5/20/2011. HITT An electron moves from point i to point f, in the direction of a uniform electric field. During this displacement:

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1 5/0/011 Chapte 5 In the last lectue: CapacitanceII we calculated the capacitance C f a system f tw islated cnducts. We als calculated the capacitance f sme simple gemeties. In this chapte we will cve the fllwing tpics: Methds f cnnecting capacits (in seies, in paallel). Euivalent capacitance. Enegy sted in a capacit. Behavi f an insulat (a.k.a. dielectic) when placed in the electic field ceated in the space between the plates f a capacit. Gauss law in the pesence f dielectics. (5 1) HITT n electn mves fm pint i t pint f, in the diectin f a unifm electic field. Duing this displacement: i. the wk dne by the field is psitive and the ptential enegy f the electnfield system inceases B. the wk dne by the field is negative and the ptential enegy f the electnfield system inceases C. the wk dne by the field is psitive and the ptential enegy f the electnfield system deceases D. the wk dne by the field is negative and the ptential enegy f the electnfield system deceases E. the wk dne by the field is psitive and the ptential enegy f the electnfield system des nt change j Euivalent Capacit Cnside the cmbinatin f capacits shwn in the figue t the left and t the ight (uppe pat). We will substitute these cmbinatins f capacit with a single capacit Ce "electically euivalent" that is t the capacit gup it substitutes. This means that if we apply the same vltage acss the capacits in fig.a and fig.b (eithe ight left) by cnnecting t a battey, the same chage is pvided by the battey. ltenatively, if we place the same chage n plates f the capacits in fig.a and fig.b (eithe ight left), the vltage acss them is identical. This can be stated in the fllwing manne: If we place the capacit cmbinatin and the euivalent capacit in sepaate black bxes, by ding electical mesuements we cannt distinguish between the tw. (5 9) Capacits in paallel The fig.a we shw thee capacits cnnected in paallel. This means that the plate f each capacit is cnnected t the teminals f a battey f vltage. Ce = C1 C C3 We will substitute the paallel cmbinatin f fig.a with a single euivalent capacit shwn in fig.b which is als cnnected t an identical battey The thee capacits have the same ptential diffeence acss thei plates. The chage n C1 is: 1 = C1. The chage n C is: = C. The chage n C3 is: 3 = C3. The net chage = 1 3 = ( C1 C C3 ) ( C1 C C3 ) The euivalent capacitance Ce = = = C1 C C3 F a paallel cmbinatin f n capacits is given by the expessin: Ce = C1 C... Cn = C j n j= 1 (5 10) (5 11) Capacits in seies The fig.a we shw thee capacits cnnected in seies. This means that ne capacit is cnnected afte the the. The cmbinatin is cnnected t the teminals f a battey f vltage. We will substitute the seies cmbinatin f fig.a with a single euivalent capacit shwn in fig.b which is als cnnected t an identical battey. The thee capacits have the same chage n thei plates. The vltage acss C is: = / C The vltage acss C is: = / C. The vltage acss C is: = / C The net vltage acss the cmbinatin = Thus we have: = C1 C C3 The euivalent capacitance Ce = = = C 1 C C 3 C C C C e Me cmplex capacit systems In geneal a capacit system may cnsist f smalle capacit gups that can be identified as cnnected "in paallel" "in seies" In the example f the figue C and C in fig.a ae cnnected in paallel. They can be substituted by the euivalent capacit C = C C as 1 1 shwn in fig.b. Capacits C and C in fig.b ae cnnected in seies. They can be substituted by a single capacit C C 13 is given by the euatin: (5 1) = C C C as shwn in fig.c 1

