CPCITNCE ND DIELECTRIC 24 LERNING GOL By tuying thi chpter, you will lern:?the energy ue in cmer flh unit i tore in cpcitor, which conit of two cloely pce conuctor tht crry oppoite chrge. If the mount of chrge on the conuctor i oule, y wht fctor oe the tore energy incree? The nture of cpcitor, n how to clculte quntity tht meure their ility to tore chrge. How to nlyze cpcitor connecte in network. How to clculte the mount of energy tore in cpcitor. Wht ielectric re, n how they mke cpcitor more effective. When you et n ol-fhione pring mouetrp or pull ck the tring of n rcher ow, you re toring mechnicl energy eltic potentil energy. cpcitor i evice tht tore electric potentil energy n electric chrge. To mke cpcitor, jut inulte two conuctor from ech other. To tore energy in thi evice, trnfer chrge from one conuctor to the other o tht one h negtive chrge n the other h n equl mount of poitive chrge. Work mut e one to move the chrge through the reulting potentil ifference etween the conuctor, n the work one i tore electric potentil energy. Cpcitor hve tremenou numer of prcticl ppliction in evice uch electronic flh unit for photogrphy, pule ler, ir g enor for cr, n rio n televiion receiver. We ll encounter mny of thee ppliction in lter chpter (prticulrly Chpter 3, in which we ll ee the crucil role plye y cpcitor in the lternting-current circuit tht perve our technologicl ociety). In thi chpter, however, our emphi i on the funmentl propertie of cpcitor. For prticulr cpcitor, the rtio of the chrge on ech conuctor to the potentil ifference etween the conuctor i contnt, clle the cpcitnce. The cpcitnce epen on the ize n hpe of the conuctor n on the inulting mteril (if ny) etween them. Compre to the ce in which there i only vcuum etween the conuctor, the cpcitnce incree when n inulting mteril ( ielectric) i preent. Thi hppen ecue reitriution of chrge, clle polriztion, tke plce within the inulting mteril. tuying polriztion will give u e inight into the electricl propertie of mtter. Cpcitor lo give u new wy to think out electric potentil energy. The energy tore in chrge cpcitor i relte to the electric fiel in the pce etween the conuctor. We will ee tht electric potentil energy cn e regre eing tore in the fiel itelf. The ie tht the electric fiel i itelf torehoue of energy i t the hert of the theory of electromgnetic wve n our moern unertning of the nture of light, to e icue in Chpter 32. 8
86 CHPTER 24 Cpcitnce n Dielectric 24. Cpcitor n Cpcitnce 87 24. ny two conuctor n inulte from ech nother form cpcitor. Conuctor ONLINE 2 E Conuctor..6 Electric Potentil: ulittive Introuction.2. n.2.3 Electric Potentil, Fiel n, Force 24. Cpcitor n Cpcitnce ny two conuctor eprte y n inultor (or vcuum) form cpcitor (Fig. 24.). In mot prcticl ppliction, ech conuctor initilly h zero net chrge n electron re trnferre from one conuctor to the other; thi i clle chrging the cpcitor. Then the two conuctor hve chrge with equl mgnitue n oppoite ign, n the net chrge on the cpcitor whole remin zero. We will ume throughout thi chpter tht thi i the ce. When we y tht cpcitor h chrge, or tht chrge i tore on the cpcitor, we men tht the conuctor t higher potentil h chrge n the conuctor t lower potentil h chrge 2 (uming tht i poitive). Keep thi in min in the following icuion n exmple. In circuit igrm cpcitor i repreente y either of thee ymol: In either ymol the verticl line (tright or curve) repreent the conuctor n the horizontl line repreent wire connecte to either conuctor. One common wy to chrge cpcitor i to connect thee two wire to oppoite terminl of ttery. Once the chrge n 2 re etlihe on the conuctor, the ttery i iconnecte. Thi give fixe potentil ifference V etween the conuctor (tht i, the potentil of the poitively chrge conuctor with repect to the negtively chrge conuctor ) tht i jut equl to the voltge of the ttery. The electric fiel t ny point in the region etween the conuctor i proportionl to the mgnitue of chrge on ech conuctor. It follow tht the potentil ifference V etween the conuctor i lo proportionl to. If we oule the mgnitue of chrge on ech conuctor, the chrge enity t ech point oule, the electric fiel t ech point oule, n the potentil ifference etween conuctor oule; however, the rtio of chrge to potentil ifference oe not chnge. Thi rtio i clle the cpcitnce C of the cpcitor: C V (efinition of cpcitnce) (24.) The I unit of cpcitnce i clle one fr ( F), in honor of the 9th-century Englih phyicit Michel Fry. From Eq. (24.), one fr i equl to one coulom per volt C/V 2 : F fr C/V coulom/volt CUTION Cpcitnce v. coulom Don t confue the ymol C for cpcitnce (which i lwy in itlic) with the revition C for coulom (which i never itlicize). The greter the cpcitnce C of cpcitor, the greter the mgnitue of chrge on either conuctor for given potentil ifference V n hence the greter the mount of tore energy. (Rememer tht potentil i potentil energy per unit chrge.) Thu cpcitnce i meure of the ility of cpcitor to tore energy. We will ee tht the vlue of the cpcitnce epen only on the hpe n ize of the conuctor n on the nture of the inulting mteril etween them. (The ove remrk out cpcitnce eing inepenent of n V o not pply to certin pecil type of inulting mteril. We won t icu thee mteril in thi ook, however.) Clculting Cpcitnce: Cpcitor in Vcuum We cn clculte the cpcitnce C of given cpcitor y fining the potentil ifference V etween the conuctor for given mgnitue of chrge