Capacitance and Dielectrics

Transcription

1 2.2 This is the Nerest One He 803 P U Z Z L E R Mny electronic components crry wrning lel like this one. Wht is there insie these evices tht mkes them so ngerous? Why wouln t you e sfe if you unplugge the equipment efore opening the cse? (George Semple) Cpcitnce n Dielectrics c h p t e r Chpter Outline 26. Definition of Cpcitnce 26.2 Clculting Cpcitnce 26.3 Comintions of Cpcitors 26.4 Energy Store in Chrge Cpcitor 26.5 Cpcitors with Dielectrics 26.6 (Optionl) Electric Dipole in n Electric Fiel 26.7 (Optionl) An Atomic Description of Dielectrics 803

2 804 CHAPTER 26 Cpcitnce n Dielectrics In this chpter, we iscuss cpcitors evices tht store electric chrge. Cpcitors re commonly use in vriety of electric circuits. For instnce, they re use to tune the frequency of rio receivers, s filters in power supplies, to eliminte sprking in utomoile ignition systems, n s energy-storing evices in electronic flsh units. A cpcitor consists of two conuctors seprte y n insultor. We shll see tht the cpcitnce of given cpcitor epens on its geometry n on the mteril clle ielectric tht seprtes the conuctors DEFINITION OF CAPACITANCE Consier two conuctors crrying chrges of equl mgnitue ut of opposite sign, s shown in Figure 26.. Such comintion of two conuctors is clle cpcitor. The conuctors re clle pltes. A potentil ifference V exists etween the conuctors ue to the presence of the chrges. Becuse the unit of potentil ifference is the volt, potentil ifference is often clle voltge. We shll use this term to escrie the potentil ifference cross circuit element or etween two points in spce. Wht etermines how much chrge is on the pltes of cpcitor for given voltge? In other wors, wht is the cpcity of the evice for storing chrge t prticulr vlue of V? Experiments show tht the quntity of chrge Q on cpcitor is linerly proportionl to the potentil ifference etween the conuctors; tht is, Q V. The proportionlity constnt epens on the shpe n seprtion of the conuctors. 2 We cn write this reltionship s Q C V if we efine cpcitnce s follows: Definition of cpcitnce The cpcitnce C of cpcitor is the rtio of the mgnitue of the chrge on either conuctor to the mgnitue of the potentil ifference etween them: C Q V (26.) Q Q Note tht y efinition cpcitnce is lwys positive quntity. Furthermore, the potentil ifference V is lwys expresse in Eqution 26. s positive quntity. Becuse the potentil ifference increses linerly with the store chrge, the rtio Q /V is constnt for given cpcitor. Therefore, cpcitnce is mesure of cpcitor s ility to store chrge n electric potentil energy. From Eqution 26., we see tht cpcitnce hs SI units of couloms per volt. The SI unit of cpcitnce is the fr (F), which ws nme in honor of Michel Fry: F C/V The fr is very lrge unit of cpcitnce. In prctice, typicl evices hve cpcitnces rnging from microfrs (0 6 F) to picofrs (0 2 F). For prcticl purposes, cpcitors often re lele mf for microfrs n mmf for micromicrofrs or, equivlently, pf for picofrs. Figure 26. A cpcitor consists of two conuctors crrying chrges of equl mgnitue ut opposite sign. Although the totl chrge on the cpcitor is zero (ecuse there is s much excess positive chrge on one conuctor s there is excess negtive chrge on the other), it is common prctice to refer to the mgnitue of the chrge on either conuctor s the chrge on the cpcitor. 2 The proportionlity etween V n Q cn e prove from Coulom s lw or y experiment.

3 26.2 Clculting Cpcitnce 805 Q Q A collection of cpcitors use in vriety of pplictions. Let us consier cpcitor forme from pir of prllel pltes, s shown in Figure Ech plte is connecte to one terminl of ttery (not shown in Fig. 26.2), which cts s source of potentil ifference. If the cpcitor is initilly unchrge, the ttery estlishes n electric fiel in the connecting wires when the connections re me. Let us focus on the plte connecte to the negtive terminl of the ttery. The electric fiel pplies force on electrons in the wire just outsie this plte; this force cuses the electrons to move onto the plte. This movement continues until the plte, the wire, n the terminl re ll t the sme electric potentil. Once this equilirium point is ttine, potentil ifference no longer exists etween the terminl n the plte, n s result no electric fiel is present in the wire, n the movement of electrons stops. The plte now crries negtive chrge. A similr process occurs t the other cpcitor plte, with electrons moving from the plte to the wire, leving the plte positively chrge. In this finl configurtion, the potentil ifference cross the cpcitor pltes is the sme s tht etween the terminls of the ttery. Suppose tht we hve cpcitor rte t 4 pf. This rting mens tht the cpcitor cn store 4 pc of chrge for ech volt of potentil ifference etween the two conuctors. If 9-V ttery is connecte cross this cpcitor, one of the conuctors ens up with net chrge of 36 pc n the other ens up with net chrge of 36 pc. Figure 26.2 Are = A A prllel-plte cpcitor consists of two prllel conucting pltes, ech of re A, seprte y istnce. When the cpcitor is chrge, the pltes crry equl mounts of chrge. One plte crries positive chrge, n the other crries negtive chrge CALCULATING CAPACITANCE We cn clculte the cpcitnce of pir of oppositely chrge conuctors in the following mnner: We ssume chrge of mgnitue Q, n we clculte the potentil ifference using the techniques escrie in the preceing chpter. We then use the expression C Q /V to evlute the cpcitnce. As we might expect, we cn perform this clcultion reltively esily if the geometry of the cpcitor is simple. We cn clculte the cpcitnce of n isolte sphericl conuctor of rius R n chrge Q if we ssume tht the secon conuctor mking up the cpcitor is concentric hollow sphere of infinite rius. The electric potentil of the sphere of rius R is simply k e Q /R, n setting V 0 t infinity s usul, we hve C Q V Q k e Q /R R k e 40R (26.2) This expression shows tht the cpcitnce of n isolte chrge sphere is proportionl to its rius n is inepenent of oth the chrge on the sphere n the potentil ifference. QuickL Roll some socks into lls n stuff them into shoeox. Wht etermines how mny socks fit in the ox? Relte how hr you push on the socks to V for cpcitor. How oes the size of the ox influence its sock cpcity?

4 806 CHAPTER 26 Cpcitnce n Dielectrics The cpcitnce of pir of conuctors epens on the geometry of the conuctors. Let us illustrte this with three fmilir geometries, nmely, prllel pltes, concentric cyliners, n concentric spheres. In these exmples, we ssume tht the chrge conuctors re seprte y vcuum. The effect of ielectric mteril plce etween the conuctors is trete in Section Prllel-Plte Cpcitors Two prllel metllic pltes of equl re A re seprte y istnce, s shown in Figure One plte crries chrge Q, n the other crries chrge Q. Let us consier how the geometry of these conuctors influences the cpcity of the comintion to store chrge. Recll tht chrges of like sign repel one nother. As cpcitor is eing chrge y ttery, electrons flow into the negtive plte n out of the positive plte. If the cpcitor pltes re lrge, the ccumulte chrges re le to istriute themselves over sustntil re, n the mount of chrge tht cn e store on plte for given potentil ifference increses s the plte re is increse. Thus, we expect the cpcitnce to e proportionl to the plte re A. Now let us consier the region tht seprtes the pltes. If the ttery hs constnt potentil ifference etween its terminls, then the electric fiel etween the pltes must increse s is ecrese. Let us imgine tht we move the pltes closer together n consier the sitution efore ny chrges hve h chnce to move in response to this chnge. Becuse no chrges hve move, the electric fiel etween the pltes hs the sme vlue ut extens over shorter istnce. Thus, the mgnitue of the potentil ifference etween the pltes V E (Eq. 25.6) is now smller. The ifference etween this new cpcitor voltge n the terminl voltge of the ttery now exists s potentil ifference cross the wires connecting the ttery to the cpcitor. This potentil ifference results in n electric fiel in the wires tht rives more chrge onto the pltes, incresing the potentil ifference etween the pltes. When the potentil ifference etween the pltes gin mtches tht of the ttery, the potentil ifference cross the wires flls ck to zero, n the flow of chrge stops. Thus, moving the pltes closer together cuses the chrge on the cpcitor to increse. If is increse, the chrge ecreses. As result, we expect the evice s cpcitnce to e inversely proportionl to. Q Q Figure 26.3 () () The electric fiel etween the pltes of prllel-plte cpcitor is uniform ner the center ut nonuniform ner the eges. () Electric fiel pttern of two oppositely chrge conucting prllel pltes. Smll pieces of thre on n oil surfce lign with the electric fiel. ()

5 26.2 Clculting Cpcitnce 807 We cn verify these physicl rguments with the following erivtion. The surfce chrge ensity on either plte is If the pltes re very close together (in comprison with their length n with), we cn ssume tht the electric fiel is uniform etween the pltes n is zero elsewhere. Accoring to the lst prgrph of Exmple 24.8, the vlue of the electric fiel etween the pltes is E Becuse the fiel etween the pltes is uniform, the mgnitue of the potentil ifference etween the pltes equls E (see Eq. 25.6); therefore, V E Q 0A Sustituting this result into Eqution 26., we fin tht the cpcitnce is 0 Q /A. Q 0A C Q V Q Q/0A C 0A (26.3) Tht is, the cpcitnce of prllel-plte cpcitor is proportionl to the re of its pltes n inversely proportionl to the plte seprtion, just s we expect from our conceptul rgument. A creful inspection of the electric fiel lines for prllel-plte cpcitor revels tht the fiel is uniform in the centrl region etween the pltes, s shown in Figure However, the fiel is nonuniform t the eges of the pltes. Figure 26.3 is photogrph of the electric fiel pttern of prllel-plte cpcitor. Note the nonuniform nture of the electric fiel t the ens of the pltes. Such en effects cn e neglecte if the plte seprtion is smll compre with the length of the pltes. Quick Quiz 26. Mny computer keyor uttons re constructe of cpcitors, s shown in Figure When key is pushe own, the soft insultor etween the movle plte n the fixe plte is compresse. When the key is presse, the cpcitnce () increses, () ecreses, or (c) chnges in wy tht we cnnot etermine ecuse the complicte electric circuit connecte to the keyor utton my cuse chnge in V. Key Movle plte Soft insultor Fixe plte Figure 26.4 B One type of computer keyor utton. EXAMPLE 26. Prllel-Plte Cpcitor A prllel-plte cpcitor hs n re A m 2 n plte seprtion.00 mm. Fin its cpcitnce F.77 pf Solution From Eqution 26.3, we fin tht C A 0 ( C 2 /Nm 2 ) m m Exercise 3.00 mm? Answer Wht is the cpcitnce for plte seprtion of pf.

