Chapter 2: Electric Energy and Capacitance

Transcription

1 Chapter : Electrc Energy and Capactance Potental One goal of physcs s to dentfy basc forces n our world, such as the electrc force as studed n the prevous lectures. Expermentally, we dscovered that the electrc force s conservatve and thus has assocated electrc potental energy. Therefore, we can apply the prncple of the conservaton of mechancal energy for the case of the electrc force. Ths extremely powerful prncple allows us to solve problems for whch calculatons based on the force alone would be very dffcult.

2 Chapter : Electrc Energy and Capactance Potental.1. Electrc Potental and Electrc Potental Dfference.. Potental Dfference n a Unform Electrc Feld.3. Electrc Potental and Potental Energy Due to Pont Charges.4. Electrc Potental Due to Contnuous Charge Dstrbutons.5. Electrc Potental of a Charged Isolated Conductor.6. Capactance. Combnatons of Capactors.7. Energy Stored n a Charged Capactor.8. Capactors wth Delectrcs

3 .1. Electrc Potental and Electrc Potental Dfference:.1.1. Electrc Potental Energy: Key ssue: Newton s law (for the gravtatonal force) and Coulomb s law (for the electrostatc force) are mathematcally dentcal. Therefore, the general features for the gravtatonal force s applcable for the electrostatc force, e.g. the propertes of a conservatve force We can assgn an electrc potental energy U: U U U W W: work done by the electrostatc force on the partcles f The reference confguraton (U 0) of a system of charged partcles: all partcles are nfntely separated from each other W W appled Thus, the potental energy of the system: W : work done by the electrostatc force durng the move n from nfnty U W

4 Checkpont 1: A proton moves from pont to f n a unform electrc feld drected as shown: (a) does E do the postve or negatve work on the proton? (b) does the electrc potental energy of the proton ncrease or decrease? d F (a) Work done by the electrc feld E: W (b) F. d Ed Ed cosθ Ed < U W > 0 0 U ncreases

5 .1.. Electrc Potental and Electrc Potental Dfference: U depends on : U ~ U (unt : J/C) s the potental energy per unt charge and t s characterstc only of the electrc feld, called the electrc potental Electrc potental dfference between two ponts: f U f U U f W The potental dfference between two ponts s the negatve of the work done by the electrostatc force to move a unt charge from one pont to other

6 We set U 0 at nfnty: W We ntroduce a new unt for : 1 volt 1 joule per coulomb We adopt a new conventonal unt for E: N 1.C 1J 1 N/C 1 C 1J 1N.m We also defne one electron volt that s the energy eual to the work reured to move 1 e - through a potental dfference of exactly one volt: 1e J 1/m

7 Work done by an appled force: Suppose we move a partcle of charge from pont to f n an electrc feld by applyng a force to t: The work-knetc energy theorem gves: K K f K Wappled + W K + U W appled (or you can use: ) If K 0 (the partcle s statonary before and after the move): W appled W U U U f W appled W appled

8 .1.3. Eupotental Surfaces: Concept: Adjacent ponts wth the same electrc potental form an eupotental surface. No net work s done on a charged partcle by an electrc feld between two ponts on the same eupotental surface. Those surfaces are always perpendcular to electrc feld lnes (.e., to E ).

9 .1.4. Fndng the Potental from the Feld: Problem: Calculate the potental dfference between two ponts and f We have: dw F. ds dw E d s. W 0 0 f E. ds f W 0 f E. ds If we choose 0: f E. ds

10 Checkpont 3 (page 633): The fgure shows a set of parallel eupotental surfaces and 5 paths along whch we shall move an electron. (a) the drecton of E (b) for each path, the work we do postve, negatve or zero? (c) Rank the paths accordng to the work we do, greatest frst. (a) from the left to the rght (b) negatve: 4; postve: 1,,3,5 (c) 3,1--5,4 f W appled W 0 f E. ds E

11 .. Potental Dfference n a Unform Electrc Feld: Problem: Fnd the potental dfference between two ponts f - : f W 0 The test charge 0 moves along the path parallel to the feld lnes, so: f E. ds f Ed Electrc feld lnes always pont n the drecton of decreasng electrc potental Potental dfference between two ponts does not depend on the path connectng them (electrostatc force s a conservatve force) 0 d Check: move 0 followng cf: f E(cos 45 ) 0 sn 45 Ed