n then uing Eq. (24.). For now we ll conier only cpcitor in vcuum; tht i, we ll ume tht the conuctor tht mke up the cpcitor re eprte y empty pce. The implet form of cpcitor conit of two prllel conucting plte, ech with re, eprte y itnce tht i mll in comprion with their imenion (Fig. 24.2). When the plte re chrge, the electric fiel i lmot completely loclize in the region etween the plte (Fig. 24.2). we icue in Exmple 22.8 (ection 22.4), the fiel etween uch plte i eentilly uniform, n the chrge on the plte re uniformly itriute over their oppoing urfce. We cll thi rrngement prllel-plte cpcitor. We worke out the electric-fiel mgnitue E for thi rrngement in Exmple 2.3 (ection 2.) uing the principle of uperpoition of electric fiel n gin in Exmple 22.8 (ection 22.4) uing Gu lw. It woul e goo ie to review thoe exmple. We foun tht E /P 0, where i the mgnitue (olute vlue) of the urfce chrge enity on ech plte. Thi i equl to the mgnitue of the totl chrge on ech plte ivie y the re of the plte, or /, o the fiel mgnitue E cn e expree The fiel i uniform n the itnce etween the plte i, o the potentil ifference (voltge) etween the two plte i From thi we ee tht the cpcitnce C of prllel-plte cpcitor in vcuum i (cpcitnce of prllel-plte cpcitor in vcuum) (24.2) The cpcitnce epen only on the geometry of the cpcitor; it i irectly proportionl to the re of ech plte n inverely proportionl to their eprtion. The quntitie n re contnt for given cpcitor, n P 0 i univerl contnt. Thu in vcuum the cpcitnce C i contnt inepenent of the chrge on the cpcitor or the potentil ifference etween the plte. If one of the cpcitor plte i flexile, the cpcitnce C chnge the plte eprtion chnge. Thi i the operting principle of conener microphone (Fig. 24.3). When mtter i preent etween the plte, it propertie ffect the cpcitnce. We will return to thi topic in ection 24.4. Menwhile, we remrk tht if the pce contin ir t tmopheric preure inte of vcuum, the cpcitnce iffer from the preiction of Eq. (24.2) y le thn 0.06%. In Eq. (24.2), if i in qure meter n in meter, C i in fr. The unit of re C 2 /N # m 2, o we ee tht P 0 C P V 0 E P 0 P 0 V E P 0 F C 2 /N # m C 2 /J Becue V J/C (energy per unit chrge), thi i conitent with our efinition F C/V. Finlly, the unit of P cn e expree C 2 /N # m 2 0 F/m, o P 0 8.8 3 0 22 F/m 24.2 chrge prllel-plte cpcitor. () rrngement of the cpcitor plte Potentil ifference V Wire Wire () ie view of the electric fiel E Plte, re When the eprtion of the plte i mll compre to their ize, the fringing of the fiel i light. Plte, re 24.3 Inie conener microphone i cpcitor with one rigi plte n one flexile plte. The two plte re kept t contnt potentil ifference V. oun wve cue the flexile plte to move ck n forth, vrying the cpcitnce C n cuing chrge to flow to n from the cpcitor in ccornce with the reltionhip C /V. Thu oun wve i converte to chrge flow tht cn e mplifie n recore igitlly. E
88 CHPTER 24 Cpcitnce n Dielectric 24. Cpcitor n Cpcitnce 89 24.4 commercil cpcitor i lele with the vlue of it cpcitnce. For thee cpcitor, C 2200 mf, 000 mf, n 470 mf. Exmple 24. ize of -F cpcitor Thi reltionhip i ueful in cpcitnce clcultion, n it lo help u to verify tht Eq. (24.2) i imenionlly conitent. One fr i very lrge cpcitnce, the following exmple how. In mny ppliction the mot convenient unit of cpcitnce re the microfr mf 0 26 F 2 n the picofr pf 0 22 F 2. For exmple, the flh unit in point-n-hoot cmer ue cpcitor of few hunre microfr (Fig. 24.4), while cpcitnce in rio tuning circuit re typiclly from 0 to 00 picofr. For ny cpcitor in vcuum, the cpcitnce C epen only on the hpe, imenion, n eprtion of the conuctor tht mke up the cpcitor. If the conuctor hpe re more complex thn thoe of the prllel-plte cpcitor, the expreion for cpcitnce i more complicte thn in Eq. (24.2). In the following exmple we how how to clculte C for two other conuctor geometrie. prllel-plte cpcitor h cpcitnce of.0 F. If the plte EVLUTE: Thi correpon to qure out 0 km (out re.0 mm prt, wht i the re of the plte? 6 mile) on ie! Thi re i out thir lrger thn Mnhttn Iln. Clerly thi i not very prcticl eign for cpcitor. In fct, it now poile to mke -F cpcitor few centimeter on ie. The trick i to hve n pproprite utnce IDENTIFY: Thi prolem ue the reltionhip mong the cpcitnce, plte eprtion, n plte re (our trget vrile) for etween the plte rther thn vcuum. We ll explore thi further prllel-plte cpcitor. in ection 24.4. ET UP: We re given the vlue of C n for prllel-plte cpcitor, o we ue Eq. (24.2) n olve for the trget vrile. EXECUTE: From Eq. (24.2), the re i C.0 F 2.0 3 023 m 2 P 0 8.8 3 0 22 F/m. 3 0 8 m 2 Exmple 24.2 Propertie of prllel-plte cpcitor The plte of prllel-plte cpcitor in vcuum re.00 mm () The chrge on the cpcitor i prt n 2.00 m 2 in re. potentil ifference of 0,000 V 0.0 kv 2 CV i pplie cro the cpcitor. Compute () the cpcitnce; 3.4 3 0 29 C/V 2.00 3 0 4 V 2 () the chrge on ech plte; n (c) the mgnitue of the electric fiel in the pce etween them. 3.4 3 0 2 C 3.4 mc The plte t higher potentil h chrge 3.4 mc n the other plte h chrge 23.4 mc. IDENTIFY: We re given the plte re, the plte pcing, n (c) The electric-fiel mgnitue i the potentil ifference V for thi prllel-plte cpcitor. Our E trget vrile re the cpcitnce C, chrge, n electric-fiel P 0 P 0 3.4 3 0 2 C 8.8 3 0 22 C 2 /N # m 2 22.00 m 2 2 mgnitue E. 2.00 3 0 6 N/C ET UP: We ue Eq. (24.2) to clculte C n then fin the chrge EVLUTE: n lterntive wy