2 5/0/011 C = ε / d (a) C/ (b)c (c) C aea is dubled (d) 0 (e) Need me infmatin

3 5/0/011 What is the euivalent capacitance between the pints and B? uestin. 1 µf B. µf C. 4 µf D. 10µF E. Nne f these B What wuld a 10 battey d, i.e. hw much chage will it pvide, when it is cnnected acss and B? 40µC ' ' Chage O d' ' ' ' B ltage Enegy sted an an electic field Cnside a capacit C which is has a chage. We can calculate the wk W euied t chage the capacit by assuming that we tansfe a chage d fm the negative plate t the psitive plate. We assume that the capacit chage is and the cespnding vltage. The wk dw euied f the chage tansfe is given by: dw = d = d C We cntinue this pcess till the capacit chage is 1 eual t. The ttal wk W = d = d C 1 C W = If we substitute C we get: C = = W = W = C 0 Wk W can als be calculated by detemining the aea f tiangle OB which is eual t d. ea = W = (5 13) 0 E d C U = = C Ptential enegy sted in a capacit The wk W spent t chage a capacit is sted in the fm f ptential enegy U = W that can be etieved when is capacit is dischaged. Thus Enegy density C U = = = C We can ask the uestin: whee is the ptential enegy f a chaged capacit sted? The answe is cunte intuitive. The enegy is sted in the space between the capacit plates whee a unifm electic field E = / d is geneated by the capacit chages. ε u = E In the wds the electic field can ste enegy in empty space! U We define as enegy densiy (symbl u ) the ptential enegy pe unit vlume. u = The vlume between the plates is: = d whee is the plate aea U C ε ε Thus the enegy density u = = = = = E d d d d This esult, deived f the paallel plate capacit hlds in geneal (5 14) 3

5/0/011 ' ' C = κc ai ' Capacit with a dielectic In 1837 Michael Faaday investigated what happens t the capacitance C f a capacit when the gap between the plates is cmpletely filled with an insulat

The fact κ is knwn as the dielectic cnstant f the mateial. Faaday's expeiment can be caied ut in tw ways: 1.

In this case the plates ae islated fm the battey This is shwn in fig.b (5 15) ' ' C = κc ai ' Fig.

κ (5 16) cnduct dielectic In a egin cmpletely filled with an insulat f dielectic cnstant κ, all electstatic euatins cntaining the cnstant ε ae t be mdified by eplacing ε with κε Example 1 : Electic

4 5/0/011 ' ' C = κc ai ' Capacit with a dielectic In 1837 Michael Faaday investigated what happens t the capacitance C f a capacit when the gap between the plates is cmpletely filled with an insulat (a.k.a. dielectic) Faaday discveed that the new capacitance is given by : C = κc ai Hee C ai is the capacitance befe the insetin f the dielectic between the plates. The fact κ is knwn as the dielectic cnstant f the mateial. Faaday's expeiment can be caied ut in tw ways: 1. With the vltage acss the plates emaining cnstant In this case a battey emains cnnected t the plates. This is shwn in fig.a. With the chage f the plates emaining cnstant. In this case the plates ae islated fm the battey This is shwn in fig.b (5 15) ' ' C = κc ai ' Fig.a : Capacit vltage emains cnstant This is bacause the battey emains cnnected t the plates fte the dielectic is inseted between the capacit plates the plate chage changes fm t = κ κ The new capacitance C = = = κ = κcai Fig.b : Capacit chage emains cnstant This is bacause the plates ae islated fte the dielectic is inseted between the capacit plates the plate vltage changes fm t = κ The new capacitance C = = = κ = κcai / κ (5 16) cnduct dielectic In a egin cmpletely filled with an insulat f dielectic cnstant κ, all electstatic euatins cntaining the cnstant ε ae t be mdified by eplacing ε with κε Example 1 : Electic field f a pint chage inside 1 a dielectic: E = 4πκε Example : The electic field utside an islated cnduct immesed in a dielectic becmes: σ E = κε (5 17) U = pe csθ (5 18) Dielectics : n atmic view Dielectics ae classified int "pla" and "nnpla" Pla dielectics cnsist f mlecules that have a nnze electic diple mment even at ze electic field due t the asymmetic distibutin f chage within the mlecule (e.g. H O ). t ze electic field (see fig.a) the electic diple mments ae andmly iented. When an extenal electic field E is applied (see fig.b) the electic diple mments tend t align pefeentially alng the diectin f E because this cnfiguatin cespnds t a minimum f the ptential enegy and thus is a psitin f stable euilibium. Themal andm mtin ppses the alignment and thus deing is incmplete. Even s, the patial alignment pduced by the extelal electic field geneates an intenal electic field that appses E. Thus the net electic field E is weake than E nnplal dielectic n the the hand cnsists f mlecules that in the absence f an electic field have ze electic diple mment (see fig.a). If we place the dielectic between the plates f a capacit the extenal electic field E induces an electic diple mment p that becmes aligned with E (see fig.b). The aligned mlecules d nt ceate any net chage inside the dielectic. net chage appeas at the left and ight sufaces f the dielectic ppsite t the capacit plates. These chages cme fm negative and psitive ends f the electic diples. These induced suface chages have sign ppsite t that f f the ppsing plate chages. Thus the induced chages ceate an electic field E which ppses the applied field E (see fig.c). s a esult is that the net electic field E between the capacit plates is weake. E E = (5 19) κ S S ˆn ˆn Gauss' law and dielectics In chapte 3 we fmulated Gauss' law assuming that the chages existed in vacuum. ε E d = ε Φ = In this sectin we will wite Gauss' law in a fm which is suitable f cases in which dielectics ae pesent. Cnside fist the paallel plate capacit shwn in fig.a. We will the Gaussian suface S. The flux Φ = E = ε E = Nw we fill the space between the plates ε with an insulat f dielectic cnstant κ (see fig.b). We will apply Gauss' law f the same suface S. Inside S in additin t the plate chage we als have the induced chage n the suface f the dielectic. Φ = E = ε E E = (es.1) Fm Faaday's expeiments we have: E = = (es.) ε κ κ ε If we cmpae es.1 with es. we have: = ε κ E d = κ (5 0) 4