6 808 CHAPTER 26 Cpcitnce n Dielectrics Cylinricl n Sphericl Cpcitors From the efinition of cpcitnce, we cn, in principle, fin the cpcitnce of ny geometric rrngement of conuctors. The following exmples emonstrte the use of this efinition to clculte the cpcitnce of the other fmilir geometries tht we mentione: cyliners n spheres. EXAMPLE 26.2 The Cylinricl Cpcitor A soli cylinricl conuctor of rius n chrge Q is coxil with cylinricl shell of negligile thickness, rius, n chrge Q (Fig. 26.5). Fin the cpcitnce of this cylinricl cpcitor if its length is. Solution It is ifficult to pply physicl rguments to this configurtion, lthough we cn resonly expect the cpcitnce to e proportionl to the cyliner length for the sme reson tht prllel-plte cpcitnce is proportionl to plte re: Store chrges hve more room in which to e istriute. If we ssume tht is much greter thn n, we cn neglect en effects. In this cse, the electric fiel is perpeniculr to the long xis of the cyliners n is confine to the region etween them (Fig. 26.5). We must first clculte the potentil ifference etween the two cyliners, which is given in generl y where E is the electric fiel in the region r. In Chpter 24, we showe using Guss s lw tht the mgnitue of the electric fiel of cylinricl chrge istriution hving liner chrge ensity is E r 2k e/r (Eq. 24.7). The sme result pplies here ecuse, ccoring to Guss s lw, the chrge on the outer cyliner oes not contriute to the electric fiel insie it. Using this result n noting from Figure 26.5 tht E is long r, we fin tht V V E r r 2k e Sustituting this result into Eqution 26. n using the fct tht we otin Q /, C Q V Q 2k e Q V V E s ln (26.4) where V is the mgnitue of the potentil ifference, given r r 2k e ln 2k e ln y V V V 2k e ln (/), positive quntity. As preicte, the cpcitnce is proportionl to the length of the cyliners. As we might expect, the cpcitnce lso epens on the rii of the two cylinricl conuctors. From Eqution 26.4, we see tht the cpcitnce per unit length of comintion of concentric cylinricl conuctors is C 2k e ln (26.5) An exmple of this type of geometric rrngement is coxil cle, which consists of two concentric cylinricl conuctors seprte y n insultor. The cle crries electricl signls in the inner n outer conuctors. Such geometry is especilly useful for shieling the signls from ny possile externl influences. Figure 26.5 () Gussin surfce Q () A cylinricl cpcitor consists of soli cylinricl conuctor of rius n length surroune y coxil cylinricl shell of rius. () En view. The she line represents the en of the cylinricl gussin surfce of rius r n length. r () Q EXAMPLE 26.3 The Sphericl Cpcitor A sphericl cpcitor consists of sphericl conucting shell of rius n chrge Q concentric with smller conucting sphere of rius n chrge Q (Fig. 26.6). Fin the cpcitnce of this evice. Solution As we showe in Chpter 24, the fiel outsie sphericlly symmetric chrge istriution is ril n given y the expression k e Q /r 2. In this cse, this result pplies to the fiel etween the spheres ( r ). From

7 26.3 Comintions of Cpcitors 809 Guss s lw we see tht only the inner sphere contriutes to this fiel. Thus, the potentil ifference etween the spheres is Q V V E r r k e Q k e Q r r 2 k eq r Q The mgnitue of the potentil ifference is V V V k e Q ( ) Sustituting this vlue for V into Eqution 26., we otin Figure 26.6 A sphericl cpcitor consists of n inner sphere of rius surroune y concentric sphericl shell of rius. The electric fiel etween the spheres is irecte rilly outwr when the inner sphere is positively chrge. C Q V k e ( ) (26.6) Exercise Show tht s the rius of the outer sphere pproches infinity, the cpcitnce pproches the vlue /k e 40. Quick Quiz 26.2 Wht is the mgnitue of the electric fiel in the region outsie the sphericl cpcitor escrie in Exmple 26.3? COMBINATIONS OF CAPACITORS Two or more cpcitors often re comine in electric circuits. We cn clculte the equivlent cpcitnce of certin comintions using methos escrie in this section. The circuit symols for cpcitors n tteries, s well s the color coes use for them in this text, re given in Figure The symol for the cpcitor reflects the geometry of the most common moel for cpcitor pir of prllel pltes. The positive terminl of the ttery is t the higher potentil n is represente in the circuit symol y the longer verticl line. Prllel Comintion Two cpcitors connecte s shown in Figure 26.8 re known s prllel comintion of cpcitors. Figure 26.8 shows circuit igrm for this comintion of cpcitors. The left pltes of the cpcitors re connecte y conucting wire to the positive terminl of the ttery n re therefore oth t the sme electric potentil s the positive terminl. Likewise, the right pltes re connecte to the negtive terminl n re therefore oth t the sme potentil s the negtive terminl. Thus, the iniviul potentil ifferences cross cpcitors connecte in prllel re ll the sme n re equl to the potentil ifference pplie cross the comintion. In circuit such s tht shown in Figure 26.8, the voltge pplie cross the comintion is the terminl voltge of the ttery. Situtions cn occur in which Figure 26.7 Cpcitor symol Bttery symol Switch symol Circuit symols for cpcitors, tteries, n switches. Note tht cpcitors re in lue n tteries n switches re in re.

8 80 CHAPTER 26 Cpcitnce n Dielectrics C V = V 2 = V C C eq = C C 2 Q C 2 C 2 Q 2 V V V Figure 26.8 () () A prllel comintion of two cpcitors in n electric circuit in which the potentil ifference cross the ttery terminls is V. () The circuit igrm for the prllel comintion. (c) The equivlent cpcitnce is C eq C C 2. () (c) the prllel comintion is in circuit with other circuit elements; in such situtions, we must etermine the potentil ifference cross the comintion y nlyzing the entire circuit. When the cpcitors re first connecte in the circuit shown in Figure 26.8, electrons re trnsferre etween the wires n the pltes; this trnsfer leves the left pltes positively chrge n the right pltes negtively chrge. The energy source for this chrge trnsfer is the internl chemicl energy store in the ttery, which is converte to electric potentil energy ssocite with the chrge seprtion. The flow of chrge ceses when the voltge cross the cpcitors is equl to tht cross the ttery terminls. The cpcitors rech their mximum chrge when the flow of chrge ceses. Let us cll the mximum chrges on the two cpcitors Q n Q 2. The totl chrge Q store y the two cpcitors is Q Q Q 2 (26.7) Tht is, the totl chrge on cpcitors connecte in prllel is the sum of the chrges on the iniviul cpcitors. Becuse the voltges cross the cpcitors re the sme, the chrges tht they crry re Q C V Q 2 C 2 V Suppose tht we wish to replce these two cpcitors y one equivlent cpcitor hving cpcitnce C eq, s shown in Figure 26.8c. The effect this equivlent cpcitor hs on the circuit must e exctly the sme s the effect of the comintion of the two iniviul cpcitors. Tht is, the equivlent cpcitor must store Q units of chrge when connecte to the ttery. We cn see from Figure 26.8c tht the voltge cross the equivlent cpcitor lso is V ecuse the equivlent cpc-

9 26.3 Comintions of Cpcitors 8 itor is connecte irectly cross the ttery terminls. Thus, for the equivlent cpcitor, Q C eq V Sustituting these three reltionships for chrge into Eqution 26.7, we hve C eq V C V C 2 V C eq C C 2 prllel comintion If we exten this tretment to three or more cpcitors connecte in prllel, we fin the equivlent cpcitnce to e C eq C C 2 C 3 (prllel comintion) (26.8) Thus, the equivlent cpcitnce of prllel comintion of cpcitors is greter thn ny of the iniviul cpcitnces. This mkes sense ecuse we re essentilly comining the res of ll the cpcitor pltes when we connect them with conucting wire. Series Comintion Two cpcitors connecte s shown in Figure 26.9 re known s series comintion of cpcitors. The left plte of cpcitor n the right plte of cpcitor 2 re connecte to the terminls of ttery. The other two pltes re connecte to ech other n to nothing else; hence, they form n isolte conuctor tht is initilly unchrge n must continue to hve zero net chrge. To nlyze this comintion, let us egin y consiering the unchrge cpcitors n follow wht hppens just fter ttery is connecte to the circuit. When the ttery is con- C V V 2 C 2 C eq Q Q Q Q V V Figure 26.9 () () A series comintion of two cpcitors. The chrges on the two cpcitors re the sme. () The cpcitors replce y single equivlent cpcitor. The equivlent cpcitnce cn e clculte from the reltionship C eq C C 2 ()

10 82 CHAPTER 26 Cpcitnce n Dielectrics necte, electrons re trnsferre out of the left plte of C n into the right plte of C 2. As this negtive chrge ccumultes on the right plte of C 2, n equivlent mount of negtive chrge is force off the left plte of C 2, n this left plte therefore hs n excess positive chrge. The negtive chrge leving the left plte of C 2 trvels through the connecting wire n ccumultes on the right plte of C. As result, ll the right pltes en up with chrge Q, n ll the left pltes en up with chrge Q. Thus, the chrges on cpcitors connecte in series re the sme. From Figure 26.9, we see tht the voltge V cross the ttery terminls is split etween the two cpcitors: V V V 2 (26.9) where V n V 2 re the potentil ifferences cross cpcitors C n C 2, respectively. In generl, the totl potentil ifference cross ny numer of cpcitors connecte in series is the sum of the potentil ifferences cross the iniviul cpcitors. Suppose tht n equivlent cpcitor hs the sme effect on the circuit s the series comintion. After it is fully chrge, the equivlent cpcitor must hve chrge of Q on its right plte n chrge of Q on its left plte. Applying the efinition of cpcitnce to the circuit in Figure 26.9, we hve V Becuse we cn pply the expression Q C V to ech cpcitor shown in Figure 26.9, the potentil ifference cross ech is Q C eq V Q C V 2 Q C 2 Sustituting these expressions into Eqution 26.9 n noting tht we hve Q C eq Q C Q C 2 Cnceling Q, we rrive t the reltionship C eq C C 2 series comintion V Q /C eq, When this nlysis is pplie to three or more cpcitors connecte in series, the reltionship for the equivlent cpcitnce is C eq C C 2 C 3 series comintion (26.0) This emonstrtes tht the equivlent cpcitnce of series comintion is lwys less thn ny iniviul cpcitnce in the comintion. EXAMPLE 26.4 Equivlent Cpcitnce Fin the equivlent cpcitnce etween n for the comintion of cpcitors shown in Figure All cpcitnces re in microfrs. Solution Using Equtions 26.8 n 26.0, we reuce the comintion step y step s inicte in the figure. The.0-F n 3.0-F cpcitors re in prllel n comine c-

11 26.4 Energy Store in Chrge Cpcitor 83 coring to the expression C eq C C F. The 2.0-F n 6.0-F cpcitors lso re in prllel n hve n equivlent cpcitnce of 8.0 F. Thus, the upper rnch in Figure 26.0 consists of two 4.0-F cpcitors in series, which comine s follows: C eq C C 2 C eq /2.0 F 2.0 F 4.0 F 4.0 F 2.0 F The lower rnch in Figure 26.0 consists of two 8.0-F cpcitors in series, which comine to yiel n equivlent cpcitnce of 4.0 F. Finlly, the 2.0-F n 4.0-F cpcitors in Figure 26.0c re in prllel n thus hve n equivlent cpcitnce of 6.0 F. Exercise Consier three cpcitors hving cpcitnces of 3.0 F, 6.0 F, n 2 F. Fin their equivlent cpcitnce when they re connecte () in prllel n () in series. Answer () 2 F; ().7 F () Figure 26.0 () To fin the equivlent cpcitnce of the cpcitors in prt (), we reuce the vrious comintions in steps s inicte in prts (), (c), n (), using the series n prllel rules escrie in the text. (c) () ENERGYSTORED IN A CHARGED CAPACITOR Almost everyone who works with electronic equipment hs t some time verifie tht cpcitor cn store energy. If the pltes of chrge cpcitor re connecte y conuctor, such s wire, chrge moves etween the pltes n the connecting wire until the cpcitor is unchrge. The ischrge cn often e oserve s visile sprk. If you shoul ccientlly touch the opposite pltes of chrge cpcitor, your fingers ct s pthwy for ischrge, n the result is n electric shock. The egree of shock you receive epens on the cpcitnce n on the voltge pplie to the cpcitor. Such shock coul e ftl if high voltges re present, such s in the power supply of television set. Becuse the chrges cn e store in cpcitor even when the set is turne off, unplugging the television oes not mke it sfe to open the cse n touch the components insie. Consier prllel-plte cpcitor tht is initilly unchrge, such tht the initil potentil ifference cross the pltes is zero. Now imgine tht the cpcitor is connecte to ttery n evelops mximum chrge Q. (We ssume tht the cpcitor is chrge slowly so tht the prolem cn e consiere s n electrosttic system.) When the cpcitor is connecte to the ttery, electrons in the wire just outsie the plte connecte to the negtive terminl move into the plte to give it negtive chrge. Electrons in the plte connecte to the positive terminl move out of the plte into the wire to give the plte positive chrge. Thus, chrges move only smll istnce in the wires. To clculte the energy of the cpcitor, we shll ssume ifferent process one tht oes not ctully occur ut gives the sme finl result. We cn mke this

