in the plarize material have an internal energy given by b h -- PEz 0 ----- ( P ( ) E ( ) ( h ) + P ( 3) E ( 3) ) (.75) r b -- ρ s ---- + Π ---- + Π - ρ s -- ---- Π ---- + ---- Π ---- ---- + ---- (.76) The ttal energy is f + b ---- Ẽ Ẽ (.77) r, fr this ne-imensinal prblem, ----- ( h ) ---- Π + + ---- (.78) Energy erivatin f tractins. In respnse t an infinitesimal increase in plate separatin frm h t h+ δ, the ttal energy will change by an amunt -δ h (.79) which must be balance by the wrk (frce times istance, tδ ). Namely, -δ h tδ (.80) Slving fr t gives the result in E. (.6). Similar lgic gives E. (.66).
DRFT 99.0.8 Energy cnsieratins Slutin summary. In summary, the cmplete slutin is Regins an 5: CONDUCTORS D ) 0, P ) 0, E ) 0, T ( ) 0, Φ 0 (lwer) Φ (upper) (.8) Regin : LOER OID D (), P ( ) 0, E ( ) ρ, T ( ), (.8) s Φ ---- z Regin 3: DIELECTRIC D ( 3) (.83) ( ) ρ s ---- + Π ---- P 3 (.84) E ( 3) Π -----, T ( 3) (.85) ---- ( + Π) Φ ---- h ----- + Π - ( h ) + z --- Charge-vltage relatin: (.86) ρ Π s ( h ) +, (.87) Energy: f h ---- Π +, (.88) ----- ( h ) ---- Π + + ---- (.89) Tractins: t( ) (.90) t( 3) ρ s ---- Π + ---- ρ s ---- + + Π ---- (.9)
Energy cnsieratins 99.0.8 DRFT Limiting case #: n ielectric. Cnsier the limiting case where the ielectric is actually just free space. In ther wrs, cnsier an Π 0 (.9) Then the slutins reuce t Regins an 5: CONDUCTORS D ( ) 0, P ( ) 0, E ( ) 0, Φ 0 (lwer) Φ (upper) (.93) Regin : LOER OID D (), P ( ) 0, E ( ) ρ s, Φ ---- z (.94) Regin 3: DIELECTRIC D ( 3), P ( 3) 0, E ( 3) ρ s, Φ ---- z (.95) Charge-vltage relatin: ------ ---- h, (.96) Energy: f ρ s h --, (.97) h -- ---- (.98) Tractins: t( ) ρ s, t( 3) (.99) 0 3
DRFT 99.0.8 Energy cnsieratins Limiting case #, h with n mechanical plarizatin: Cnsier the limiting case where the entire space between the plates is taken up by the ielectric. Furthermre, cnsier n mechanical plarizatin. In ther wrs, cnsier h an Π 0 (.00) Then the slutin reuces t Regins an 5: CONDUCTORS D ( ) 0, P ( ) 0, E ( ) 0, Φ 0 (lwer) Φ (upper) (.0) Regin : N/ Regin 3: DIELECTRIC D ( 3) ( ) ρ s ---- P 3 E ( 3) ρ s Φ ---- z Charge-vltage relatin: (.0) (.03) (.04) (.05) --, (.06) Energy: f ρ s h --, (.07) ----- ρ s h -- ---- (.08) Tractins t( ). See E. (.). (.09) 4
Energy cnsieratins 99.0.8 DRFT ρ s ---- ρ s ---- + t( 3) ---- (.0) Nte: t ( ) applies t ne sie f regin () an t( 3) applies t the ther sie f regin (). In this limiting case, the thickness f regin () vanishes, s the tw tractins n regin () must be ae t give ρ s + ---- t( 3) t( ) + t ( 3 ) ρ s --- (.) e shul be able t btain the same result by irect applicatin f the jump euatins. Specifically, the tractin n the -3 interface must be t ( 3 ) ρ ( 3) E( 3) s avg (.) The surface charge at the -3 interface is given by the jump laws f E. (.37) an (.38). Bth free charge an plarizatin charge exist simultaneusly at the -3 interface, an bth types f charge cntribute t the tractin. Thus, the apprpriate charge ensity is [[ D] ] ρf s [[ P] ] ρb s acrss jump surfaces. (.3) acrss jump surfaces. (.4) ρ f s + ρb s [[ D] ] [[ P] ] [[ E] ] (.5) Thus, n the -3 interface, 3 ( ) [ E ( 3) E ( ) ] The average electric fiel is ------ (.6) E( 3) avg -- [ E ( ) + E ( 3) ] ρ s (.7) Thus, substituting these relatins int E. (.) gives t( 3) ρ s ---, (.8) which is in agreement with E. (.). The parts f this tractin ue t free an plarizatin charges are respectively 5
DRFT t( 3), free 99.0.8 Energy cnsieratins ρ s ρ ----- an t( 3) (.9) ----- s, plar ---- Nte that the free charge tens t pull the interface upwar in the psitive z-irectin while the plarizatin charge pulls wnwar. Cnseuently, the net electrical tractin is lwer than the free charge tractin by a factr f. 6