to get the reult in prt (c) i to on ech plte uing the given potentil ifference V n recll tht the electric fiel i equl in mgnitue to the potentil Eq. (24.). Once we hve, we fin the electric fiel etween the grient [Eq. (23.22)]. ince the fiel etween the plte i uniform, plte uing the reltionhip E /P 0. EXECUTE: () From Eq. (24.2), E V.00 3 04 V.00 3 0 23 m 2.00 3 06 V/m C P 0 8.8 3 022 F/m 22.00 m 2 2 (Rememer tht the newton per coulom n the volt per meter re.00 3 0 23 m equivlent unit.) 3.4 3 0 29 F 0.0034 mf Exmple 24.3 phericl cpcitor Two concentric phericl conucting hell re eprte y vcuum. The inner hell h totl chrge n outer riu r, n the outer hell h chrge 2 n inner riu r (Fig. 24.). (The inner hell i ttche to the outer hell y thin inulting ro tht hve negligile effect on the cpcitnce.) Fin the cpcitnce of thi phericl cpcitor. IDENTIFY: Thi in t prllel-plte cpcitor, o we cn t ue the reltionhip evelope for tht prticulr geometry. Inte, we ll go ck to the funmentl efinition of cpcitnce: the mgnitue of the chrge on either conuctor ivie y the potentil ifference etween the conuctor. ET UP: We ue Gu lw to fin the electric fiel etween the phericl conuctor. From thi vlue we etermine the potentil ifference V etween the two conuctor; we then ue Eq. (24.) to fin the cpcitnce C /V. EXECUTE: Uing the me proceure in Exmple 22. (ection 22.4), we tke our Guin urfce phere with riu r etween the two phere n concentric with them. Gu lw, Eq. (22.8), tte tht the electric flux through thi urfce i equl to the totl chrge encloe within the urfce, ivie y P 0 : C E # encl P 0 By ymmetry, E i contnt in mgnitue n prllel to t every point on thi urfce, o the integrl in Gu lw i equl 24. phericl cpcitor. Exmple 24.4 r r Inner hell, chrge r Guin urfce Outer hell, chrge 2 cylinricl cpcitor long cylinricl conuctor h riu r n liner chrge enity l. It i urroune y coxil cylinricl conucting hell with inner riu r n liner chrge enity 2l (Fig. 24.6). Clculte the cpcitnce per unit length for thi cpcitor, uming tht there i vcuum in the pce etween cyliner. IDENTIFY: in Exmple 24.3, we ue the funmentl efinition of cpcitnce. to E 24pr 2 2. The totl chrge encloe i encl, o we hve The electric fiel etween the phere i jut tht ue to the chrge on the inner phere; the outer phere h no effect. We foun in Exmple 22. tht the chrge on conucting phere prouce zero fiel inie the phere, which lo tell u tht the outer conuctor mke no contriution to the fiel etween the conuctor. The ove expreion for E i the me tht for point chrge, o the expreion for the potentil cn lo e tken to e the me for point chrge, V /4pP 0 r. Hence the potentil of the inner (poitive) conuctor t r r with repect to tht of the outer (negtive) conuctor t r r i V V 2 V Finlly, the cpcitnce i E 24pr 2 2 P 0 E 4pP 0 r 2 4pP 0 r 2 4pP 0 r 2 r 2 4pP 0 r 2 r r r C r r 4pP V 0 r 2 r n exmple, if r 9. cm n r 0. cm, 0.09 m 20.0 m 2 C 4p 8.8 3 0 22 F/m 2 0.00 m. 3 0 20 F 0 pf EVLUTE: We cn relte thi reult to the cpcitnce of prllel-plte cpcitor. The quntity 4pr r i intermeite etween the 2 2 re 4pr n 4pr of the two phere; in fct, it the geometric men of thee two re, which we cn enote y gm. The itnce etween phere i r 2 r, o we cn rewrite the ove reult C P 0 gm/. Thi i exctly the me form for prllel plte: C P 0 /. The point i tht if the itnce etween phere i very mll in comprion to their rii, they ehve like prllel plte with the me re n pcing. 4pP 0 r ET UP: We firt fin expreion for the potentil ifference etween the cyliner n the chrge in length L of the cyliner; we then fin the cpcitnce of length L uing Eq. (24.). Our trget vrile i thi cpcitnce ivie y L. EXECUTE: To fin the potentil ifference etween the cyliner, we ue reult tht we worke out in Exmple 23.0 (ection 23.3). There we foun tht t point outie chrge V Continue
820 CHPTER 24 Cpcitnce n Dielectric 24.2 Cpcitor in erie n Prllel 82 24.6 long cylinricl cpcitor. The liner chrge enity l i ume to e poitive in thi figure. The mgnitue of chrge in length L of either cyliner i ll. l r r 2l cyliner itnce r from the xi, the potentil ue to the cyliner i V where r 0 i the (ritrry) riu t which V 0. We cn ue thi me reult for the potentil etween the cyliner in the preent prolem ecue, ccoring to Gu lw, the chrge on the outer cyliner oen t contriute to the fiel etween cyliner (ee Exmple 24.3). In our ce, we tke the riu r 0 to e r, the riu of the inner urfce of the outer cyliner, o tht the outer conucting cyliner i t V 0. Then the potentil t the outer urfce of the inner cyliner (where r r 2 i jut equl to the L l ln r 0 2pP 0 r potentil V of the inner (poitive) cyliner with repect to the outer (negtive) cyliner, or Thi potentil ifference i poitive (uming tht l i poitive, in Fig. 24.6) ecue the inner cyliner i t higher potentil thn the outer. The totl chrge in length L i ll, o from Eq. (24.) the cpcitnce C of length L i C V V The cpcitnce per unit length i l 2pP 0 ln r r ll l ln r 2pP 0 L ln r /r 2 2pP 0 r C L 2pP 0 ln r /r 2 utituting P 0 8.8 3 0 22 F/m 8.8 pf/m, we get C L.6 pf /m ln r /r 2 EVLUTE: We ee tht the cpcitnce of the coxil cyliner i etermine entirely y the imenion, jut for the prllel-plte ce. Orinry coxil cle re me like thi ut with n inulting mteril inte of vcuum etween the inner n outer conuctor. typicl cle for TV ntenn n VCR connection h cpcitnce per unit length of 69 pf/m. point onto the ottom plte of C 2. The totl chrge on the lower plte of C n the upper plte