12 84 CHAPTER 26 Cpcitnce n Dielectrics QuickL Here s how to fin out whether your clcultor hs cpcitor to protect vlues or progrms uring ttery chnges: Store numer in your clcultor s memory, remove the clcultor ttery for moment, n then quickly replce it. Ws the numer tht you store preserve while the ttery ws out of the clcultor? (You my wnt to write own ny criticl numers or progrms tht re store in the clcultor efore trying this!) ssumption ecuse the energy in the finl configurtion oes not epen on the ctul chrge-trnsfer process. We imgine tht we rech in n gr smll mount of positive chrge on the plte connecte to the negtive terminl n pply force tht cuses this positive chrge to move over to the plte connecte to the positive terminl. Thus, we o work on the chrge s we trnsfer it from one plte to the other. At first, no work is require to trnsfer smll mount of chrge q from one plte to the other. 3 However, once this chrge hs een trnsferre, smll potentil ifference exists etween the pltes. Therefore, work must e one to move itionl chrge through this potentil ifference. As more n more chrge is trnsferre from one plte to the other, the potentil ifference increses in proportion, n more work is require. Suppose tht q is the chrge on the cpcitor t some instnt uring the chrging process. At the sme instnt, the potentil ifference cross the cpcitor is V q/c. From Section 25.2, we know tht the work necessry to trnsfer n increment of chrge q from the plte crrying chrge q to the plte crrying chrge q (which is t the higher electric potentil) is W V q q C q Energy store in chrge cpcitor This is illustrte in Figure 26.. The totl work require to chrge the cpcitor from q 0 to some finl chrge q Q is W Q 0 q C q Q C The work one in chrging the cpcitor ppers s electric potentil energy U store in the cpcitor. Therefore, we cn express the potentil energy store in chrge cpcitor in the following forms: U Q 2 2C 2 Q V 2C(V )2 (26.) This result pplies to ny cpcitor, regrless of its geometry. We see tht for given cpcitnce, the store energy increses s the chrge increses n s the potentil ifference increses. In prctice, there is limit to the mximum energy 0 q q Q 2 2C V q q Figure 26. A plot of potentil ifference versus chrge for cpcitor is stright line hving slope /C. The work require to move chrge q through the potentil ifference V cross the cpcitor pltes is given y the re of the she rectngle. The totl work require to chrge the cpcitor to finl chrge Q is the tringulr re uner the stright line, W 2 Q V. (Don t forget tht V J/C; hence, the unit for the re is the joule.) 3 We shll use lowercse q for the vrying chrge on the cpcitor while it is chrging, to istinguish it from uppercse Q, which is the totl chrge on the cpcitor fter it is completely chrge.

13 26.4 Energy Store in Chrge Cpcitor 85 (or chrge) tht cn e store ecuse, t sufficiently gret vlue of V, ischrge ultimtely occurs etween the pltes. For this reson, cpcitors re usully lele with mximum operting voltge. Quick Quiz 26.3 You hve three cpcitors n ttery. How shoul you comine the cpcitors n the ttery in one circuit so tht the cpcitors will store the mximum possile energy? We cn consier the energy store in cpcitor s eing store in the electric fiel crete etween the pltes s the cpcitor is chrge. This escription is resonle in view of the fct tht the electric fiel is proportionl to the chrge on the cpcitor. For prllel-plte cpcitor, the potentil ifference is relte to the electric fiel through the reltionship V E. Furthermore, its cpcitnce is C 0A/ (Eq. 26.3). Sustituting these expressions into Eqution 26., we otin U 2 0A (E 2 2 ) 2 (0A)E 2 (26.2) Becuse the volume V (volume, not voltge!) occupie y the electric fiel is A, the energy per unit volume u E U/V U/A, known s the energy ensity, is Energy store in prllel-plte cpcitor u E 20E 2 (26.3) Energy ensity in n electric fiel Although Eqution 26.3 ws erive for prllel-plte cpcitor, the expression is generlly vli. Tht is, the energy ensity in ny electric fiel is proportionl to the squre of the mgnitue of the electric fiel t given point. This nk of cpcitors stores electricl energy for use in the prticle ccelertor t FermiL, locte outsie Chicgo. Becuse the electric utility compny cnnot provie lrge enough urst of energy to operte the equipment, these cpcitors re slowly chrge up, n then the energy is rpily umpe into the ccelertor. In this sense, the setup is much like fireprotection wter tnk on top of uiling. The tnk collects wter n stores it for situtions in which lot of wter is neee in short time.

14 86 CHAPTER 26 Cpcitnce n Dielectrics EXAMPLE 26.5 Rewiring Two Chrge Cpcitors Two cpcitors C n C 2 (where C C 2 ) re chrge to the sme initil potentil ifference V i, ut with opposite polrity. The chrge cpcitors re remove from the ttery, n their pltes re connecte s shown in Figure The switches S n S 2 re then close, s shown in Figure () Fin the finl potentil ifference V f etween n fter the switches re close. Solution Let us ientify the left-hn pltes of the cpcitors s n isolte system ecuse they re not connecte to the right-hn pltes y conuctors. The chrges on the lefthn pltes efore the switches re close re Q i () C S S 2 Q 2i C 2 Figure 26.2 Q C f S S 2 Q 2f C 2 () Q i C V i n Q 2i C 2 V i The negtive sign for Q 2i is necessry ecuse the chrge on the left plte of cpcitor C 2 is negtive. The totl chrge Q in the system is () After the switches re close, the totl chrge in the system remins the sme: (2) The chrges reistriute until the entire system is t the sme potentil V f. Thus, the finl potentil ifference cross C must e the sme s the finl potentil ifference cross C 2. To stisfy this requirement, the chrges on the cpcitors fter the switches re close re Diviing the first eqution y the secon, we hve (3) Comining Equtions (2) n (3), we otin Q 2 f Q C 2 C C 2 Using Eqution (3) to fin Q f in terms of Q, we hve Finlly, using Eqution 26. to fin the voltge cross ech cpcitor, we fin tht V f Q f C Q Q i Q 2i (C C 2 )V i Q Q f Q 2f Q f C V f n Q 2f C 2 V f Q f Q 2f C V f C 2 V f C C 2 Q f C C 2 Q 2f Q Q f Q 2 f C C 2 Q 2f Q 2f Q 2f C C 2 Q f C C 2 Q 2f C C 2 Q C 2 C C 2 Q Q C C C 2 C C C C 2 Q C C 2 As note erlier, V f V 2 f V f. To express V f in terms of the given quntities C, C 2, n V i, we sustitute the vlue of Q from Eqution () to otin () Fin the totl energy store in the cpcitors efore n fter the switches re close n the rtio of the finl energy to the initil energy. Solution Before the switches re close, the totl energy store in the cpcitors is U i 2 C (V i ) 2 2 C 2(V i ) 2 After the switches re close, the totl energy store in the cpcitors is U f 2 C (V f ) 2 2 C 2(V f ) 2 2 (C C 2 )(V f ) 2 Using Eqution (), we cn express this s U f 2 Therefore, the rtio of the finl energy store to the initil energy store is U f U i V 2 f Q 2 f C 2 2 (C C 2 ) Q C C V f C C 2 C C 2 V i Q 2 (C C 2 ) Q C 2 C C 2 C 2 2 (C C 2 ) 2 (V i ) 2 2 (C C 2 ) 2 (C C 2 )(V i ) 2 2 (C C 2 )(V i ) 2 (C C 2 ) 2 (V i ) 2 (C C 2 ) Q C C 2 Q 2 C C 2 C C 2 C C 2 2

15 26.4 Energy Store in Chrge Cpcitor 87 This rtio is less thn unity, inicting tht the finl energy is less thn the initil energy. At first, you might think tht the lw of energy conservtion hs een violte, ut this is not the cse. The missing energy is rite wy in the form of electromgnetic wves, s we shll see in Chpter 34. Quick Quiz 26.4 You chrge prllel-plte cpcitor, remove it from the ttery, n prevent the wires connecte to the pltes from touching ech other. When you pull the pltes prt, o the following quntities increse, ecrese, or sty the sme? () C; () Q ; (c) E etween the pltes; () V ; (e) energy store in the cpcitor. Quick Quiz 26.5 Repet Quick Quiz 26.4, ut this time nswer the questions for the sitution in which the ttery remins connecte to the cpcitor while you pull the pltes prt. One evice in which cpcitors hve n importnt role is the efirilltor (Fig. 26.3). Up to 360 J is store in the electric fiel of lrge cpcitor in efirilltor when it is fully chrge. The efirilltor cn eliver ll this energy to ptient in out 2 ms. (This is roughly equivlent to times the power output of 60-W lightul!) The suen electric shock stops the firilltion (rnom contrctions) of the hert tht often ccompnies hert ttcks n helps to restore the correct rhythm. A cmer s flsh unit lso uses cpcitor, lthough the totl mount of energy store is much less thn tht store in efirilltor. After the flsh unit s cpcitor is chrge, tripping the cmer s shutter cuses the store energy to e sent through specil lightul tht riefly illumintes the suject eing photogrphe. we To lern more out efirilltors, visit Figure 26.3 In hospitl or t n emergency scene, you might see ptient eing revive with efirilltor. The efirilltor s ples re pplie to the ptient s chest, n n electric shock is sent through the chest cvity. The im of this technique is to restore the hert s norml rhythm pttern.

16 88 CHAPTER 26 Cpcitnce n Dielectrics 26.5 CAPACITORS WITH DIELECTRICS A ielectric is nonconucting mteril, such s ruer, glss, or wxe pper. When ielectric is inserte etween the pltes of cpcitor, the cpcitnce increses. If the ielectric completely fills the spce etween the pltes, the cpcitnce increses y imensionless fctor, which is clle the ielectric constnt. The ielectric constnt is property of mteril n vries from one mteril to nother. In this section, we nlyze this chnge in cpcitnce in terms of electricl prmeters such s electric chrge, electric fiel, n potentil ifference; in Section 26.7, we shll iscuss the microscopic origin of these chnges. We cn perform the following experiment to illustrte the effect of ielectric in cpcitor: Consier prllel-plte cpcitor tht without ielectric hs chrge Q 0 n cpcitnce C 0. The potentil ifference cross the cpcitor is V 0 Q 0 /C 0. Figure 26.4 illustrtes this sitution. The potentil ifference is mesure y voltmeter, which we shll stuy in greter etil in Chpter 28. Note tht no ttery is shown in the figure; lso, we must ssume tht no chrge cn flow through n iel voltmeter, s we shll lern in Section Hence, there is no pth y which chrge cn flow n lter the chrge on the cpcitor. If ielectric is now inserte etween the pltes, s shown in Figure 26.4, the voltmeter inictes tht the voltge etween the pltes ecreses to vlue V. The voltges with n without the ielectric re relte y the fctor s follows:. V V 0 Becuse V V 0, we see tht Becuse the chrge Q 0 on the cpcitor oes not chnge, we conclue tht the cpcitnce must chnge to the vlue The cpcitnce of fille cpcitor is greter thn tht of n empty one y fctor. C Q 0 V Q 0 V 0 / C C 0 (26.4) Tht is, the cpcitnce increses y the fctor when the ielectric completely fills the region etween the pltes. 4 For prllel-plte cpcitor, where C 0 0A/ (Eq. 26.3), we cn express the cpcitnce when the cpcitor is fille with ielectric s 0A C (26.5) Q 0 From Equtions 26.3 n 26.5, it woul pper tht we coul mke the cpcitnce very lrge y ecresing, the istnce etween the pltes. In prctice, the lowest vlue of is limite y the electric ischrge tht coul occur through the ielectric meium seprting the pltes. For ny given seprtion, the mximum voltge tht cn e pplie to cpcitor without cusing ischrge epens on the ielectric strength (mximum electric fiel) of the ielectric. If the mgnitue of the electric fiel in the ielectric excees the ielectric strength, then the insulting properties rek own n the ielectric egins to conuct. Insulting mterils hve vlues of greter thn unity n ielectric strengths V 0 4 If the ielectric is introuce while the potentil ifference is eing mintine constnt y ttery, the chrge increses to vlue Q Q 0. The itionl chrge is supplie y the ttery, n the cpcitnce gin increses y the fctor.

