Chapter 19 8/30/2010 ( ) Let s review what we have learned in PHY College Physics I. Electric Potential Energy and the Electric Potential

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1 8/3/ Chapte 9 Electic Ptential Enegy and the Electic Ptential Gals Chapte 9 T undestand electical ptential enegy. T deine electicalptential. T study euiptential suaces. T study capacits and dielectics. T study enegy cnsevatin invlving electical ptential enegy Gavity Electicity 9. Ptential enegy Field: Fce: Mass M Chage (± M ˆ ˆ g G E k e m g E F g F e This is the easiest way t pictue ield The cncepts electic ptential enegy and electic ptential ae exactly the same as the cncepts gavitatinal ptential enegy and gavitatinal ptential. Let s eview what we have leaned in PHY53---Cllege Physics I. Wk and Gavitatinal Ptential Enegy Cnside blck mass m at initial height y i Wk dne by the gavitatinal ce W gav ( F cs θ s (mg cs θ s but : Thus : s yi y, cs θ, W mg y y gav ( i mgyi mgy Electic and gavitatinal ces ae cnsevative U g GPE U e EPE PE mgy This uantity is called ptential enegy: W gavity PE PE i ΔPE Imptant: wk is elated t the dieence in PE s! W ΔU mgh mgh g GPE GPE W EPE EPE ΔU F Δy E( y y e

2 8/3/ Wk and Ptential Enegy ssume a unim ield between the tw plates s the chage mves m t, wk is dne n it W Fd E x (x x i ΔPE - W - E x Δx Only a unim ield Ptential Enegy Wk dne by gavity mving m m t : ΔU G U U W G Nea the suace eath (assume U at y ( ( F g m g ŷ g In geneal, at an abitay pint (assume U, at G Mm ˆ Fg (F cnsevative ces nly U mg y + g U G Mm U + g U U : cnstant depending n eeence pint Only ptential enegy dieence ΔU has physical signiicance Enegy and Chage Mvement psitive chage gains electical ptential enegy when it is mved in a diectin ppsite the electic ield negative chage lses electical ptential enegy when it mves in the diectin ppsite the electic ield When the electic ield is diected dwnwad, pint is at a lwe ptential than pint psitive test chage that mves m t lses electic ptential enegy It will gain the same amunt kinetic enegy as it lses in ptential enegy 9. The electic ptential dieence Deinitin electic ptential The electic ptential V at a given pint is the electic ptential enegy U e a small test chage at that pint divided by the chage itsel. U V e SI Unit electic ptential : jule/culmb vlt (V. Ptential Dieence V V U U W Example : Hw psitive and negative chages acceleate Thee pints,, and C ae lcated alng a hizntal line as shwn. psitive test chage is eleased at and acceleates twad C. What will a negative chage d when it is eleased m pint? (a cceleates twad C, (b Remains statinay, (c cceleates twad. Example 3: Opeating a Headlight The wattage the headlight in this dawing is 6 W. Detemine the numbe chage each caying.6x 9 C, that pass between the teminals the V ca battey when the headlight buns ne hu. Pwe enegy/time Ttal chage (numbe paticles x (chage each paticle S ttal enegy used is 6Wx36s.6x 5 J Diectin electic ield is m high ptential t lw ptential. U U U (V V.6x 5 J (.6x 5 Jule /( Vlt.8x 4 C (.8x 4 C / (.6x 9 C.x 3

3 8/3/ Nw we can include electic ptential enegy Ue as pat the ttal enegy that an bject can have: E mv + Iω + mgh + kx + U e Example 4 The Cnsevatin Enegy paticle has a mass.8x 5 kg and a chage +3.x 5 C. It is eleased m pint and acceleates hizntally until it eaches pint. The nly ce acting n the paticle is the electic ce, and the electic ptential at is 5V geate than at. (a What is the speed the paticle at pint? (b I the same paticle had a negative chage and wee eleased m pint, what wuld be its speed at? Electn vlt (ev anthe unit enegy. The amunt enegy that an electn changes when it mves thugh a ptential dieence ne vlt is called electn vlt (ev 9 ev (.6 C (V 9.6 J ssume a pint paticle N tatinal kinetic enegy Igne gavitatinal ptential enegy and elastic ptential enegy, We have nly kinetic enegy and electic ptential enegy. (a v mv ( V V ( V V m mv + U mv + U 5 5 ( C( 5 V (.8 kg 9.m s 3. mv mv + U U (b v ( V V m mv mv + ( V V 5 5 ( 3. C( 5 V (.8 kg 9.m s Check yu undestanding ( n in, stating m est, acceleates m pint t pint due t a ptential dieence between the tw pints. Des the electic ptential enegy the in at pint depend n (a the magnitude its chage and (b its mass? Des the speed the in at depend n (c the magnitude its chage and (d its mass? 9.3 Electic Ptential Pint Chages psitive pint chage + at the igin will ceate an electic ptential as the igue belw indicated. The wk dne n a test chage when it mves in this electic ptential caused by the ixed pint chage + is given by euatin belw. W k k ( ptn and an electn ae eleased m est at the midpint between the plates a chaged paallel plate capacit. Except these paticles, nthing else is between the plates. Igne the attactin between the ptn and the electn, and decide which paticle stikes a capacit plate ist. (a Electn will stike the plate ist. (b The ptn will stike the plate ist. (c th paticles will stike at the same time. W V V k k 3