of C 2 together mut lwy e zero ecue thee plte ren t connecte to nything except ech other. Thu in erie connection the mgnitue of chrge on ll plte i the me. Referring to Fig. 24.8, we cn write the potentil ifference etween point n c, c n, n n n o V c V V C c V 2 C 2 V V V V 2 C C 2 2 V C C 2 (24.3) Following common convention, we ue the ymol V, V 2, n V to enote the potentil ifference V c (cro the firt cpcitor), V c (cro the econ cpcitor), n V (cro the entire comintion of cpcitor), repectively. The equivlent cpcitnce C eq of the erie comintion i efine the cpcitnce of ingle cpcitor for which the chrge i the me for the comintion, when the potentil ifference V i the me. In other wor, the comintion cn e replce y n equivlent cpcitor of cpcitnce C eq. For uch cpcitor, hown in Fig. 24.8, C eq V or C eq V (24.4) 24.8 erie connection of two cpcitor. () Two cpcitor in erie Cpcitor in erie: The cpcitor hve the me chrge. Their potentil ifference : V c V c V. V V 2 C V c V Chrge i the me V for the C eq iniviul 2 cpcitor. c C V 2 2 c V 2 () The equivlent ingle cpcitor Equivlent cpcitnce i le thn the iniviul cpcitnce: V C eq C C 2 Comining Eq. (24.3) n (24.4), we fin Tet Your Unertning of ection 24. cpcitor h vcuum in the pce etween the conuctor. If you oule the mount of chrge on ech conuctor, wht hppen to the cpcitnce? (i) It incree; (ii) it ecree; (iii) it remin the me; (iv) the nwer epen on the ize or hpe of the conuctor. C eq C C 2 We cn exten thi nlyi to ny numer of cpcitor in erie. We fin the following reult for the reciprocl of the equivlent cpcitnce: 24.7 n ortment of commercilly ville cpcitor. 24.2 Cpcitor in erie n Prllel Cpcitor re mnufcture with certin tnr cpcitnce n working voltge (Fig. 24.7). However, thee tnr vlue my not e the one you ctully nee in prticulr ppliction. You cn otin the vlue you nee y comining cpcitor; mny comintion re poile, ut the implet comintion re erie connection n prllel connection. Cpcitor in erie Figure 24.8 i chemtic igrm of erie connection. Two cpcitor re connecte in erie (one fter the other) y conucting wire etween point n. Both cpcitor re initilly unchrge. When contnt poitive potentil ifference V i pplie etween point n, the cpcitor ecome chrge; the figure how tht the chrge on ll conucting plte h the me mgnitue. To ee why, note firt tht the top plte of C cquire poitive chrge. The electric fiel of thi poitive chrge pull negtive chrge up to the ottom plte of C until ll of the fiel line tht egin on the top plte en on the ottom plte. Thi require tht the ottom plte hve chrge 2. Thee negtive chrge h to come from the top plte of C 2, which ecome poitively chrge with chrge. Thi poitive chrge then pull negtive chrge 2 from the connection t C eq C C 2 C 3 c (cpcitor in erie) (24.) The reciprocl of the equivlent cpcitnce of erie comintion equl the um of the reciprocl of the iniviul cpcitnce. In erie connection the equivlent cpcitnce i lwy le thn ny iniviul cpcitnce. CUTION Cpcitor in erie The mgnitue of chrge i the me on ll plte of ll the cpcitor in erie comintion; however, the potentil ifference of the iniviul cpcitor re not the me unle their iniviul cpcitnce re the me. The potentil ifference of the iniviul cpcitor to give the totl potentil ifference cro the erie comintion: V totl V V 2 V 3 c. Cpcitor in Prllel The rrngement hown in Fig. 24.9 i clle prllel connection. Two cpcitor re connecte in prllel etween point n. In thi ce the upper plte of the two cpcitor re connecte y conucting wire to form n equipotentil urfce, n the lower plte form nother. Hence in prllel connection the potentil ifference for ll iniviul cpcitor i the me n i equl to V V. The chrge n re not necerily equl, however, 2 24.9 prllel connection of two cpcitor. () Two cpcitor in prllel Cpcitor in prllel: The cpcitor hve the me potentil V. The chrge on ech cpcitor epen on it cpcitnce: C V, 2 C 2 V. V V C C 2 2 () The equivlent ingle cpcitor Chrge i the um of the iniviul chrge: V C eq 2 2 Equivlent cpcitnce: C eq C C 2
822 CHPTER 24 Cpcitnce n Dielectric 24.2 Cpcitor in erie n Prllel 823 Prolem-olving trtegy 24. ince chrge cn rech ech cpcitor inepenently from the ource (uch ttery) of the voltge V. The chrge re The totl chrge of the comintion, n thu the totl chrge on the equivlent cpcitor, i o C V n 2 C 2 V 2 C C 2 2 V V C C 2 (24.6) The prllel comintion i equivlent to ingle cpcitor with the me totl chrge 2 n potentil ifference V the comintion (Fig. 24.9). The equivlent cpcitnce of the comintion, C eq, i the me the cpcitnce /V of thi ingle equivlent cpcitor. o from Eq. (24.6), C eq C C 2 In the me wy we cn how tht for ny numer of cpcitor in prllel, C eq C C 2 C 3 c (cpcitor in prllel) (24.7) The equivlent cpcitnce of prllel comintion equl the um of the iniviul cpcitnce. In prllel connection the equivlent cpcitnce i lwy greter thn ny iniviul cpcitnce. CUTION Cpcitor in prllel The potentil ifference re the me for ll the cpcitor in prllel comintion; however, the chrge on iniviul cpcitor re not the me unle their iniviul cpcitnce re the me. The chrge on the iniviul cpcitor to give the totl chrge on the prllel comintion: totl 2 3 c. [Compre thee ttement to thoe in the Cution prgrph following Eq. (24.).] Equivlent Cpcitnce IDENTIFY the relevnt concept: The concept of equivlent cpcitnce i ueful whenever two or more cpcitor re connecte. ET UP the prolem uing the following tep:. Mke rwing of the cpcitor rrngement. 