12 .3. Electrc Potental and Potental Energy Due to Pont Charges: f Key euaton: f E. ds.3.1. Potental Due to a Pont Charge: Choose the zero potental at nfnty Move 0 along a feld lne extendng radally from pont P to nfnty, so θ 0 0 P Edr R 1 P k dr r R k R

13 In a general case: k r A postvely charged partcle produces a postve electrc potental, a negatvely charged partcle produces a negatve electrc potental Potentals (r) at ponts n the xy plane due to a postve pont charge at O

14 .3.. Potental Due to A Group of Pont Charges: Usng the superposton prncple: n 1 k n 1 r (an algebrac sum, not a vector sum) Checkpont 4: The fgure shows three arrangements of two protons. Rank the arrangements accordng to the net electrc potental produced at pont P by the protons, greatest frst. Use of the formula above gves the same rank

15 .3.3. Potental Due to an Electrc Dpole: Problem: Calculate at pont P If r >> d: ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( r r r r k r r k ) ( ) ( ) ( ) ( cos r r r d r r + + and θ d p r p k ; cos θ

16 Induced Dpole Moment: + No external electrc feld: n some molecules, the centers of the postve and negatve charges concde, thus no dpole moment s set up + Presence of an external E: the feld dstorts the electron orbts and hence separates the centers of postve and negatve charge, the electrons tend to be shfted n a drecton opposte the feld. Ths shft sets up a dpole moment, the so-called nduced dpole moment. The atom or molecule s called to be polarzed by the feld. Charged electrons postve nucleus

17 .3.4. Calculatng the Feld from the Potental: Problem: 0 moves through a dsplacement ds from one eupotental surface to the adjacent surface: d ds W d E(cos ) ds : the gradent of the electrc potental n the E s Ecosθ s the component of drecton of : 0 0 θ E cosθ ds d ds ds E s drecton E n the s Partal Dervatve

18 In Cartesan coordnates: E : Nabla symbol gradent or : k z j y x ˆ ˆ ˆ + + z E y E x E z y x ; ;

19 Two Dmensonal Feld and Potental A unformly charged rod (x,y) Eupotental curves A dpole (x,y)

20 .3.5. Electrc Potental Energy of a System of Pont Charges: U U Problem: Calculate the electrc potental energy of a system of charges due to the electrc feld produced by those same charges Consder a smple case: two pont charges held a fxed dstance r f U W W appled We defne: The electrc potental energy of a system of fxed pont charges s eual to the work that must be done by an external agent to assemble the system, brngng each charge n from an nfnte dstance The work by an appled force to brng n from nfnty: W appled ( f ) W U k 1 appled system r

21 If the system conssts of three charges, we calculate U for each par of charges and sum the terms algebracally: ( 1 1 U U U13 + U3 k + + r r 1 13 r 3 3 ).4. Electrc Potental Due to Contnuous Charge Dstrbutons Method: We calculate the potental d at pont P due to a dfferental element d, then ntegrate over the entre charge dstrbuton d k d r d k d r

22 .4.1. Lne of Charge: d k d r L 0 d λdx k k λdx ( ) 1/ x + d λdx ( ) x + d 1/ (see Appendx E, ntegral 17, page A-11) kλ ln L + ( ) L + d d 1/

23 .4.. Charged Dsk: d σ ( πr')( dr') d d σ (πr')( dr') k k r z + R' R 1 σ (πr')( dr') 4πε 0 z 0 + R' σ z + R z ε 0

24 .5. Electrc Potental of a Charged Isolated Conductor: Usng Gauss law, we prove the followng concluson: An excess charge placed on an solated conductor wll dstrbute tself on the surface of that conductor so that all ponts of the conductor (on the surface or nsde) have the same potental. f Electrc feld at the surface s perpendcular to the surface and t s zero nsde the f Eds conductor, so: f

25 A large spark jumps to a car s body and then exsts by movng across the nsulatng left front tre, leavng the person nsde unharmed metal cage

