Chapter 23: Electrostatic Energy and Capacitance

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hpte 3: Electosttic Enegy nd pcitnce pcito pcitos nd pcitnce ny two conductos septed y n insulto (o vcuum) fom cpcito In pctice ech conducto initilly hs zeo net chge nd electons e tnsfeed fom one

1 hpte 3: Electosttic Enegy nd pcitnce pcito pcitos nd pcitnce ny two conductos septed y n insulto (o vcuum) fom cpcito In pctice ech conducto initilly hs zeo net chge nd electons e tnsfeed fom one conducto to the othe (chging the conducto) Then two conductos hve chge with equl mgnitude nd opposite sign, lthough the net chge is still zeo When cpcito hs o stoes chge, the conducto with the highe potentil hs chge + nd the othe - if >

2 pcitnce pcitos nd pcitnce One wy to chge cpcito is to connect these conductos to opposite teminls of ttey, which gives fixed potentil diffeence V etween conductos ( -side fo positive chge nd -side fo negtive chge). Then once the chge nd e estlished, the ttey is disconnected. If the mgnitude of the chge is douled, the electic field ecomes twice stonge nd V is twice lge. Then the tio /V is still constnt nd it is clled the cpcitnce. - V units F fd /V coulom/volt When cpcito hs o stoes chge, the conducto with the highe potentil hs chge + nd the othe - if >

3 lculting pcitnce Pllel-plte cpcito in vcuum hge density: Electic field: Potentil diff.: V V pcitnce: V σ E E dl E d σ ε ε V Ed ε V ε d The cpcitnce depends only on the geomety of the cpcito. It is popotionl to the e. It is invesely popotionl to the seption d When mtte is pesent etween the pltes, its popeties ffect the cpcitnce. l Ed d

4 Units lculting pcitnce F /N m (Note [ε ] /N m ) µf -6 F, pf - F ε 8.85 x - F/m Exmple 4.: Size of -F cpcito d mm,. F d 3 (. F)(. m) F/m ε m

5 lculting pcitnce Exmple 4.: Popeties of pllel cpcito pllel- plte cpcito in vcuum d 5. mm,. m, V, V. kv E ε d 3.54 V (8.85 F/m)(. m 3 5. m σ 5 F.354 µ F ( µ ε ε (8.85. N/ ) /V)(. 4 V) / N m )(. m )

6 lculting pcitnce Exmple 4.3: spheicl cpcito Fom Guss s lw: E d V V t evey point on sphee s encl ε E is constnt in mgnitude nd pllel to d E(4 ε π ) E Gussin sufce 4πε This fom is the sme s tht fo point chge 4πε V V 4πε 4πε πε V 4πε 4

7 lculting pcitnce Exmple 4.4: cylindicl cpcito (length L) - Oute metl id V λ πε V ln λl λ ln πε fom Exmple 3. πε L ln Signl wie line chge density λ

8 pcitos in Seies nd Pllel pcitos in seies

9 pcitos in Seies nd Pllel pcitos in seies (cont d) V V + c + V c V V c V V c V Vc V V V V V + + V + The equivlent cpcitnce eq of the seies comintion is defined s the cpcitnce of single cpcito fo which the chge is the sme s fo the comintion, when the potentil diffeence V is the sme. eq V eq V eq + eq i i

10 pcitos in Seies nd Pllel pcitos in pllel V V V V ( + + ) V V + The pllel comintion is equivlent to single cpcito with the sme totl chge + nd potentil diffeence. + eq eq i i

11 pcitos in Seies nd Pllel pcito netwoks

12 pcitos in Seies nd Pllel pcito netwoks (cont d)

13 B pcitos in Seies nd Pllel pcito netwoks B 3 B 4 3 B 5 4

14 Enegy Stoge nd Electic-field Enegy Wok done to chge cpcito onside pocess to chge cpcito up to with the finl potentil diffeence V. V Let q nd v e the chge nd potentil diffeence t n intemedite stge duing the chging pocess. υ q t this stge the wok dw equied to tnsfe n dditionl element of chge dq is: dw υdq qdq The totl wok needed to incese the cpcito chge q fom zeo to is: W W dw qdq

15 Enegy Stoge nd Electic-field Enegy Potentil enegy of chged cpcito Define the potentil enegy of n unchged cpcito to e zeo. Then W in the pevious slide is equl to the potentil enegy U of the chged cpcito U V V The totl wok W equied to chge the cpcito is equl to the totl chge multiplied y the vege potentil diffeence (/)V duing the chging pocess

16 Enegy Stoge nd Electic-field Enegy Electic-field enegy We cn think of the ove enegy stoed in the field in the egion etween the pltes. Define the enegy density u to e the enegy pe unit volume u V d field volume ε d ε E This eltion woks fo ny electic field

17 Enegy Stoge nd Electic-field Enegy Exmple 4.9: Two wys to clculte enegy stoed onside the spheicl cpcito in Exmple 4.3. The enegy stoed in this cpcito is: 4πε U 8 πε 4 E πε The electic field etween two conducting sphee: The electic field inside the inne sphee is zeo. The electic field outside the inne sufce of the oute sphee is zeo E u ε π πε ε ε d d udv U πε πε π ε π

18 Enegy Stoge nd Electic-field Enegy Exmple : Stoed enegy R dv U E u E R ) ( πε πε ε ε πε

19 Dielectic mteils Dielectics Expeimentlly it is found tht when non-conducting mteil (dielectics) etween the conducting pltes of cpcito, the cpcitnce inceses fo the sme stoed chge. Define the dielectic constnt κ ( K in the textook) s: κ When the chge is constnt, V E V E κ κ Mteil κ Mteil κ Mic 3-6 Myl 3. vcuum i( tm).59 Teflon. Polyethelene.5 V V V / V Plexigls 3.4 Wte 8.4 /

Dielectics Induced chge nd poliztion onside two oppositely chged pllel pltes with vcuum etween the pltes. Now inset dielectic mteil of dielectic constnt κ.