2. Ientify whether the cpcitor re connecte in erie or in prllel. With more complicte comintion, you cn ometime ientify prt tht re imple erie or prllel connection. 3. Keep in min tht when we y cpcitor h chrge, we lwy men tht the plte t higher potentil h chrge n the other plte h chrge 2. EXECUTE the olution follow:. When cpcitor re connecte in erie, in Fig. 24.8, they lwy hve the me chrge, uming tht they were unchrge efore they were connecte. The potentil ifference re not equl unle the cpcitnce re equl. The totl potentil ifference cro the comintion i the um of the iniviul potentil ifference. 2. When cpcitor re connecte in prllel, in Fig. 24.9, the potentil ifference V i lwy the me for ll of the iniviul cpcitor. The chrge on the iniviul cpcitor re not equl unle the cpcitnce re equl. The totl chrge on the comintion i the um of the iniviul chrge. 3. For more complicte comintion, fin the prt tht re imple erie or prllel connection n replce them with their equivlent cpcitnce, in tep-y-tep reuction. If you then nee to fin the chrge or potentil ifference for n iniviul cpcitor, you my hve to retrce your pth to the originl cpcitor. EVLUTE your nwer: Check whether your reult mke ene. If the cpcitor re connecte in erie, the equivlent cpcitnce C eq mut e mller thn ny of the iniviul cpcitnce. By contrt, if the cpcitor re connecte in prllel, C eq mut e greter thn ny of the iniviul cpcitnce. Exmple 24. Cpcitor in erie n in prllel In Fig. 24.8 n 24.9, let C 6.0 mf, C 2 3.0 mf, n V 8 V. Fin the equivlent cpcitnce, n fin the chrge n potentil ifference for ech cpcitor when the two cpcitor re connecte () in erie n () in prllel. IDENTIFY: Thi prolem ue the ie icue in thi ection out cpcitor connection. ET UP: In oth prt, one of the trget vrile i the equivlent cpcitnce C eq. For the erie comintion in prt (), it i given y Eq. (24.); for the prllel comintion in prt (), C eq i given y Eq. (24.6). In ech prt we fin the chrge n potentil ifference uing the efinition of cpcitnce, Eq. (24.), n the rule outline in the Prolem-olving trtegy 24.. EXECUTE: () Uing Eq. (24.) for the equivlent cpcitnce of the erie comintion (Fig. 24.8), we fin C eq C C 2 6.0 mf C 3.0 mf eq 2.0 mf The chrge on ech cpcitor in erie i the me the chrge on the equivlent cpcitor: C eq V 2.0 mf 28 V 2 36 mc The potentil ifference cro ech cpcitor i inverely proportionl to it cpcitnce: Exmple 24.6 V c V C V c V 2 C 2 36 mc 6.0 mf 6.0 V 36 mc 3.0 mf 2.0 V cpcitor network Fin the equivlent cpcitnce of the comintion hown in Fig. 24.0. IDENTIFY: The five cpcitor in Fig. 24.0 re neither ll in erie nor ll in prllel. We cn, however, ientify portion of the () () () To fin the equivlent cpcitnce of the prllel comintion (Fig. 24.9), we ue Eq. (24.6): C eq C C 2 6.0 mf 3.0 mf 9.0 mf The potentil ifference cro ech of the two cpcitor in prllel i the me tht cro the equivlent cpcitor, 8 V. The chrge n 2 re irectly proportionl to the cpcitnce C n C 2, repectively: C V 6.0 mf 28 V 2 08 mc 2 C 2 V 3.0 mf 28 V 2 4 mc EVLUTE: Note tht the equivlent cpcitnce C eq for the erie comintion in prt () i inee le thn either C or C 2, while for the prllel comintion in prt () the equivlent cpcitnce i inee greter thn either C or C 2. It intructive to compre the potentil ifference n chrge in ech prt of the exmple. For two cpcitor in erie, in prt (), the chrge i the me on either cpcitor n the lrger potentil ifference pper cro the cpcitor with the mller cpcitnce. Furthermore, V c V c V 8 V, it mut. By contrt, for two cpcitor in prllel, in prt (), ech cpcitor h the me potentil ifference n the lrger chrge pper on the cpcitor with the lrger cpcitnce. Cn you how tht the totl chrge 2 on the prllel comintion i equl to the chrge C eq V on the equivlent cpcitor? rrngement tht re either in erie or prllel, which we comine to fin the net equivlent cpcitnce. ET UP: We ue Eq. (24.) to nlyze portion of the network tht re erie connection n Eq. (24.7) to nlyze portion tht re prllel connection. 24.0 () cpcitor network etween point n. () The 2-mF n 6-mF cpcitor in erie in () re replce y n equivlent 4-mF cpcitor. (c) The 3-mF, -mf, n 4-mF cpcitor in prllel in () re replce y n equivlent 8-mF cpcitor. () Finlly, the 8-mF n 9-mF cpcitor in erie in (c) re replce y n equivlent 6-mF cpcitor. (c) 2 mf 3 mf mf 3 mf mf 4 mf 8 mf 6 mf Replce thee erie cpcitor 9 mf 9 mf y n equivlent cpcitor...... replce thee prllel cpcitor y n equivlent cpcitor... 9 mf ()... replce thee erie cpcitor y n equivlent cpcitor. 6 mf Continue
824 CHPTER 24 Cpcitnce n Dielectric 24.3 Energy torge in Cpcitor n Electric-Fiel Energy 82 EXECUTE: We firt replce the 2-mF n 6-mF erie comintion y it equivlent cpcitnce; clling tht Cr, we ue Eq. (24.): Cr 2 mf 6 mf Cr 4 mf Thi give u the equivlent comintion hown in Fig. 24.0. Next we fin the equivlent cpcitnce of the three cpcitor in prllel, uing Eq. (24.7). Clling their equivlent cpcitnce C, we hve C 3 mf mf 4 mf 8 mf Tet Your Unertning of ection 24.2 You wnt to connect 4-mF cpcitor n n 8-mF cpcitor. () With which type of connection will the 4-mF cpcitor hve greter potentil ifference cro it thn the 8-mF cpcitor? (i) erie; (ii) prllel; (iii) either erie or prllel; (iv) neither erie nor prllel. () With which type of connection will the 4-mF cpcitor hve greter chrge thn the 8-mF cpcitor? (i) erie; (ii) prllel; (iii) either erie or prllel; (iv) neither erie nor prllel. 