26 Homework: 1, 6, 18, 4, 35, 43, 64 (page )

27 Overvew One goal of Physcs s to provde the basc scence for practcal devces desgned by engneers In ths part, we study such a practcal devce called a capactor n whch electrcal energy can be stored

28 .6. Capactance. Capactors n Parallel and n Seres:.6.1. Capactance: Some concepts: A capactor conssts of two solated conductors of any shape. These two conductors are called plates. A parallel-plate capactor has two parallel conductng plates of area A separated by a dstance d (symbol: ) When a capactor s charged, ts plates have eual but opposte charges of + and

29 All ponts on a plate have the same potental, s the potental dfference between the two plates For electrcal devces: symbol (not ) often represents a potental dfference When a capactor s charged: C C: Capactance of the capactor The capactance descrbes how much charge an arrangement of conductors can hold for a gven voltage appled and ts value depends on the geometry (delectrc, plate area, dstance) of the plates Unt: 1 farad 1 F 1 Coulomb/ol 1 C/ 1 µf 10-6 F; 1 pf 10-1 F

30 Chargng a capactor: Place the capactor n an electrc crcut wth a battery whch s a devce that mantans a certan potental dfference between ts termnals In schematc dagram (b) of a crcut: a battery B; a capactor C; a swtch S; nterconnectng wres When S s closed: the electrc feld drves electrons from capactor plate h to the postve termnal of B, plate h loosng e- becomes postvely charged; the feld drves electrons from negatve termnal of B to plate l, so t gans electrons and becomes negatvely charged between the plates s eual to that of battery B Checkpont 1: Does the capactance C of a capactor ncrease, decrease, or reman the same (a) when the charge on t s doubled and (b) when the potental dfference across t s trpled? (a) the same; (b) the same

31 .6.. Calculatng the Capactance: Goal: Calculate the capactance of a capactor once we know ts geometry step 1: Calculatng the electrc feld Use Gauss law for a Gaussan surface as ndcated: ε0 C EdA step : Calculatng the potental dfference f ε0 EA If the ntegraton path from the negatve to the postve plates + f Eds Eds

32 (a) A Parallel-Plate Capactor: Usng the formula above, we can use: nstead of ε 0 EA + d Eds E 0 C ε 0 d C ds A Ed 1 ε F/m pf/m ε C /(N.m )

33 (b) A Cylndrcal Capactor: A cylndrcal capactor of length L ncludes two coaxal cylnders of rad a and b We choose a Gaussan surface as ndcated: ε 0EA ε0e(πrl) E πε 0 rl ds dr : ntegrated radally nward + a dr b Eds ln πε 0 L r πε L a b 0 L C πε 0 ln( b / a)

34 (c) A Sphercal Capactor: A capactor conssts of two concentrc sphercal shells of rad a and b + ε 0EA ε0e(4πr E 1 4πε dr Eds πε 0 r 4πε a 0 r 4 b 0 1 a ) 1 b C 4πε 0 ab b a

35 (d) An Isolated Sphere: A capactor conssts only a sngle solated sphercal conductor of radus R, assumng that the mssng plate s a conductng sphere of nfnte radus C ab πε 0 4πε b a 4 0 C 4πε 0 R 1 a a b Checkpont : For capactors charged by the same battery, does the charge stored by the capactor ncrease, decrease, or reman the same n each of the followng stuatons? (a) The plate separaton of a parallel-plate capactor s ncreased. (b) The radus of the nner cylnder of a cylndrcal capactor s ncreased. (c) The radus of the outer sphercal shell of a sphercal capactor s ncreased. (a) decreases; (b) ncreases; (c) decreases

36 Capactance Summary Unt: 1 farad 1 F 1 Coulomb/ol 1 C/ 1 µf 10-6 F; 1 pf 10-1 F Parallel-Plate Capactor: Cylndrcal Capactor: Sphercal Capactor: C C C ε 0 πε d 0 4πε 0 b A L ln( b ab a / a) Isolated Sphere: C 4πε 0R 1 ε F/m pf/m

37 .6.3. Capactors n Parallel and n Seres: If there s a combnaton of capactors n a crcut, we can replace that wth an euvalent capactor, whch has the same capactance as the actual combnaton of capactors (a) Capactors n Parallel: A potental dfference s appled across each capactor. The total charge stored on the capactors s the sum of the charges stored on all capactors: ( C1 + C + C3 ) C e C1 + C + C3 If we have n capactors n parallel: n C e C 1