17 26.5 Cpcitors with Dielectrics 89 Dielectric C 0 Q 0 C Q 0 V 0 V Figure 26.4 () A chrge cpcitor () efore n () fter insertion of ielectric etween the pltes. The chrge on the pltes remins unchnge, ut the potentil ifference ecreses from V 0 to V V 0 /. Thus, the cpcitnce increses from C 0 to C 0. () greter thn tht of ir, s Tle 26. inictes. Thus, we see tht ielectric provies the following vntges: Increse in cpcitnce Increse in mximum operting voltge Possile mechnicl support etween the pltes, which llows the pltes to e close together without touching, therey ecresing n incresing C TABLE 26. Dielectric Constnts n Dielectric Strengths of Vrious Mterils t Room Temperture Dielectric Dielectric Mteril Constnt Strength (V/m) Air (ry) Bkelite Fuse qurtz Neoprene ruer Nylon Pper Polystyrene Polyvinyl chlorie Porcelin Pyrex glss Silicone oil Strontium titnte Teflon Vcuum Wter 80 The ielectric strength equls the mximum electric fiel tht cn exist in ielectric without electricl rekown. Note tht these vlues epen strongly on the presence of impurities n flws in the mterils.

18 820 CHAPTER 26 Cpcitnce n Dielectrics () () () Kirlin photogrph crete y ropping steel ll into high-energy electric fiel. Kirlin photogrphy is lso known s electrophotogrphy. () Sprks from sttic electricity ischrge etween fork n four electroes. Mny sprks were use to crete this imge ecuse only one sprk forms for given ischrge. Note tht the ottom prong ischrges to oth electroes t the ottom right. The light of ech sprk is crete y the excittion of gs toms long its pth. Types of Cpcitors Commercil cpcitors re often me from metllic foil interlce with thin sheets of either prffin-impregnte pper or Mylr s the ielectric mteril. These lternte lyers of metllic foil n ielectric re rolle into cyliner to form smll pckge (Fig. 26.5). High-voltge cpcitors commonly consist of numer of interwoven metllic pltes immerse in silicone oil (Fig. 26.5). Smll cpcitors re often constructe from cermic mterils. Vrile cpcitors (typiclly 0 to 500 pf) usully consist of two interwoven sets of metllic pltes, one fixe n the other movle, n contin ir s the ielectric. Often, n electrolytic cpcitor is use to store lrge mounts of chrge t reltively low voltges. This evice, shown in Figure 26.5c, consists of metllic foil in contct with n electrolyte solution tht conucts electricity y virtue of the motion of ions contine in the solution. When voltge is pplie etween the foil n the electrolyte, thin lyer of metl oxie (n insultor) is forme on the foil, Metl foil Pltes Cse Electrolyte Figure 26.5 Pper () () (c) Oil Contcts Metllic foil oxie lyer Three commercil cpcitor esigns. () A tuulr cpcitor, whose pltes re seprte y pper n then rolle into cyliner. () A high-voltge cpcitor consisting of mny prllel pltes seprte y insulting oil. (c) An electrolytic cpcitor.

19 26.5 Cpcitors with Dielectrics 82 n this lyer serves s the ielectric. Very lrge vlues of cpcitnce cn e otine in n electrolytic cpcitor ecuse the ielectric lyer is very thin, n thus the plte seprtion is very smll. Electrolytic cpcitors re not reversile s re mny other cpcitors they hve polrity, which is inicte y positive n negtive signs mrke on the evice. When electrolytic cpcitors re use in circuits, the polrity must e ligne properly. If the polrity of the pplie voltge is opposite tht which is intene, the oxie lyer is remove n the cpcitor conucts electricity inste of storing chrge. Quick Quiz 26.6 If you hve ever trie to hng picture, you know it cn e ifficult to locte wooen stu in which to nchor your nil or screw. A crpenter s stu-finer is siclly cpcitor with its pltes rrnge sie y sie inste of fcing one nother, s shown in Figure When the evice is move over stu, oes the cpcitnce increse or ecrese? Stu Cpcitor pltes Figure 26.6 () Wll or Stu-finer A stu-finer. ()The mterils etween the pltes of the cpcitor re the wllor n ir. () When the cpcitor moves cross stu in the wll, the mterils etween the pltes re the wllor n the woo. The chnge in the ielectric constnt cuses signl light to illuminte. () EXAMPLE 26.6 A Pper-Fille Cpcitor A prllel-plte cpcitor hs pltes of imensions 2.0 cm y 3.0 cm seprte y.0-mm thickness of pper. () Fin its cpcitnce. Solution hve C 0A F Becuse 3.7 for pper (see Tle 26.), we () Wht is the mximum chrge tht cn e plce on the cpcitor? Solution 3.7( C 2 /Nm 2 ) m m 20 pf From Tle 26. we see tht the ielectric strength of pper is V/m. Becuse the thickness of the pper is.0 mm, the mximum voltge tht cn e pplie efore rekown is Hence, the mximum chrge is Q mx C V mx ( F)(6 0 3 V) Exercise V mx E mx (6 0 6 V/m)( m) V Wht is the mximum energy tht cn e store in the cpcitor? Answer J C

20 822 CHAPTER 26 Cpcitnce n Dielectrics EXAMPLE 26.7 Energy Store Before n After A prllel-plte cpcitor is chrge with ttery to chrge Q 0, s shown in Figure The ttery is then remove, n sl of mteril tht hs ielectric constnt is inserte etween the pltes, s shown in Figure Fin the energy store in the cpcitor efore n fter the ielectric is inserte. Solution The energy store in the sence of the ielectric is (see Eq. 26.): U 0 Q 2 0 2C 0 After the ttery is remove n the ielectric inserte, the chrge on the cpcitor remins the sme. Hence, the energy store in the presence of the ielectric is U Q 2 0 2C But the cpcitnce in the presence of the ielectric is C C 0, so U ecomes U Q 2 0 U 0 2C 0 Becuse, the finl energy is less thn the initil energy. We cn ccount for the missing energy y noting tht the ielectric, when inserte, gets pulle into the evice (see the following iscussion n Figure 26.8). An externl gent must o negtive work to keep the ielectric from ccelerting. This work is simply the ifference U U 0. (Alterntively, the positive work one y the system on the externl gent is U 0 U.) Exercise Suppose tht the cpcitnce in the sence of ielectric is 8.50 pf n tht the cpcitor is chrge to potentil ifference of 2.0 V. If the ttery is isconnecte n sl of polystyrene is inserte etween the pltes, wht is U 0 U? Answer 373 pj. C 0 Q 0 Q 0 () () V 0 Dielectric Figure 26.7 As we hve seen, the energy of cpcitor not connecte to ttery is lowere when ielectric is inserte etween the pltes; this mens tht negtive work is one on the ielectric y the externl gent inserting the ielectric into the cpcitor. This, in turn, implies tht force tht rws it into the cpcitor must e cting on the ielectric. This force origintes from the nonuniform nture of the electric fiel of the cpcitor ner its eges, s inicte in Figure The horizontl component of this fringe fiel cts on the inuce chrges on the surfce of the ielectric, proucing net horizontl force irecte into the spce etween the cpcitor pltes. Quick Quiz 26.7 A fully chrge prllel-plte cpcitor remins connecte to ttery while you slie ielectric etween the pltes. Do the following quntities increse, ecrese, or sty the sme? () C; () Q ; (c) E etween the pltes; () V ; (e) energy store in the cpcitor.

21 26.6 Electric Dipole in n Electric Fiel 823 Q Q Figure 26.8 The nonuniform electric fiel ner the eges of prllel-plte cpcitor cuses ielectric to e pulle into the cpcitor. Note tht the fiel cts on the inuce surfce chrges on the ielectric, which re nonuniformly istriute. Optionl Section 26.6 ELECTRIC DIPOLE IN AN ELECTRIC FIELD We hve iscusse the effect on the cpcitnce of plcing ielectric etween the pltes of cpcitor. In Section 26.7, we shll escrie the microscopic origin of this effect. Before we cn o so, however, we nee to expn upon the iscussion of the electric ipole tht we egn in Section 23.4 (see Exmple 23.6). The electric ipole consists of two chrges of equl mgnitue ut opposite sign seprte y istnce 2, s shown in Figure The electric ipole moment of this configurtion is efine s the vector p irecte from q to q long the line joining the chrges n hving mgnitue 2q: p 2q (26.6) Now suppose tht n electric ipole is plce in uniform electric fiel E, s shown in Figure We ientify E s the fiel externl to the ipole, istinguishing it from the fiel ue to the ipole, which we iscusse in Section The fiel E is estlishe y some other chrge istriution, n we plce the ipole into this fiel. Let us imgine tht the ipole moment mkes n ngle with the fiel. The electric forces cting on the two chrges re equl in mgnitue ut opposite in irection s shown in Figure (ech hs mgnitue F qe). Thus, the net force on the ipole is zero. However, the two forces prouce net torque on the ipole; s result, the ipole rottes in the irection tht rings the ipole moment vector into greter lignment with the fiel. The torque ue to the force on the positive chrge out n xis through O in Figure is F sin, where sin is the moment rm of F out O. This force tens to prouce clockwise rottion. The torque out O on the negtive chrge lso is F sin ; here gin, the force tens to prouce clockwise rottion. Thus, the net torque out O is 2F sin Becuse F qe n p 2q, we cn express s 2qE sin pe sin (26.7) q Figure 26.9 F Figure q O q An electric ipole in uniform externl electric fiel. The ipole moment p is t n ngle to the fiel, cusing the ipole to experience torque. p θ q An electric ipole consists of two chrges of equl mgnitue ut opposite sign seprte y istnce of 2. The electric ipole moment p is irecte from q to q. E F

22 824 CHAPTER 26 Cpcitnce n Dielectrics It is convenient to express the torque in vector form s the cross prouct of the vectors p n E: Torque on n electric ipole in n externl electric fiel p E (26.8) We cn etermine the potentil energy of the system of n electric ipole in n externl electric fiel s function of the orienttion of the ipole with respect to the fiel. To o this, we recognize tht work must e one y n externl gent to rotte the ipole through n ngle so s to cuse the ipole moment vector to ecome less ligne with the fiel. The work one is then store s potentil energy in the system of the ipole n the externl fiel. The work W require to rotte the ipole through n ngle is W (Eq. 0.22). Becuse n ecuse the work is trnsforme into potentil energy U, we fin pe sin tht, for rottion from i to f, the chnge in potentil energy is f U f U i i i f f pe cos pe(cos i cos f) i The term tht contins cos i is constnt tht epens on the initil orienttion of the ipole. It is convenient for us to choose i 90, so tht cos i cos Furthermore, let us choose U i 0 t i 90 s our reference of potentil energy. Hence, we cn express generl vlue of U U f s p sin pe i f sin U pe cos (26.9) We cn write this expression for the potentil energy of ipole in n electric fiel s the ot prouct of the vectors p n E: Potentil energy of ipole in n electric fiel U p E (26.20) To evelop conceptul unerstning of Eqution 26.9, let us compre this expression with the expression for the potentil energy of n oject in the grvittionl fiel of the Erth, U mgh (see Chpter 8). The grvittionl expression inclues prmeter ssocite with the oject we plce in the fiel its mss m. Likewise, Eqution 26.9 inclues prmeter of the oject in the electric fiel its ipole moment p. The grvittionl expression inclues the mgnitue of the grvittionl fiel g. Similrly, Eqution 26.9 inclues the mgnitue of the electric fiel E. So fr, these two contriutions to the potentil energy expressions pper nlogous. However, the finl contriution is somewht ifferent in the two cses. In the grvittionl expression, the potentil energy epens on how high we lift the oject, mesure y h. In Eqution 26.9, the potentil energy epens on the ngle through which we rotte the ipole. In oth cses, we re mking chnge in the system. In the grvittionl cse, the chnge involves moving n oject in trnsltionl sense, wheres in the electricl cse, the chnge involves moving n oject in rottionl sense. In oth cses, however, once the chnge is me, the system tens to return to the originl configurtion when the oject is relese: the oject of mss m flls ck to the groun, n the ipole egins to rotte ck towr the configurtion in which it ws ligne with the fiel. Thus, prt from the type of motion, the expressions for potentil energy in these two cses re similr.