24.3 Energy torge in Cpcitor n Electric-Fiel Energy Mny of the mot importnt ppliction of cpcitor epen on their ility to tore energy. The electric potentil energy tore in chrge cpcitor i jut equl to the mount of work require to chrge it tht i, to eprte oppoite chrge n plce them on ifferent conuctor. When the cpcitor i ichrge, thi tore energy i recovere work one y electricl force. We cn clculte the potentil energy U of chrge cpcitor y clculting the work W require to chrge it. uppoe tht when we re one chrging the cpcitor, the finl chrge i n the finl potentil ifference i V. From Eq. (24.) thee quntitie re relte y Let q n v e the chrge n potentil ifference, repectively, t n intermeite tge uring the chrging proce; then v q/c. t thi tge the work W require to trnfer n itionl element of chrge q i W v q q q C The totl work W neee to incree the cpcitor chrge q from zero to finl vlue i W W 3 W C 3 0 Thi give u the impler equivlent comintion hown in Fig. 24.0c. Finlly, we fin the equivlent cpcitnce C eq of thee two cpcitor in erie (Fig. 24.0): EVLUTE: The equivlent cpcitnce of the network i 6 mf; tht i, if potentil ifference V i pplie cro the terminl of the network, the net chrge on the network i 6 mf time V. How i thi net chrge relte to the chrge on the iniviul cpcitor in Fig. 24.0? 0 q q 2 2C (work to chrge cpcitor) (24.8) Thi i lo equl to the totl work one y the electric fiel on the chrge when the cpcitor ichrge. Then q ecree from n initil vlue to zero the element of chrge q fll through potentil ifference v tht vry from V own to zero. If we efine the potentil energy of n unchrge cpcitor to e zero, then W in Eq. (24.8) i equl to the potentil energy U of the chrge cpcitor. The finl tore chrge i CV, o we cn expre U (which i equl to W) U 2 2C 2 CV 2 2 V C eq 8 mf 9 mf V C C eq 6 mf (potentil energy tore in cpcitor) (24.9) When i in coulom, C in fr (coulom per volt), n V in volt ( joule per coulom), U i in joule. The lt form of Eq. (24.9), U 2 V, how tht the totl work W require to chrge the cpcitor i equl to the totl chrge multiplie y the verge potentil ifference 2 V uring the chrging proce. The expreion U 2 2 /C 2 in Eq. (24.9) how tht chrge cpcitor i the electricl nlog of tretche pring with eltic potentil energy U 2 kx2. The chrge i nlogou to the elongtion x, n the reciprocl of the cpcitnce, /C, i nlogou to the force contnt k. The energy upplie to cpcitor in the chrging proce i nlogou to the work we o on pring when we tretch it. Eqution (24.8) n (24.9) tell u tht cpcitnce meure the ility of cpcitor to tore oth energy n chrge. If cpcitor i chrge y connecting it to ttery or other ource tht provie fixe potentil ifference V, then increing the vlue of C give greter chrge CV n greter mount of tore energy U 2 CV 2. If inte the gol i to trnfer given quntity of chrge from one conuctor to nother, Eq. (24.8) how tht the work W require i inverely proportionl to C; the greter the cpcitnce, the eier it i to give cpcitor fixe mount of chrge. ppliction of Cpcitor: Energy torge Mot prcticl ppliction of cpcitor tke vntge of their ility to tore n relee energy. In electronic flh unit ue y photogrpher, the? energy tore in cpcitor (ee Fig. 24.4) i relee y epreing the cmer hutter utton. Thi provie conucting pth from one cpcitor plte to the other through the flh tue. Once thi pth i etlihe, the tore energy i rpily converte into rief ut intene flh of light. n extreme exmple of the me principle i the Z mchine t ni Ntionl Lortorie in New Mexico, which i ue in experiment in controlle nucler fuion (Fig. 24.). nk of chrge cpcitor relee more thn million joule of energy in jut few illionth of econ. For tht rief pce of time, the power output of the Z mchine i 2.9 3 0 4 W, or out 80 time the electric output of ll the electric power plnt on erth comine! In other ppliction, the energy i relee more lowly. pring in the upenion of n utomoile, help mooth out the rie y oring the energy from uen jolt n releing tht energy grully; in n nlogou wy, cpcitor in n electronic circuit cn mooth out unwnte vrition in voltge ue to power urge. n jut the preence of pring give mechnicl ytem nturl frequency t which it repon mot trongly to n pplie perioic force, o the preence of cpcitor give n electric circuit nturl frequency for current ocilltion. Thi ie i ue in tune circuit uch thoe in rio n televiion receiver, which repon to roct ignl t one prticulr frequency n ignore ignl t other frequencie. We ll icu thee circuit in etil in Chpter 3. The energy-torge propertie of cpcitor lo hve ome uneirle prcticl effect. jcent pin on the unerie of computer chip ct like cpcitor, n the property tht mke cpcitor ueful for moothing out voltge vrition ct to retr the rte t which the potentil of the chip pin cn e chnge. Thi tenency limit how rpily the chip cn perform computtion, n effect tht ecome more importnt computer chip ecome mller n re puhe to operte t fter pee. Electric-Fiel Energy We cn chrge cpcitor y moving electron irectly from one plte to nother. Thi require oing work gint the electric fiel etween the plte. Thu we cn think of the energy eing tore in the fiel in the region etween the 24. The Z mchine ue lrge numer of cpcitor in prllel to give tremenou equivlent cpcitnce C (ee ection 24.2). Hence lrge mount of energy U 2CV 2 cn e tore with even moet potentil ifference V. The rc hown here re prouce when the cpcitor ichrge their energy into trget, which i no lrger thn pool of thre. Thi het the trget to temperture higher thn 2 3 0 9 K.