kept inside the integal t descibe the mst geneal case in which κ is nt cnstant ve the Gaussian suface HITT paallelplate capacit has a plate aea f 0.m and a plate sepaatin f 0.1 mm.

5 5/0/011 Gauss' law in the pesence f dielectics ε κe d = Even thugh the euatin abve was deived f the paallel plate capacit is tue in geneal. Nte 1: The flux integal nw invlves κ E Nte : The chage that is used is the plate chage, als knwn as "fee chage" Using the euatin abve we can igne the induced chage Nte 3 : The dielectic cnstant κ is kept inside the integal t descibe the mst geneal case in which κ is nt cnstant ve the Gaussian suface HITT paallelplate capacit has a plate aea f 0.m and a plate sepaatin f 0.1 mm. T btain an electic field f.0 x 10 6 /m between the plates, the magnitude f the chage n each plate shuld be:. 8.9 x 10 7 C B. 1.8 x 10 6 C C. 3.5 x 10 6 C D. 7.1 x 10 6 C E. 1.4 x 10 5 C (5 1) Questin Questin paallelplate capacit has a plate aea f 0.3m and a plate sepaatin f 0.1mm. If the chage n each plate has a magnitude f 5x10 6 C then the fce exeted by ne plate n the the has a magnitude f abut:. 0 B. 5N C. 9N D. 1 x10 4 N E. 9 x 10 5 N paallelplate capacit has a plate aea f 0.3m and a plate sepaatin f 0.1mm. If the chage n each plate has a magnitude f 5x10 6 C then the fce exeted by ne plate n the the has a magnitude f abut:. 0 B. 5N C. 0. 9N D. 1 x10 4 N E. 9 x 10 5 N F = E = ε 6 ( 5 10 ) = 4.71N = The electic field = σ/ε why? 5

6 5/0/011 Summay Capacit netwks; Seies, paallel and the me cmplex cnfiguatins. Euivalent capacitance. Enegy sted n a capacit Dielectic between the capacit plates. Dielectic cnstant and dielectic stength Pla and nnpla mlecules 6

A) (0.46 î ) N B) (0.17 î ) N

A) (0.46 î ) N B) (0.17 î ) N Phys10 Secnd Maj-14 Ze Vesin Cdinat: xyz Thusday, Apil 3, 015 Page: 1 Q1. Thee chages, 1 = =.0 μc and Q = 4.0 μc, ae fixed in thei places as shwn in Figue 1. Find the net electstatic fce n Q due t 1 and.