23 26.6 Electric Dipole in n Electric Fiel 825 Molecules re si to e polrize when seprtion exists etween the verge position of the negtive chrges n the verge position of the positive chrges in the molecule. In some molecules, such s wter, this conition is lwys present such molecules re clle polr molecules. Molecules tht o not possess permnent polriztion re clle nonpolr molecules. We cn unerstn the permnent polriztion of wter y inspecting the geometry of the wter molecule. In the wter molecule, the oxygen tom is one to the hyrogen toms such tht n ngle of 05 is forme etween the two ons (Fig. 26.2). The center of the negtive chrge istriution is ner the oxygen tom, n the center of the positive chrge istriution lies t point miwy long the line joining the hyrogen toms (the point lele in Fig. 26.2). We cn moel the wter molecule n other polr molecules s ipoles ecuse the verge positions of the positive n negtive chrges ct s point chrges. As result, we cn pply our iscussion of ipoles to the ehvior of polr molecules. Microwve ovens tke vntge of the polr nture of the wter molecule. When in opertion, microwve ovens generte rpily chnging electric fiel tht cuses the polr molecules to swing ck n forth, soring energy from the fiel in the process. Becuse the jostling molecules collie with ech other, the energy they sor from the fiel is converte to internl energy, which correspons to n increse in temperture of the foo. Another househol scenrio in which the ipole structure of wter is exploite is wshing with sop n wter. Grese n oil re me up of nonpolr molecules, which re generlly not ttrcte to wter. Plin wter is not very useful for removing this type of grime. Sop contins long molecules clle surfctnts. In long molecule, the polrity chrcteristics of one en of the molecule cn e ifferent from those t the other en. In surfctnt molecule, one en cts like nonpolr molecule n the other cts like polr molecule. The nonpolr en cn ttch to grese or oil molecule, n the polr en cn ttch to wter molecule. Thus, the sop serves s chin, linking the irt n wter molecules together. When the wter is rinse wy, the grese n oil go with it. A symmetric molecule (Fig ) hs no permnent polriztion, ut polriztion cn e inuce y plcing the molecule in n electric fiel. A fiel irecte to the left, s shown in Figure 26.22, woul cuse the center of the positive chrge istriution to shift to the left from its initil position n the center of the negtive chrge istriution to shift to the right. This inuce polriztion is the effect tht preomintes in most mterils use s ielectrics in cpcitors. O H 05 H Figure 26.2 The wter molecule, H 2 O, hs permnent polriztion resulting from its ent geometry. The center of the positive chrge istriution is t the point. () Figure () () A symmetric molecule hs no permnent polriztion. () An externl electric fiel inuces polriztion in the molecule. E EXAMPLE 26.8 The H 2 O Molecule The wter (H 2 O) molecule hs n electric ipole moment of C m. A smple contins 0 2 wter molecules, with the ipole moments ll oriente in the irection of n electric fiel of mgnitue N/C. How much work is require to rotte the ipoles from this orienttion to one in which ll the ipole moments re perpeniculr to the fiel ( 0) Solution ( 90)? The work require to rotte one molecule 90 is equl to the ifference in potentil energy etween the 90 orienttion n the 0 orienttion. Using Eqution 26.9, we otin W U 90 U 0 (pe cos 90) (pe cos 0) pe ( Cm)( N/C) J Becuse there re 0 2 molecules in the smple, the totl work require is W totl (0 2 )( J) J

24 826 CHAPTER 26 Cpcitnce n Dielectrics Optionl Section 26.7 AN ATOMIC DESCRIPTION OF DIELECTRICS In Section 26.5 we foun tht the potentil ifference V 0 etween the pltes of cpcitor is reuce to V 0 / when ielectric is introuce. Becuse the potentil ifference etween the pltes equls the prouct of the electric fiel n the seprtion, the electric fiel is lso reuce. Thus, if E 0 is the electric fiel without the ielectric, the fiel in the presence of ielectric is Figure () () E 0 () Polr molecules re rnomly oriente in the sence of n externl electric fiel. () When n externl fiel is pplie, the molecules prtilly lign with the fiel. E E 0 (26.2) Let us first consier ielectric me up of polr molecules plce in the electric fiel etween the pltes of cpcitor. The ipoles (tht is, the polr molecules mking up the ielectric) re rnomly oriente in the sence of n electric fiel, s shown in Figure When n externl fiel E 0 ue to chrges on the cpcitor pltes is pplie, torque is exerte on the ipoles, cusing them to prtilly lign with the fiel, s shown in Figure We cn now escrie the ielectric s eing polrize. The egree of lignment of the molecules with the electric fiel epens on temperture n on the mgnitue of the fiel. In generl, the lignment increses with ecresing temperture n with incresing electric fiel. If the molecules of the ielectric re nonpolr, then the electric fiel ue to the pltes prouces some chrge seprtion n n inuce ipole moment. These inuce ipole moments ten to lign with the externl fiel, n the ielectric is polrize. Thus, we cn polrize ielectric with n externl fiel regrless of whether the molecules re polr or nonpolr. With these ies in min, consier sl of ielectric mteril plce etween the pltes of cpcitor so tht it is in uniform electric fiel E 0, s shown in Figure The electric fiel ue to the pltes is irecte to the right n polrizes the ielectric. The net effect on the ielectric is the formtion of n inuce positive surfce chrge ensity in on the right fce n n equl negtive surfce chrge ensity in on the left fce, s shown in Figure These inuce surfce chrges on the ielectric give rise to n inuce electric fiel E in in the irection opposite the externl fiel E 0. Therefore, the net electric fiel E in the E 0 E 0 E in σσ in σ in Figure () () When ielectric is polrize, the ipole moments of the molecules in the ielectric re prtilly ligne with the externl fiel E 0. () This polriztion cuses n inuce negtive surfce chrge on one sie of the ielectric n n equl inuce positive surfce chrge on the opposite sie. This seprtion of chrge results in reuction in the net electric fiel within the ielectric. ()

25 26.7 An Atomic Description of Dielectrics 827 ielectric hs mgnitue (26.22) In the prllel-plte cpcitor shown in Figure 26.25, the externl fiel E 0 is relte to the chrge ensity on the pltes through the reltionship E 0 /0. The inuce electric fiel in the ielectric is relte to the inuce chrge ensity in through the reltionship E in in/0. Becuse E E 0 / /0, sustitution into Eqution gives, E E 0 E in in (26.23) Becuse this expression shows tht the chrge ensity in inuce on the ielectric is less thn the chrge ensity on the pltes. For instnce, if we see tht the inuce chrge ensity is two-thirs the chrge ensity on the pltes. If no ielectric is present, then n in 0 s expecte. However, if the ielectric is replce y n electricl conuctor, for which E 0, then Eqution inictes tht E 0 E in ; this correspons to in. Tht is, the surfce chrge inuce on the conuctor is equl in mgnitue ut opposite in sign to tht on the pltes, resulting in net electric fiel of zero in the conuctor. in 3, σ Figure σσ in σ in σ Inuce chrge on ielectric plce etween the pltes of chrge cpcitor. Note tht the inuce chrge ensity on the ielectric is less thn the chrge ensity on the pltes. EXAMPLE 26.9 Effect of Metllic Sl A prllel-plte cpcitor hs plte seprtion n plte re A. An unchrge metllic sl of thickness is inserte miwy etween the pltes. () Fin the cpcitnce of the evice. Solution We cn solve this prolem y noting tht ny chrge tht ppers on one plte of the cpcitor must inuce chrge of equl mgnitue ut opposite sign on the ner sie of the sl, s shown in Figure Consequently, the net chrge on the sl remins zero, n the electric fiel insie the sl is zero. Hence, the cpcitor is equivlent to two cpcitors in series, ech hving plte seprtion ( )/2, s shown in Figure Using the rule for ing two cpcitors in series (Eq. 26.0), we otin C C C 2 C 0A 0A ( )/2 Note tht C pproches infinity s pproches. Why? () Show tht the cpcitnce is unffecte if the metllic sl is infinitesimlly thin. 0A ( )/2 Solution In the result for prt (), we let : 0: C lim :0 0A which is the originl cpcitnce. Figure σ ( )/2 σ σ ( )/2 σ () 0A ( )/2 ( )/2 () A prllel-plte cpcitor of plte seprtion prtilly fille with metllic sl of thickness. () The equivlent circuit of the evice in prt () consists of two cpcitors in series, ech hving plte seprtion ( )/2. ()

26 828 CHAPTER 26 Cpcitnce n Dielectrics (c) Show tht the nswer to prt () oes not epen on where the sl is inserte. Solution Let us imgine tht the sl in Figure is move upwr so tht the istnce etween the upper ege of the sl n the upper plte is. Then, the istnce etween the lower ege of the sl n the lower plte is. As in prt (), we fin the totl cpcitnce of the series comintion: C C C 2 0A C 0A 0A 0A This is the sme result s in prt (). It is inepenent of the vlue of, so it oes not mtter where the sl is locte. 0A 0A EXAMPLE 26.0 A Prtilly Fille Cpcitor A prllel-plte cpcitor with plte seprtion hs cpcitnce C 0 in the sence of ielectric. Wht is the cpcitnce when sl of ielectric mteril of ielectric constnt n thickness 26.27)? Figure is inserte etween the pltes (Fig. () () () A prllel-plte cpcitor of plte seprtion prtilly fille with ielectric of thickness /3. () The equivlent circuit of the cpcitor consists of two cpcitors connecte in series. κ κ C C 2 Solution In Exmple 26.9, we foun tht we coul insert metllic sl etween the pltes of cpcitor n consier the comintion s two cpcitors in series. The resulting cpcitnce ws inepenent of the loction of the sl. Furthermore, if the thickness of the sl pproches zero, then the cpcitnce of the system pproches the cpcitnce when the sl is sent. From this, we conclue tht we cn insert n infinitesimlly thin metllic sl nywhere etween the pltes of cpcitor without ffecting the cpcitnce. Thus, let us imgine sliing n infinitesimlly thin metllic sl long the ottom fce of the ielectric shown in Figure We cn then consier this system to e the series comintion of the two cpcitors shown in Figure 26.27: one hving plte seprtion /3 n fille with ielectric, n the other hving plte seprtion 2/3 n ir etween its pltes. From Equtions 26.5 n 26.3, the two cpcitnces re C Using Eqution 26.0 for two cpcitors comine in series, we hve C /3 C C 2 0A 2/3 0A 30A C 3 0A /3 n C A Becuse the cpcitnce without the ielectric is we see tht C 3 30A 2 2 C 0 0A 2/3 C 0 0A/,