4 8/3/ Ptential Enegy Pint Chages Fm pevius page we can see that the electic ptential a pint chage can be deined as, V k (9.6 V at. Yu may ecall that electic ield due t a pint chage is given by, E k (8.3 Example 5 The ptential a pint chage Using a ze eeence ptential at ininity, detemine the ptential due t a pint chage 4.x 8 C at a spt. m away m the chage when the chage is (a psitive and (b negative. (a 9 8 k (8.89 ( + 4. V. + 3 Vlts (b V 3 Vlts Cmpae the abve tw mula, we can tell that they ae elated. EXMPLE: What is the electic ptential enegy between tw ptns in the Uanium nucleus? The 9 ptns in the nucleus 38 U ae abut 6 m apat..6 x -9 C U 4πε 9 9 (8.99 Nm / C (.6 C 5 6. m J.4 ev 4keV 6 m,, m This is a huge enegy. The atmic binding enegy an electn is nly abut ev. Why desn t the Culmb ce split the nucleus apat? ecause the tw ptns ae held tgethe in the nucleus by an attactive stng nuclea ce. Ptential Dieence The ptential dieence between pints and is deined as the change in the ptential enegy (inal value minus initial value a chage mved m t divided by the size the chage V V ΔU / Ptential dieence is nt the same as ptential enegy nthe way t elate the enegy and the ptential dieence: Δ U th electic ptential enegy and ptential dieence ae scala uantities Ptential Enegy Cmpaed t Ptential Electic ptential is chaacteistic the ield nly Independent any test chage that may be placed in the ield Electic ptential enegy is chaacteistic the chage-ield system Due t an inteactin between the ield and the chage placed in the ield th, ptential and ptential enegy ae deined within a cnstant (can set cnstant t ze e.g. Ptential at distance ininiy is ze (athe than sme cnstant value Electic Ptential a Pint Chage The pint ze electic ptential is taken t be at an ininite distance m the chage. ptential exists at sme pint in space whethe nt thee is a test chage at that pint. The ptential ceated by a pint chage at any distance m the chage is V k The electic ield is pptinal t / The electic ptential is pptinal t / 4

5 8/3/ +5 μc -3 μc Tw chages + 5 μc, -3μC ae placed at eual distancesm the pint. What is the electic ptential at pint? a V < b V c V > V( whee 4πε Ptential m a pint chage V( The electic ield the pint chage has spheical symmety. The ptential depends nly n the distance m the pint chage, as is expected m spheical symmety. 9.4 Euiptential suaces and thei elatin t the electic ield 9.4 Euiptential Suaces and Thei Relatin t the Electic Field n euiptential suace is a suace n which the electic ptential is the same eveywhee. k V The net electic ce des n wk n a chage as it mves n an euiptential suace. The electic ield ceated by any chage gup chages is pependicula t the assciated euiptential suaces and pints in the diectin deceasing ptential. 9.4 Euiptential Suaces and Thei Relatin t the Electic Field This is a css-sectinal view the euiptential suaces (blue and the electic ield (ed an electic diple. Tpgaphic Maps Euiptential Cuves ll pints n an euiptential cuve ae at the same ptential. Each cuve epesented by V(x,y cnst 5

6 8/3/ Ptential dieence in a unim electic ield The elatin between electic ield and electic ptential can be undestd thugh the llwing example: a paallel plate capacit. V V W ut wk dne is F s, and F is E, s we have EΔs E Δs E Δ s Electic ptential due t many pint chages Just add up the cntibutin m each chage. This Pinciple Supepsitin applies t bth the electic ield and the electic ptential. ut it s much easie t apply a scala uantity such as the ptential. This gives a vey easy way t calculate the electic ield cmplicated chage distibutins: ist ind the ttal ptential V(, and then calculate m V. ne pint chage V ( k cllectin pint chages 3 V( k Supepsitin theem Chages CRETE Ptential Landscapes Electic Ptential Enegy Tw Chages V is the electic ptential due t at sme pint P The wk euied t bing m ininity t P withut acceleatin is V This wk is eual t the ptential enegy the tw paticle system U V k e I the chages have the same sign, U is psitive Psitive wk must be dne t ce the tw chages nea ne anthe The like chages wuld epel I the chages have ppsite signs, U is negative The ce wuld be attactive Wk must be dne t hld back the unlike chages m acceleating as they ae bught clse tgethe Ptential enegy chage in ptential V geneated by chage Ptential enegy chage 3 in ptential V + V geneated by chages and 6