826 CHPTER 24 Cpcitnce n Dielectric 24.3 Energy torge in Cpcitor n Electric-Fiel Energy 827 plte. To evelop thi reltionhip, let fin the energy per unit volume in the pce etween the plte of prllel-plte cpcitor with plte re n eprtion. We cll thi the energy enity, enote y u. From Eq. (24.9) the totl tore potentil energy i 2 CV 2 n the volume etween the plte i jut ; hence the energy enity i u Energy enity 2 CV 2 (24.0) From Eq. (24.2) the cpcitnce C i given y C P 0 /. The potentil ifference V i relte to the electric fiel mgnitue E y V E. If we ue thee expreion in Eq. (24.0), the geometric fctor n cncel, n we fin In the finl tte, when the chrge re no longer moving, oth upper plte re t the me potentil; they re connecte y conucting wire n o form ingle equipotentil urfce. Both lower plte re lo t the me potentil, ifferent from tht of the upper plte. The finl potentil ifference V etween the plte i therefore the me for oth cpcitor, we woul expect for prllel connection. The cpcitor chrge re C V 2 C 2 V When we comine thee with the preceing eqution for conervtion of chrge, we fin V 0 960 mc C C 2 8.0 mf 4.0 mf 80 V 640 mc 2 320 mc () The finl energy of the ytem i the um of the energie tore in ech cpcitor: U finl 2 V 2 2 V 2 0 V 2 960 3 026 C 280 V 2 0.038 J EVLUTE: The finl energy i le thn the originl energy U initil 0.08 J; the ifference h een converte to energy of ome other form. The conuctor ecome little wrmer ecue of their reitnce, n ome energy i rite electromgnetic wve. We ll tuy the circuit ehvior of cpcitor in etil in Chpter 26 n 3. u 2 P 0 E 2 (electric energy enity in vcuum) (24.) Exmple 24.8 Electric-fiel energy lthough we hve erive thi reltionhip only for prllel-plte cpcitor, it turn out to e vli for ny cpcitor in vcuum n inee for ny electric fiel configurtion in vcuum. Thi reult h n intereting impliction. We think of vcuum pce with no mtter in it, ut vcuum cn neverthele hve electric fiel n therefore energy. Thu empty pce nee not e truly empty fter ll. We will ue thi ie n Eq. (24.) in Chpter 32 in connection with the energy trnporte y electromgnetic wve. CUTION Electricl-fiel energy i electric potentil energy It common miconception tht electric-fiel energy i new kin of energy, ifferent from the electric potentil energy ecrie efore. Thi i not the ce; it i imply ifferent wy of interpreting electric potentil energy. We cn regr the energy of given ytem of chrge eing hre property of ll the chrge, or we cn think of the energy eing property of the electric fiel tht the chrge crete. Either interprettion le to the me vlue of the potentil energy. uppoe you wnt to tore.00 J of electric potentil energy in EXECUTE: () The eire energy enity i u.00 J/m 3. We volume of.00 m 3 in vcuum. () Wht i the mgnitue of the olve Eq. (24.) for E: require electric fiel? () If the fiel mgnitue i 0 time lrger, how much energy i tore per cuic meter? 2u 2.00 J/m 3 2 E Å P 0 Å 8.8 3 0 22 C 2 /N # m 2 4.7 3 0 N/C 4.7 3 0 V/m IDENTIFY: We ue the reltionhip etween the electric-fiel () Eqution (24.) how tht u i proportionl to E 2. If E mgnitue E n the energy enity u, which equl the electricfiel energy ivie y the volume occupie y the fiel. incree y fctor of 0, u incree y fctor of 0 2 00, n the energy enity i 00 J/m 3. ET UP: In prt () we ue the given informtion to fin u, then we EVLUTE: The vlue of E foun in prt () i izle, correponing to potentil ifference of nerly hlf million volt over ue Eq. (24.) to fin the require vlue of E. Thi me eqution give u the reltionhip etween chnge in E n the correponing chnge in u. nitue in prcticl inultor cn e gret thi or even itnce of meter. We will ee in ection 24.4 tht the fiel mg- lrger. Exmple 24.7 Trnferring chrge n energy etween cpcitor In Fig. 24.2 we chrge cpcitor of cpcitnce C 8.0 mf y connecting it to ource of potentil ifference V 0 20 V (not hown in the figure). The witch i initilly open. Once C i chrge, the ource of potentil ifference i iconnecte. () Wht i the chrge 0 on C if witch i left open? () Wht i the energy tore in C if witch i left open? (c) The cpcitor of cpcitnce C 2 4.0 mf i initilly unchrge. fter we cloe witch, wht i the potentil ifference cro ech cpcitor, n wht i the chrge on ech cpcitor? () Wht i the totl energy of the ytem fter we cloe witch? IDENTIFY: Initilly we hve ingle cpcitor with given potentil ifference etween it plte. fter the witch i cloe, one wire connect the upper plte of the two cpcitor n nother wire connect the lower plte; in other wor, the cpcitor re connecte in prllel. ET UP: In prt () n () we fin the chrge n tore energy for cpcitor C uing Eq. (24.) n (24.9), repectively. In prt (c) we ue the chrcter of the prllel connection to etermine how the chrge 0 i hre etween the two cpcitor. In prt () we gin ue Eq. (24.9) to fin the energy tore in cpcitor n C 2 ; the totl energy i the um of thee vlue. C 24.2 When the witch i cloe, the chrge cpcitor C i connecte to n unchrge cpcitor C 2. The center prt of the witch i n inulting hnle; chrge cn flow only etween the two upper terminl n etween the two lower terminl. C 8.0 mf 0 V0 20 V 0 EXECUTE: () The chrge on i C 0 C V 0 8.0 mf 220 V 2 960 mc () The energy initilly tore in the cpcitor i U initil 2 0 V 0 2 960 3 026 C 220 V 2 0.08 J (c) When the witch i cloe, the poitive chrge 0 ecome itriute over the upper plte