27 Summry 829 SUMMARY A cpcitor consists of two conuctors crrying chrges of equl mgnitue ut opposite sign. The cpcitnce C of ny cpcitor is the rtio of the chrge Q on either conuctor to the potentil ifference V etween them: C Q V (26.) This reltionship cn e use in situtions in which ny two of the three vriles re known. It is importnt to rememer tht this rtio is constnt for given configurtion of conuctors ecuse the cpcitnce epens only on the geometry of the conuctors n not on n externl source of chrge or potentil ifference. The SI unit of cpcitnce is couloms per volt, or the fr (F), n F C/V. Cpcitnce expressions for vrious geometries re summrize in Tle If two or more cpcitors re connecte in prllel, then the potentil ifference is the sme cross ll of them. The equivlent cpcitnce of prllel comintion of cpcitors is C eq C C 2 C 3 (26.8) If two or more cpcitors re connecte in series, the chrge is the sme on ll of them, n the equivlent cpcitnce of the series comintion is given y C eq C C 2 C 3 (26.0) These two equtions enle you to simplify mny electric circuits y replcing multiple cpcitors with single equivlent cpcitnce. Work is require to chrge cpcitor ecuse the chrging process is equivlent to the trnsfer of chrges from one conuctor t lower electric potentil to nother conuctor t higher potentil. The work one in chrging the cpcitor to chrge Q equls the electric potentil energy U store in the cpcitor, where U Q 2 2C 2 Q V 2C(V )2 (26.) TABLE 26.2 Cpcitnce n Geometry Geometry Cpcitnce Eqution Isolte chrge sphere of rius R (secon chrge conuctor C 40R 26.2 ssume t infinity) Prllel-plte cpcitor of plte 26.3 re A n plte seprtion C A 0 Cylinricl cpcitor of length C n inner n outer rii 26.4 n, respectively 2k e ln Sphericl cpcitor with inner n outer rii n, C 26.6 respectively k e ( )

28 830 CHAPTER 26 Cpcitnce n Dielectrics When ielectric mteril is inserte etween the pltes of cpcitor, the cpcitnce increses y imensionless fctor, clle the ielectric constnt: C C 0 (26.4) where C 0 is the cpcitnce in the sence of the ielectric. The increse in cpcitnce is ue to ecrese in the mgnitue of the electric fiel in the presence of the ielectric n to corresponing ecrese in the potentil ifference etween the pltes if we ssume tht the chrging ttery is remove from the circuit efore the ielectric is inserte. The ecrese in the mgnitue of E rises from n internl electric fiel prouce y ligne ipoles in the ielectric. This internl fiel prouce y the ipoles opposes the pplie fiel ue to the cpcitor pltes, n the result is reuction in the net electric fiel. The electric ipole moment p of n electric ipole hs mgnitue p 2q (26.6) The irection of the electric ipole moment vector is from the negtive chrge towr the positive chrge. The torque cting on n electric ipole in uniform electric fiel E is p E (26.8) The potentil energy of n electric ipole in uniform externl electric fiel E is U p E (26.20) Prolem-Solving Hints Cpcitors Be creful with units. When you clculte cpcitnce in frs, mke sure tht istnces re expresse in meters n tht you use the SI vlue of 0. When checking consistency of units, rememer tht the unit for electric fiels cn e either N/C or V/m. When two or more cpcitors re connecte in prllel, the potentil ifference cross ech is the sme. The chrge on ech cpcitor is proportionl to its cpcitnce; hence, the cpcitnces cn e e irectly to give the equivlent cpcitnce of the prllel comintion. The equivlent cpcitnce is lwys lrger thn the iniviul cpcitnces. When two or more cpcitors re connecte in series, they crry the sme chrge, n the sum of the potentil ifferences equls the totl potentil ifference pplie to the comintion. The sum of the reciprocls of the cpcitnces equls the reciprocl of the equivlent cpcitnce, which is lwys less thn the cpcitnce of the smllest iniviul cpcitor. A ielectric increses the cpcitnce of cpcitor y fctor (the ielectric constnt) over its cpcitnce when ir is etween the pltes. For prolems in which ttery is eing connecte or isconnecte, note whether moifictions to the cpcitor re me while it is connecte to the ttery or fter it hs een isconnecte. If the cpcitor remins connecte to the ttery, the voltge cross the cpcitor remins unchnge (equl to the ttery voltge), n the chrge is proportionl to the cpci-

29 Prolems 83 tnce, lthough it my e moifie (for instnce, y the insertion of ielectric). If you isconnect the cpcitor from the ttery efore mking ny moifictions to the cpcitor, then its chrge remins fixe. In this cse, s you vry the cpcitnce, the voltge cross the pltes chnges ccoring to the expression V Q /C. QUESTIONS. If you were ske to esign cpcitor in sitution for which smll size n lrge cpcitnce were require, wht fctors woul e importnt in your esign? 2. The pltes of cpcitor re connecte to ttery. Wht hppens to the chrge on the pltes if the connecting wires re remove from the ttery? Wht hppens to the chrge if the wires re remove from the ttery n connecte to ech other? 3. A fr is very lrge unit of cpcitnce. Clculte the length of one sie of squre, ir-fille cpcitor tht hs plte seprtion of m. Assume tht it hs cpcitnce of F. 4. A pir of cpcitors re connecte in prllel, while n ienticl pir re connecte in series. Which pir woul e more ngerous to hnle fter eing connecte to the sme voltge source? Explin. 5. If you re given three ifferent cpcitors C, C 2, C 3, how mny ifferent comintions of cpcitnce cn you prouce? 6. Wht vntge might there e in using two ienticl cpcitors in prllel connecte in series with nother ienticl prllel pir rther thn single cpcitor? 7. Is it lwys possile to reuce comintion of cpcitors to one equivlent cpcitor with the rules we hve evelope? Explin. 8. Becuse the net chrge in cpcitor is lwys zero, wht oes cpcitor store? 9. Becuse the chrges on the pltes of prllel-plte cpcitor re of opposite sign, they ttrct ech other. Hence, it woul tke positive work to increse the plte seprtion. Wht hppens to the externl work one in this process? 0. Explin why the work neee to move chrge Q through potentil ifference V is W Q V, wheres the energy store in chrge cpcitor is U 2 Q V. Where oes the 2 fctor come from?. If the potentil ifference cross cpcitor is oule, y wht fctor oes the store energy chnge? 2. Why is it ngerous to touch the terminls of highvoltge cpcitor even fter the pplie voltge hs een turne off? Wht cn e one to mke the cpcitor sfe to hnle fter the voltge source hs een remove? 3. Descrie how you cn increse the mximum operting voltge of prllel-plte cpcitor for fixe plte seprtion. 4. An ir-fille cpcitor is chrge, isconnecte from the power supply, n, finlly, connecte to voltmeter. Explin how n why the voltge reing chnges when ielectric is inserte etween the pltes of the cpcitor. 5. Using the polr molecule escription of ielectric, explin how ielectric ffects the electric fiel insie cpcitor. 6. Explin why ielectric increses the mximum operting voltge of cpcitor even though the physicl size of the cpcitor oes not chnge. 7. Wht is the ifference etween ielectric strength n the ielectric constnt? 8. Explin why wter molecule is permnently polrize. Wht type of molecule hs no permnent polriztion? 9. If ielectric-fille cpcitor is hete, how oes its cpcitnce chnge? (Neglect therml expnsion n ssume tht the ipole orienttions re temperture epenent.) PROBLEMS, 2, 3 = strightforwr, intermeite, chllenging = full solution ville in the Stuent Solutions Mnul n Stuy Guie WEB = solution poste t = Computer useful in solving prolem = Interctive Physics = pire numericl/symolic prolems Section 26. Definition of Cpcitnce. () How much chrge is on ech plte of 4.00-F cpcitor when it is connecte to 2.0-V ttery? () If this sme cpcitor is connecte to.50-v ttery, wht chrge is store? 2. Two conuctors hving net chrges of 0.0 C n 0.0 C hve potentil ifference of 0.0 V. Determine () the cpcitnce of the system n () the potentil ifference etween the two conuctors if the chrges on ech re increse to 00 C n 00 C.

30 832 CHAPTER 26 Cpcitnce n Dielectrics WEB WEB Section 26.2 Clculting Cpcitnce 3. An isolte chrge conucting sphere of rius 2.0 cm cretes n electric fiel of N/C t istnce 2.0 cm from its center. () Wht is its surfce chrge ensity? () Wht is its cpcitnce? 4. () If rop of liqui hs cpcitnce.00 pf, wht is its rius? () If nother rop hs rius 2.00 mm, wht is its cpcitnce? (c) Wht is the chrge on the smller rop if its potentil is 00 V? 5. Two conucting spheres with imeters of m n.00 m re seprte y istnce tht is lrge compre with the imeters. The spheres re connecte y thin wire n re chrge to 7.00 C. () How is this totl chrge shre etween the spheres? (Neglect ny chrge on the wire.) () Wht is the potentil of the system of spheres when the reference potentil is tken to e V 0 t r? 6. Regring the Erth n clou lyer 800 m ove the Erth s the pltes of cpcitor, clculte the cpcitnce if the clou lyer hs n re of.00 km 2. Assume tht the ir etween the clou n the groun is pure n ry. Assume tht chrge uils up on the clou n on the groun until uniform electric fiel with mgnitue of N/C throughout the spce etween them mkes the ir rek own n conuct electricity s lightning olt. Wht is the mximum chrge the clou cn hol? 7. An ir-fille cpcitor consists of two prllel pltes, ech with n re of 7.60 cm 2, seprte y istnce of.80 mm. If 20.0-V potentil ifference is pplie to these pltes, clculte () the electric fiel etween the pltes, () the surfce chrge ensity, (c) the cpcitnce, n () the chrge on ech plte. 8. A -megit computer memory chip contins mny 60.0-fF cpcitors. Ech cpcitor hs plte re of m 2. Determine the plte seprtion of such cpcitor (ssume prllel-plte configurtion). The chrcteristic tomic imeter is 0 0 m 0.00 nm. Express the plte seprtion in nnometers. 9. When potentil ifference of 50 V is pplie to the pltes of prllel-plte cpcitor, the pltes crry surfce chrge ensity of 30.0 nc/cm 2. Wht is the spcing etween the pltes? 0. A vrile ir cpcitor use in tuning circuits is me of N semicirculr pltes ech of rius R n positione istnce from ech other. As shown in Figure P26.0, secon ienticl set of pltes is enmeshe with its pltes hlfwy etween those of the first set. The secon set cn rotte s unit. Determine the cpcitnce s function of the ngle of rottion, where correspons to the mximum cpcitnce.. A 50.0-m length of coxil cle hs n inner conuctor tht hs imeter of 2.58 mm n crries chrge of 8.0 C. The surrouning conuctor hs n inner imeter of 7.27 mm n chrge of 8.0 C. () Wht is the cpcitnce of this cle? () Wht is 0 the potentil ifference etween the two conuctors? Assume the region etween the conuctors is ir. 2. A 20.0-F sphericl cpcitor is compose of two metllic spheres, one hving rius twice s lrge s the other. If the region etween the spheres is vcuum, etermine the volume of this region. 3. A smll oject with mss of 350 mg crries chrge of 30.0 nc n is suspene y thre etween the verticl pltes of prllel-plte cpcitor. The pltes re seprte y 4.00 cm. If the thre mkes n ngle of 5.0 with the verticl, wht is the potentil ifference etween the pltes? 4. A smll oject of mss m crries chrge q n is suspene y thre etween the verticl pltes of prllel-plte cpcitor. The plte seprtion is. If the thre mkes n ngle with the verticl, wht is the potentil ifference etween the pltes? 5. An ir-fille sphericl cpcitor is constructe with inner n outer shell rii of 7.00 n 4.0 cm, respectively. () Clculte the cpcitnce of the evice. () Wht potentil ifference etween the spheres results in chrge of 4.00 C on the cpcitor? 6. Fin the cpcitnce of the Erth. (Hint: The outer conuctor of the sphericl cpcitor my e consiere s conucting sphere t infinity where V pproches zero.) Section 26.3 Comintions of Cpcitors 7. Two cpcitors C 5.00 F n C F re connecte in prllel, n the resulting comintion is connecte to 9.00-V ttery. () Wht is the vlue of the equivlent cpcitnce of the comintion? Wht re () the potentil ifference cross ech cpcitor n (c) the chrge store on ech cpcitor? 8. The two cpcitors of Prolem 7 re now connecte in series n to 9.00-V ttery. Fin () the vlue of the equivlent cpcitnce of the comintion, () the voltge cross ech cpcitor, n (c) the chrge on ech cpcitor. 9. Two cpcitors when connecte in prllel give n equivlent cpcitnce of 9.00 pf n n equivlent c- R Figure P26.0