20 Dielectics Induced chge nd poliztion onside two oppositely chged pllel pltes with vcuum etween the pltes. Now inset dielectic mteil of dielectic constnt κ. E E / κ when is constnt Souce of chnge in the electic field is edistiution of positive nd negtive chge within the dielectic mteil (net chge ). This edistiution is clled poliztion nd it poduces induced chge nd field tht ptilly cncels the oiginl electic field. σ σ ind E E E ε ε E κ σ σ ind σ nd define the pemittivity ε κε κ E σ ε d d κ κε ε u κεe εe

21 Dielectics Molecul model of induced chge

22 Dielectics Molecul model of induced chge (cont d)

23 Dielectics Why slt dissolves Nomlly Nl is in igid cystl stuctue, mintined y the electosttic ttction etween the N + nd l - ions. Wte hs vey high dielectic constnt (78). This educes the field etween the toms, hence thei ttction to ech othe. The cystl lttice comes pt nd dissolves.

24 conducto Dielectics Guss s lw in dielectics σ σ ind dielectic Guss s lw: E ( σ σ ε ind ) σ σ σ ind o σ σ ind κ κ σ σ E o κe κε ε κe d encl fee ε enclosed fee chge

25 Execises Polem n i cpcito is mde y using two flt pltes ech with e septed y distnce d. () If the distnce d is hlved, how much does the cpcitnce chnges? () If the e is douled, how much does the cpcitnce chnges? (c) Fo given stoed chge, to doule the mount of enegy stoed how much should the distnce d e chnged? Now metl sl of thickness (< d) nd of the sme e is inseted etween the two pltes in pllel to the pltes s shown in the figue (the sl does not touch the pltes). (d) Wht is the cpcitnce of this ngement?(hint:seil connection) (e) Expess the cpcitnce s multiple of the cpcitnce when the metl sl is not pesent. d

26 Polem Solution () () (c) (d) (e) ε d d, so is douled. ε, so is douled. ε ndu, sou d connected in seies, ech of d nd d should e douled. ε This ngement cn e consideed to e system of two cpcitos which hs gp of ( d ) / etween the pltes. Ech of these two cpcito hs the cpcitnce ε.theefoe d the equivlent cpcitnce eq is :/ eq / eq ε d d ε, theefoe eq d d

27 Polem In this polem you ty to mesue dielectic constnt of mteil. Fist pllel-plte cpcito with only i etween the pltes is chged y connecting it to ttey. The cpcito is then disconnected fom the ttey without ny of the chge leving the pltes. () Expess the cpcitnce in tems of the potentil diffeence V etween the pltes nd the chge if i is etween the pltes. () Expess the dielectic constnt κ in tems of the cpcitnce (i gp) nd the cpcitnce with mteil of the dielectic constnt κ). (c) Using the esults of () nd (), expess the tio of the potentil diffeence V/V if is the sme, whee V is the potentil diffeence etween the pltes nd dielectic mteil dielectic constnt is κ fills the spce etween them. (d) voltmete eds 45. V when plced coss the cpcito. When dielectic mteil is inseted completely filling the spce, the voltmete eds.5 V. Find the dielectic constnt of this mteil. (e) Wht is the voltmete ed if the dielectic is now pulled ptwy out so tht it fills only one-thid of the spce etween the pltes? (Use the fomul fo the pllel connection of two cpcitos.)

28 () () (c) Polem /V κ / V / V / κ V / V / κ κ V / V 45. /.5 3. (d) Fom (c) 9 (e) In the new configution the equivlent cpcito is eq /3 +,/3 whee /3 is the contiution fom the pt tht hs the dielectic mteil nd,/3 is the pt tht hs i gp. /3 ( / 3) nd,/3 (/3) ecuse the cpcitnce is popotionl to the e. eq /3 +,/3 ( / 3) + ( / 3) [(/ 3) κ + Using the esults fom (c) V V / V V eq / [(/ 3) κ + ( / 3)] /[(/ 3) κ + ( / 3)] (45. V) V ( / 3)]

Exam in physics, El-grunder (Electromagnetism), 2014-03-26, kl 9.00-15.00

Exam in physics, El-grunder (Electromagnetism), 2014-03-26, kl 9.00-15.00 Umeå Univesitet, Fysik 1 Vitly Bychkov Em in physics, El-gunde (Electomgnetism, 14--6, kl 9.-15. Hjälpmedel: Students my use ny book(s. Mino notes in the books e lso llowed. Students my not use thei lectue