7 8/3/ 9.5 Capacits and Dielectics Capacitance capacit cnsists tw cnducts. When the capacit is chaged, the cnducts cay chages eual magnitude and ppsite sign capacit is a device whse pupse is t ste electical enegy which can then be eleased in a cntlled manne duing a sht peid time. capacit cnsists spatially sepaated cnducts which can be chaged t + and - espectively. The capacitance C is deined as the ati the chage n ne cnduct the capacit t the ptential dieence between the cnducts. d + σ C σ + Δ V Chage and ptential dieence a capacit The magnitude the chage n each plate a capacit is diectly pptinal t the magnitude V the ptential dieence between the tw plates. CV Capacitance: Paallel plate capacit The capacitance a capacit is ixed, independent the chage and vltage. C Δ V Units: Faad (F F C / V Faad is vey lage Oten will see µf pf The capacitance a capacit is a measue hw much chage the capacit can ste pe unit vltage. The capacitance a device depends n the gemetic aangement the cnducts F a paallel-plate capacit whse plates ae sepaated by ai: C ε d Field ne chaged sheet σ E ε ε The capacit cnsists plates psitive and negative chage The ttal electic ield between the plates is given by σ E ε ε Paallel Plate Capacit: Field Lines V b V a ε d C E Δ The ield utside the plates is ze d ε capacitance depends nly n capacit s gemety (, d. 7

8 8/3/ Capacits in Cicuits cicuit is a cllectin bjects usually cntaining a suce electical enegy (such as a battey cnnected t elements that cnvet electical enegy t the ms + C V Capacits in Cicuits (Symbl a capacit cicuit diagam can be used t shw the path the eal cicuit piece metal in euilibium has a cnstant value ptential. Thus, the ptential a plate and attached wie is the same. The ptential dieence between the ends the wies is V, the same as the ptential dieence between the plates. What Des a Capacit D? What Des a Capacit D? Stes electical chage. Stes electical enegy. Capacits ae basic elements electical cicuits bth macscpic (as discete elements and micscpic (as pats integated cicuits. Capacits ae used when a sudden elease enegy is needed (such as in a phtgaphic lash. Electdes with capacit-like cniguatins ae used t cntl chaged paticle beams (ins, electns. Stes electical chage. Stes electical enegy. The chage is easy t see. I a cetain ptential, V, is applied t a capacit C, it must ste a chage CV: + C V (Symbl a capacit Dielectics n islated chaged capacit: bee and ate insetin a dielectic tmic Desciptin Dielectics In the pesence a dielectic, the electic E C ield, and thus the vltage between the E V plates, is educed by κ (> κ Vltmete Vltmete The chage n the plates emains unchanged, but the ptential dieence deceases Pla mlecules ae andmly iented in the absence an extenal electic ield When an extenal electic ield is applied, the mlecules patially align with the ield The chaged edges the dielectic can be mdeled as an additinal pai paallel plates establishing an electic ield E ind in the diectin ppsite t that E 8

9 8/3/ Eect n Capacitance dielectic educes the electic ield by a act κ E v E κ d Hence V E d is educed by κ [V (E /κ d V /κ] and C /V is inceased by κ [C ( κ / V C κ ε κ C paallel plate capacit with dielectic d dding a dielectic inceases the capacitance. E κ E Dielectic cnstant Enegy sage in a Capacit Hw much enegy is sted in a chaged capacit? Calculate the wk pvided (usually by a battey t chage a capacit t +/ : Calculate incemental wk ΔW needed t add chage Δ t capacit at vltage V (thee is a tick hee!: Δ W V av ( Δ The ttal wk W t chage t is then given by: V U Wttal In tems the vltage V: W CV + Enegy sted in a Capacit The ttal wk t chage t euals the enegy U sted in the capacit: V U C In tems the vltage V: Yu can d ne tw things t a capacit : hk it up t a battey, speciy V and llws put sme chage n it, speciy and V llws Lk at this! Tw ways t wite U U CV CV V C Whee is the Enegy Sted? Claim: enegy is sted in the electic ield itsel. Think the enegy needed t chage the capacit as being the enegy needed t ceate the ield. T calculate the enegy density in the ield, ist cnside the cnstant ield geneated by a paallel plate capacit, whee The electic ield is given by: U C ( ε / d σ E U E εd ε ε 9

10 8/3/ Enegy Density electic ield U E d The enegy density u in the ield is given by: ε U U u ε E vlume d Enegy sted in a capacit I the capacit cntains dielectic mateial, the enegy density inside is given by u κε E We can als deine the pemittivity a mateial as Claim: the expessin the enegy density the electstatic ield u ε E ε ε ε κε Pemittivity ee space Pemittivity a mateial Example 3 Cnsevatin Enegy and Mmentum Paticle has a mass m 3.6x 6 kg, while paticle has a mass m 6. x 6 kg. Each has the same electic chage. The paticles ae held at est initially and has an electic enegy.5 J. Suddenly, the paticles ae eleased and ly apat because the epulsive electic ce. t ne instance, the speed paticle is 7 m/s. What is the electic ptential enegy the tw paticle system? Fist use cnsevatin mmentum t ind the velcity paticle. m v + m v v v m v m v 98.7 m / s + m v + m v (3.6 / 6.( 7 Next, use cnsevatin enegy t ind the electic ptential enegy KE + U KE + U U U KE ( KE U.5J ( m 7 U.68J + m 98.7