31 Prolems 833 WEB pcitnce of 2.00 pf when connecte in series. Wht is the cpcitnce of ech cpcitor? 20. Two cpcitors when connecte in prllel give n equivlent cpcitnce of C p n n equivlent cpcitnce of C s when connecte in series. Wht is the cpcitnce of ech cpcitor? 2. Four cpcitors re connecte s shown in Figure P26.2. () Fin the equivlent cpcitnce etween points n. () Clculte the chrge on ech cpcitor if V 5.0 V. 5.0 µf µ 3.00 µf µ 6.00 µf µ Figure P µf µ 24. Accoring to its esign specifiction, the timer circuit elying the closing of n elevtor oor is to hve cpcitnce of 32.0 F etween two points A n B. () When one circuit is eing constructe, the inexpensive cpcitor instlle etween these two points is foun to hve cpcitnce 34.8 F. To meet the specifiction, one itionl cpcitor cn e plce etween the two points. Shoul it e in series or in prllel with the 34.8-F cpcitor? Wht shoul e its cpcitnce? () The next circuit comes own the ssemly line with cpcitnce 29.8 F etween A n B. Wht itionl cpcitor shoul e instlle in series or in prllel in tht circuit, to meet the specifiction? 25. The circuit in Figure P26.25 consists of two ienticl prllel metllic pltes connecte y ienticl metllic springs to 00-V ttery. With the switch open, the pltes re unchrge, re seprte y istnce 8.00 mm, n hve cpcitnce C 2.00 F. When the switch is close, the istnce etween the pltes ecreses y fctor of () How much chrge collects on ech plte n () wht is the spring constnt for ech spring? (Hint: Use the result of Prolem 35.) 22. Evlute the equivlent cpcitnce of the configurtion shown in Figure P All the cpcitors re ienticl, n ech hs cpcitnce C. k k C C C C C C S 23. Consier the circuit shown in Figure P26.23, where C 6.00 F, C F, n V 20.0 V. Cpcitor C is first chrge y the closing of switch S. Switch S is then opene, n the chrge cpcitor is connecte to the unchrge cpcitor y the closing of S 2. Clculte the initil chrge cquire y C n the finl chrge on ech. V Figure P26.22 S C C 2 S 2 Figure P Figure P26.26 shows six concentric conucting spheres, A, B, C, D, E, n F hving rii R, 2R, 3R, 4R, 5R, n 6R, respectively. Spheres B n C re connecte y conucting wire, s re spheres D n E. Determine the equivlent cpcitnce of this system. 27. A group of ienticl cpcitors is connecte first in series n then in prllel. The comine cpcitnce in prllel is 00 times lrger thn for the series connection. How mny cpcitors re in the group? 28. Fin the equivlent cpcitnce etween points n for the group of cpcitors connecte s shown in Figure P26.28 if C 5.00 F, C F, n C F. 29. For the network escrie in the previous prolem if the potentil ifference etween points n is 60.0 V, wht chrge is store on C 3? V Figure P26.25

32 834 CHAPTER 26 Cpcitnce n Dielectrics A B C D E F Figure P26.26 C C C 2 C 3 C 2 WEB energy store in the two cpcitors. () Wht potentil ifference woul e require cross the sme two cpcitors connecte in series so tht the comintion stores the sme energy s in prt ()? Drw circuit igrm of this circuit. 33. A prllel-plte cpcitor is chrge n then isconnecte from ttery. By wht frction oes the store energy chnge (increse or ecrese) when the plte seprtion is oule? 34. A uniform electric fiel E V/m exists within certin region. Wht volume of spce contins n energy equl to J? Express your nswer in cuic meters n in liters. 35. A prllel-plte cpcitor hs chrge Q n pltes of re A. Show tht the force exerte on ech plte y the other is F Q 2 /20A. (Hint: Let C 0A/x for n ritrry plte seprtion x ; then require tht the work one in seprting the two chrge pltes e W F x.) 36. Plte of prllel-plte, ir-fille cpcitor is connecte to spring hving force constnt k, n plte is fixe. They rest on tle top s shown (top view) in Figure P If chrge Q is plce on plte n chrge Q is plce on plte, y how much oes the spring expn? C 2 C 2 Figure P26.28 Prolems 28 n Fin the equivlent cpcitnce etween points n in the comintion of cpcitors shown in Figure P µfµ 7.0 µfµ 5.0 µf µ 6.0 µf µ Section 26.4 Energy Store in Chrge Cpcitor 3. () A 3.00-F cpcitor is connecte to 2.0-V ttery. How much energy is store in the cpcitor? () If the cpcitor h een connecte to 6.00-V ttery, how much energy woul hve een store? 32. Two cpcitors C 25.0 F n C F re connecte in prllel n chrge with 00-V power supply. () Drw circuit igrm n clculte the totl Figure P26.30 k 37. Review Prolem. A certin storm clou hs potentil ifference of V reltive to tree. If, uring lightning storm, 50.0 C of chrge is trnsferre through this potentil ifference n.00% of the energy is sore y the tree, how much wter (sp in the tree) initilly t 30.0 C cn e oile wy? Wter hs specific het of 4 86 J/kg C, oiling point of 00 C, n het of vporiztion of J/kg. 38. Show tht the energy ssocite with conucting sphere of rius R n chrge Q surroune y vcuum is U k e Q 2 /2R. 39. Einstein si tht energy is ssocite with mss ccoring to the fmous reltionship E mc 2. Estimte the rius of n electron, ssuming tht its chrge is istriute uniformly over the surfce of sphere of rius R n tht the mssenergy of the electron is equl to the totl energy store in the resulting nonzero electric fiel etween R n infinity. (See Prolem 38. Experimentlly, n electron nevertheless ppers to e point prticle. The electric fiel close to the electron must e escrie y quntum electroynmics, rther thn the clssicl electroynmics tht we stuy.) Figure P26.36

33 Prolems 835 Section 26.5 Cpcitors with Dielectrics 40. Fin the cpcitnce of prllel-plte cpcitor tht uses Bkelite s ielectric, if ech of the pltes hs n re of 5.00 cm 2 n the plte seprtion is 2.00 mm. 4. Determine () the cpcitnce n () the mximum voltge tht cn e pplie to Teflon-fille prllelplte cpcitor hving plte re of.75 cm 2 n plte seprtion of mm. 42. () How much chrge cn e plce on cpcitor with ir etween the pltes efore it reks own, if the re of ech of the pltes is 5.00 cm 2? () Fin the mximum chrge if polystyrene is use etween the pltes inste of ir. 43. A commercil cpcitor is constructe s shown in Figure This prticulr cpcitor is rolle from two strips of luminum seprte y two strips of prffincote pper. Ech strip of foil n pper is 7.00 cm wie. The foil is mm thick, n the pper is mm thick n hs ielectric constnt of Wht length shoul the strips e if cpcitnce of F is esire? (Use the prllel-plte formul.) 44. The supermrket sells rolls of luminum foil, plstic wrp, n wxe pper. Descrie cpcitor me from supermrket mterils. Compute orer-of-mgnitue estimtes for its cpcitnce n its rekown voltge. 45. A cpcitor tht hs ir etween its pltes is connecte cross potentil ifference of 2.0 V n stores 48.0 C of chrge. It is then isconnecte from the source while still chrge. () Fin the cpcitnce of the cpcitor. () A piece of Teflon is inserte etween the pltes. Fin its new cpcitnce. (c) Fin the voltge n chrge now on the cpcitor. 46. A prllel-plte cpcitor in ir hs plte seprtion of.50 cm n plte re of 25.0 cm 2. The pltes re chrge to potentil ifference of 250 V n isconnecte from the source. The cpcitor is then immerse in istille wter. Determine () the chrge on the pltes efore n fter immersion, () the cpcitnce n voltge fter immersion, n (c) the chnge in energy of the cpcitor. Neglect the conuctnce of the liqui. 47. A conucting sphericl shell hs inner rius n outer rius c. The spce etween these two surfces is fille with ielectric for which the ielectric constnt is etween n, n 2 etween n c (Fig. P26.47). Determine the cpcitnce of this system. 48. A wfer of titnium ioxie hs n re of.00 cm 2 n thickness of 0.00 mm. Aluminum is evporte on the prllel fces to form prllel-plte cpcitor. () Clculte the cpcitnce. () When the cpcitor is chrge with 2.0-V ttery, wht is the mgnitue of chrge elivere to ech plte? (c) For the sitution in prt (), wht re the free n inuce surfce chrge ensities? () Wht is the mgnitue E of the electric fiel? ( 73) 49. Ech cpcitor in the comintion shown in Figure P26.49 hs rekown voltge of 5.0 V. Wht is the rekown voltge of the comintion? 20.0 µf µ 20.0 µf µ Q κ 2 κ 0.0 µf µ 20.0 µf µ 20.0 µf µ (Optionl) Section 26.6 Electric Dipole in n Electric Fiel 50. A smll rigi oject crries positive n negtive 3.50-nC chrges. It is oriente so tht the positive chrge is t the point (.20 mm,.0 mm) n the negtive chrge is t the point (.40 mm,.30 mm). () Fin the electric ipole moment of the oject. The oject is plce in n electric fiel E (7 800i 4 900j) N/C. () Fin the torque cting on the oject. (c) Fin the potentil energy of the oject in this orienttion. () If the orienttion of the oject cn chnge, fin the ifference etween its mximum n its minimum potentil energies. 5. A smll oject with electric ipole moment p is plce in nonuniform electric fiel E E(x)i. Tht is, the fiel is in the x irection, n its mgnitue epens on the coorinte x. Let represent the ngle etween the ipole moment n the x irection. () Prove tht the ipole experiences net force F p(e/x) cos in the irection towr which the fiel increses. () Consier the fiel crete y sphericl lloon centere t the origin. The lloon hs rius of 5.0 cm n crries chrge of 2.00 C. Evlute E/x t the point (6 cm, 0, 0). Assume tht wter roplet t this point hs n inuce ipole moment of (6.30i) nc m. Fin the force on it. (Optionl) Section 26.7 An Atomic Description of Dielectrics 52. A etector of rition clle GeigerMuller counter consists of close, hollow, conucting cyliner with Q c Figure P26.47 Figure P26.49