38 (b) Capactors n Seres: All capactors have dentcal charges The sum of the potental dfferences across all the capactors s eual to the appled potental dfference C C C C C C C e + + n C e C If we have n capactors n seres:

39 Checkpont: A battery of potental stores charge on a combnaton of two dentcal capactors. What are the potental dfference across and the charge on ether capactor f the capactors are (a) n parallel and (b) n seres? , / /,

40 .7. Energy Stored n a Charged Capactor: When we charge a capactor, work must be done by an external agent and the work s stored n the form of electrc potental energy U n the electrc feld between the plates At any nstant, the capactor has and /C, f an extra charge d s then transferred: dw ' d' ' d' C W 1 dw C 0 ' d' C The potental energy of the capactor: U U C 1 C

41 The medcal defbrllator: A medcal devce s used to stop the fbrllaton of heart attack vctms based on the ablty of storng potental energy of a capactor Example: a 70 µf capactor n a defbrllator s charged to 5000 gvng an amount of energy U: About 00 J s sent through the vctm n a pulse of ms, so the power s: 1 U C 875 (J) U P 100 (kw) t

42

43 Energy densty: Consder a parallel-plate capactor, the unform energy densty u, the potental energy per unt volume, between the plates: The electrc feld s unform between the plates (neglectng frngng): C ε 0 d A u U C Ad Ad 1 u ε0 d E s d u 1 ε 0 E

44 .8. Capactor wth a Delectrc: Some concepts: Delectrc: an nsulatng materal such as mneral ol or plastc If we fll the space between the plates of a capactor wth a delectrc, ts capactance ncreases by a factor κ (kappa), whch s called the delectrc constant (for a vacuum, κ 1) A delectrc has a maxmum potental dfference max that can be appled between the plates, called breakdown potental. Every delectrc has a characterstc delectrc strength E max correspondng to max C' κc ar C : capactance wth a delectrc C ar : ar between the plates

45 If s mantaned (e.g., by a battery), more charge flows to the capactor wth a delectrc If s mantaned, voltage drops n the capactor wth a delectrc C C ' κc ' κ C

46 In a regon completely flled by a delectrc materal, all electrostatc euatons contanng ε 0 are to be corrected by replacng ε 0 wth κε Delectrcs: An Atomc ew (a) What happens when we put a delectrc n an external electrc feld? + For polar delectrcs: the molecules already have electrc dpole moments (e.g., water) + For nonpolar delectrcs: n a feld, molecules acure dpole moments by nducton So the molecules tend to lne up wth the feld, Ther algnment produces an electrc feld n the opposte drecton to the external feld and thus reduces the appled feld

47 A nonpolar delectrc slab An electrc feld s appled The resultant feld E nsde the delectrc The presence of a delectrc wll therefore reduce the dfference potental for a gven charge ( constant) or wll ncrease the charge on the plates..8.. Delectrcs and Gauss Law: Goal: Generalze Gauss s law f delectrcs are present We consder a parallel-plate capactor, wthout a delectrc: ε 0 ε 0 0 E da E A

48 E 0 ε 0 A If a delectrc s present: ε 0 EdA ε0ea ' E ε A The effect of the delectrc s to reduce the appled feld E 0 : or DdA E 0 ' E0 κ ' κε A κ ε 0 0 E da κ ; D ε 0 κ E : electrc dsplacement

49 Checkpont: Two dentcal parallel-plate capactors are connected n seres to a battery as shown below. If a delectrc s nserted n the lower capactor, whch of the followng ncrease for that capactor? (a) Capactance of the capactor (b) Charge on the capactor (c) oltage across the capactor (d) Energy stored n the capactor C κc 0 C U (a) the capactance C ncreases by κ, C κc (b) more charge flows to C as C euvalent ncreases; ncreases 1 also ncreases, κ/(κ+1) 1 C (c) 1 ncreases, so 1 ncreases; 1 + constant, (d) before: after: 4κ so, U U C U C 1 < ( κ + 1) C constant

50 Homework:, 6, 11, 16, 6, 31, 33, 4, 48, 51 (page )