34 836 CHAPTER 26 Cpcitnce n Dielectrics fine wire long its xis. Suppose tht the internl imeter of the cyliner is 2.50 cm n tht the wire long the xis hs imeter of mm. If the ielectric strength of the gs etween the centrl wire n the cyliner is V/m, clculte the mximum voltge tht cn e pplie etween the wire n the cyliner efore rekown occurs in the gs. 53. The generl form of Guss s lw escries how chrge cretes n electric fiel in mteril, s well s in vcuum. It is E A q where 0 is the permittivity of the mteril. () A sheet with chrge Q uniformly istriute over its re A is surroune y ielectric. Show tht the sheet cretes uniform electric fiel with mgnitue E Q /2A t nery points. () Two lrge sheets of re A crrying opposite chrges of equl mgnitue Q re smll istnce prt. Show tht they crete uniform electric fiel of mgnitue E Q /A etween them. (c) Assume tht the negtive plte is t zero potentil. Show tht the positive plte is t potentil Q /A. () Show tht the cpcitnce of the pir of pltes is A/ A0/. ADDITIONAL PROBLEMS 54. For the system of cpcitors shown in Figure P26.54, fin () the equivlent cpcitnce of the system, () the potentil ifference cross ech cpcitor, (c) the chrge on ech cpcitor, n () the totl energy store y the group µf µ 6.00 µf µ 2.00 µf µ 4.00 µf µ WEB 56. A 2.00-nF prllel-plte cpcitor is chrge to n initil potentil ifference V i 00 V n then isolte. The ielectric mteril etween the pltes is mic ( 5.00). () How much work is require to withrw the mic sheet? () Wht is the potentil ifference of the cpcitor fter the mic is withrwn? 57. A prllel-plte cpcitor is constructe using ielectric mteril whose ielectric constnt is 3.00 n whose ielectric strength is V/m. The esire cpcitnce is F, n the cpcitor must withstn mximum potentil ifference of V. Fin the minimum re of the cpcitor pltes. 58. A prllel-plte cpcitor is constructe using three ielectric mterils, s shown in Figure P You my ssume tht W. () Fin n expression for the cpcitnce of the evice in terms of the plte re A n,, 2, n 3. () Clculte the cpcitnce using the vlues A.00 cm 2, 2.00 mm, 4.90, , n κ κ /2 59. A conucting sl of thickness n re A is inserte into the spce etween the pltes of prllel-plte cpcitor with spcing s n surfce re A, s shown in Figure P The sl is not necessrily hlfwy etween the cpcitor pltes. Wht is the cpcitnce of the system? κ 2 κ 3 Figure P26.58 A / V Figure P26.54 s A Figure P Consier two long, prllel, n oppositely chrge wires of rius with their centers seprte y istnce D. Assuming the chrge is istriute uniformly on the surfce of ech wire, show tht the cpcitnce per unit length of this pir of wires is C 0 ln D 60. () Two spheres hve rii n n their centers re istnce prt. Show tht the cpcitnce of this system is C 40 2 provie tht is lrge compre with n. (Hint: Becuse the spheres re fr prt, ssume tht the

35 Prolems 837 chrge on one sphere oes not pertur the chrge istriution on the other sphere. Thus, the potentil of ech sphere is expresse s tht of symmetric chrge istriution, V k e Q /r, n the totl potentil t ech sphere is the sum of the potentils ue to ech sphere. () Show tht s pproches infinity the ove result reuces to tht of two isolte spheres in series. 6. When certin ir-fille prllel-plte cpcitor is connecte cross ttery, it cquires chrge (on ech plte) of q 0. While the ttery connection is mintine, ielectric sl is inserte into n fills the region etween the pltes. This results in the ccumultion of n itionl chrge q on ech plte. Wht is the ielectric constnt of the sl? 62. A cpcitor is constructe from two squre pltes of sies n seprtion. A mteril of ielectric constnt is inserte istnce x into the cpcitor, s shown in Figure P () Fin the equivlent cpcitnce of the evice. () Clculte the energy store in the cpcitor if the potentil ifference is V. (c) Fin the irection n mgnitue of the force exerte on the ielectric, ssuming constnt potentil ifference V. Neglect friction. () Otin numericl vlue for the force ssuming tht 5.00 cm, V V, 2.00 mm, n the ielectric is glss ( 4.50). (Hint: The system cn e consiere s two cpcitors connecte in prllel.) 64. When consiering the energy supply for n utomoile, the energy per unit mss of the energy source is n importnt prmeter. Using the following t, compre the energy per unit mss ( J/kg) for gsoline, leci tteries, n cpcitors. (The mpere A will e introuce in Chpter 27 n is the SI unit of electric current. A C/s.) Gsoline: Btu/gl; ensity 670 kg/m 3 Leci ttery: 2.0 V; 00 A h; mss 6.0 kg Cpcitor: potentil ifference t full chrge 2.0 V; cpcitnce 0.00 F; mss 0.00 kg 65. An isolte cpcitor of unknown cpcitnce hs een chrge to potentil ifference of 00 V. When the chrge cpcitor is then connecte in prllel to n unchrge 0.0-F cpcitor, the voltge cross the comintion is 30.0 V. Clculte the unknown cpcitnce. 66. A certin electronic circuit clls for cpcitor hving cpcitnce of.20 pf n rekown potentil of 000 V. If you hve supply of 6.00-pF cpcitors, ech hving rekown potentil of 200 V, how coul you meet this circuit requirement? 67. In the rrngement shown in Figure P26.67, potentil ifference V is pplie, n C is juste so tht the voltmeter etween points n res zero. This lnce occurs when C 4.00 F. If C F n C F, clculte the vlue of C 2. κ x C 4 C Figure P26.62 Prolems 62 n 63. V V 63. A cpcitor is constructe from two squre pltes of sies n seprtion, s suggeste in Figure P You my ssume tht is much less thn. The pltes crry chrges Q 0 n Q 0. A lock of metl hs with, length, n thickness slightly less thn. It is inserte istnce x into the cpcitor. The chrges on the pltes re not isture s the lock slies in. In sttic sitution, metl prevents n electric fiel from penetrting it. The metl cn e thought of s perfect ielectric, with :. () Clculte the store energy s function of x. () Fin the irection n mgnitue of the force tht cts on the metllic lock. (c) The re of the vncing front fce of the lock is essentilly equl to. Consiering the force on the lock s cting on this fce, fin the stress (force per re) on it. () For comprison, express the energy ensity in the electric fiel etween the cpcitor pltes in terms of Q 0,,, n 0. C 3 Figure P It is possile to otin lrge potentil ifferences y first chrging group of cpcitors connecte in prllel n then ctivting switch rrngement tht in effect isconnects the cpcitors from the chrging source n from ech other n reconnects them in series rrngement. The group of chrge cpcitors is then ischrge in series. Wht is the mximum potentil ifference tht cn e otine in this mnner y using ten cpcitors ech of 500 F n chrging source of 800 V? 69. A prllel-plte cpcitor of plte seprtion is chrge to potentil ifference V 0. A ielectric sl c C 2

36 838 CHAPTER 26 Cpcitnce n Dielectrics of thickness n ielectric constnt is introuce etween the pltes while the ttery remins connecte to the pltes. () Show tht the rtio of energy store fter the ielectric is introuce to the energy store in the empty cpcitor is U/U 0. Give physicl explntion for this increse in store energy. () Wht hppens to the chrge on the cpcitor? (Note tht this sitution is not the sme s Exmple 26.7, in which the ttery ws remove from the circuit efore the ielectric ws introuce.) 70. A prllel-plte cpcitor with pltes of re A n plte seprtion hs the region etween the pltes fille with two ielectric mterils s in Figure P Assume tht V L n tht V W. () Determine the cpcitnce n () show tht when 2 your result ecomes the sme s tht for cpcitor contining single ielectric, C 0A/. κ L W pcitors re isconnecte from the ttery n from ech other. They re then connecte positive plte to negtive plte n negtive plte to positive plte. Clculte the resulting chrge on ech cpcitor. 73. The inner conuctor of coxil cle hs rius of mm, n the outer conuctor s insie rius is 3.00 mm. The spce etween the conuctors is fille with polyethylene, which hs ielectric constnt of 2.30 n ielectric strength of V/m. Wht is the mximum potentil ifference tht this cle cn withstn? 74. You re optimizing coxil cle esign for mjor mnufcturer. Show tht for given outer conuctor rius, mximum potentil ifference cpility is ttine when the rius of the inner conuctor is /e where e is the se of nturl logrithms. 75. Clculte the equivlent cpcitnce etween the points n in Figure P Note tht this is not simple series or prllel comintion. (Hint: Assume potentil ifference V etween points n. Write expressions for V in terms of the chrges n cpcitnces for the vrious possile pthwys from to, n require conservtion of chrge for those cpcitor pltes tht re connecte to ech other.) κ 2 Figure P µf µ 7. A verticl prllel-plte cpcitor is hlf fille with ielectric for which the ielectric constnt is 2.00 (Fig. P26.7). When this cpcitor is positione horizontlly, wht frction of it shoul e fille with the sme ielectric (Fig. P26.7) so tht the two cpcitors hve equl cpcitnce? 2.00 µf µ 8.00 µf µ 4.00 µfµ 2.00 µf µ Figure P Determine the effective cpcitnce of the comintion shown in Figure P (Hint: Consier the symmetry involve!) C 2C () Figure P26.7 () 3C 72. Cpcitors C 6.00 F n C F re chrge s prllel comintion cross 250-V ttery. The c- C 2C Figure P26.76

37 Answers to Quick Quizzes 839 ANSWERS TO QUICK QUIZZES 26. () ecuse the plte seprtion is ecrese. Cpcitnce epens only on how cpcitor is constructe n not on the externl circuit Zero. If you construct sphericl gussin surfce outsie n concentric with the cpcitor, the net chrge insie the surfce is zero. Applying Guss s lw to this configurtion, we fin tht E 0 t points outsie the cpcitor For given voltge, the energy store in cpcitor is proportionl to C: U C(V ) 2 /2. Thus, you wnt to mximize the equivlent cpcitnce. You o this y connecting the three cpcitors in prllel, so tht the cpcitnces () C ecreses (Eq. 26.3). () Q stys the sme ecuse there is no plce for the chrge to flow. (c) E remins constnt (see Eq n the prgrph following it). () V increses ecuse V Q /C, Q is constnt (prt ), n C ecreses (prt ). (e) The energy store in the cpcitor is proportionl to oth Q n V (Eq. 26.) n thus increses. The itionl energy comes from the work you o in pulling the two pltes prt () C ecreses (Eq. 26.3). () Q ecreses. The ttery supplies constnt potentil ifference V ; thus, chrge must flow out of the cpcitor if C Q /V is to e- crese. (c) E ecreses ecuse the chrge ensity on the pltes ecreses. () V remins constnt ecuse of the presence of the ttery. (e) The energy store in the cpcitor ecreses (Eq. 26.) It increses. The ielectric constnt of woo (n of ll other insulting mterils, for tht mtter) is greter thn ; therefore, the cpcitnce increses (Eq. 26.4). This increse is sense y the stu-finer s specil circuitry, which cuses n inictor on the evice to light up () C increses (Eq. 26.4). () Q increses. Becuse the ttery mintins constnt V, Q must increse if C (Q /V ) increses. (c) E etween the pltes remins constnt ecuse V E n neither V nor chnges. The electric fiel ue to the chrges on the pltes increses ecuse more chrge hs flowe onto the pltes. The inuce surfce chrges on the ielectric crete fiel tht opposes the increse in the fiel cuse y the greter numer of chrges on the pltes. () The ttery mintins constnt V. (e) The energy store in the cpcitor increses (Eq. 26.). You woul hve to push the ielectric into the cpcitor, just s you woul hve to o positive work to rise mss n increse its grvittionl potentil energy.

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