of oth cpcitor n the negtive chrge 2 0 i itriute over the lower plte of oth cpcitor. Let n 2 e the mgnitue of the finl chrge on the two cpcitor. From conervtion of chrge, 2 0 C 2 4.0 mf Exmple 24.9 Two wy to clculte energy tore in cpcitor The phericl cpcitor ecrie in Exmple 24.3 (ection 24.) h chrge n 2 on it inner n outer conuctor. Fin the electric potentil energy tore in the cpcitor () y uing the cpcitnce C foun in Exmple 24.3 n () y integrting the electric-fiel energy enity. IDENTIFY: Thi prolem k u to think out the energy tore in cpcitor, U, in two ifferent wy: in term of the work one to put the chrge on the two conuctor, U 2 /2C, n in term of the energy in the electric fiel etween the two conuctor. Both ecription re equivlent, o oth mut give u the me nwer for U. ET UP: In Exmple 24.3 we foun the cpcitnce C n the fiel mgnitue E etween the conuctor. We fin the tore energy U in prt () uing the expreion for C in Eq. (24.9). In prt () we ue the expreion for E in Eq. (24.) to fin the electric-fiel energy enity u etween the conuctor. The fiel mgnitue epen on the itnce r from the center of the cpcitor, o u lo epen on r. Hence we cnnot fin U y imply multiplying u y the volume etween the conuctor; inte, we mut integrte u over thi volume. EXECUTE: () From Exmple 24.3, the phericl cpcitor h cpcitnce r r where n re the rii of the inner n outer conucting phere. From Eq. (24.9) the energy tore in thi cpcitor i U 2 2C 2 r 2 r 8pP 0 r r () The electric fiel in the volume etween the two conucting phere h mgnitue E /4pP 0 r 2. The electric fiel i zero inie the inner phere n i lo zero outie the inner urfce of the outer phere, ecue Guin urfce with riu r, r or r. r encloe zero net chrge. Hence the energy enity i nonzero only in the pce etween the phere r, r, r 2. In thi region, u 2 P 0E 2 2 P 0 r r C 4pP 0 r 2 r 2 2 2 4pP 0 r 2 32p 2 P 0 r 4 The energy enity i not uniform; it ecree rpily with increing itnce from the center of the cpcitor. To fin the Continue
828 CHPTER 24 Cpcitnce n Dielectric 24.4 Dielectric 829 totl electric-fiel energy, we integrte u (the energy per unit volume) over the volume etween the inner n outer conucting we mut. We emphize tht electric potentil energy cn e EVLUTE: We otin the me reult for U with either pproch, phere. Diviing thi volume up into phericl hell of riu r, regre eing ocite with either the chrge, in prt (), urfce re 4pr 2, thickne r, n volume V 4pr 2 r, we or the fiel, in prt (); regrle of which viewpoint you hve chooe, the mount of tore energy i the me. U 3 u V 3 r 24.3 common type of cpcitor ue ielectric heet to eprte the conuctor. Conuctor (metl foil) 2 8pP 0 3 r r r r r 2 2 8pP 0 r 2 r r r 2 32p 2 P 0 r 4 24pr 2 r 2 8pP 0 2 r r 2 Conuctor (metl foil) Dielectric (pltic heet) Tet Your Unertning of ection 24.3 You wnt to connect 4-mF cpcitor n n 8-mF cpcitor. With which type of connection will the 4-mF cpcitor hve greter mount of tore energy thn the 8-mF cpcitor? (i) erie; (ii) prllel; (iii) either erie or prllel; (iv) neither erie nor prllel. 24.4 Dielectric Mot cpcitor hve nonconucting mteril, or ielectric, etween their conucting plte. common type of cpcitor ue long trip of metl foil for the plte, eprte y trip of pltic heet uch Mylr. nwich of thee mteril i rolle up, forming unit tht cn provie cpcitnce of everl microfr in compct pckge (Fig. 24.3). Plcing oli ielectric etween the plte of cpcitor erve three function. Firt, it olve the mechnicl prolem of mintining two lrge metl heet t very mll eprtion without ctul contct. econ, uing ielectric incree the mximum poile potentil ifference etween the cpcitor plte. we ecrie in ection 23.3, ny inulting mteril, when ujecte to ufficiently lrge electric fiel, experience prtil ioniztion tht permit conuction through it. Thi i clle ielectric rekown. Mny ielectric mteril cn tolerte tronger electric fiel without rekown thn cn ir. Thu uing ielectric llow cpcitor to utin higher potentil ifference V n o tore greter mount of chrge n energy. Thir, the cpcitnce of cpcitor of given imenion i greter when there i ielectric mteril etween the plte thn when there i vcuum. We cn emontrte thi effect with the i of enitive electrometer, evice tht meure the potentil ifference etween two conuctor without letting ny pprecile chrge flow from one to the other. Figure 24.4 how n electrometer connecte cro chrge cpcitor, with mgnitue of chrge on ech plte n potentil ifference V 0. When we inert n unchrge heet of ielectric, uch gl, prffin, or polytyrene, etween the plte, experiment how tht the potentil ifference ecree to mller vlue V (Fig. 24.4). When we remove the ielectric, the potentil ifference return to it originl vlue V 0, howing tht the originl chrge on the plte hve not chnge. The originl cpcitnce C 0 i given y C 0 /V 0, n the cpcitnce C with the ielectric preent i C /V. The chrge i the me in oth ce, n V i le thn V 0, o we conclue tht the cpcitnce C with the ielectric preent i greter thn C 0. When the pce etween plte i completely fille y the ielectric, the rtio of C to C 0 (equl to the rtio of V 0 to V) i clle the ielectric contnt of the mteril, K: K C C 0 (efinition of ielectric contnt) (24.2) When the chrge i contnt, C 0 V 0 CV n C/C 0 V 0/V. In thi ce, Eq. (24.2) cn e rewritten V V 0 K (when i contnt) (24.3) With the ielectric preent, the potentil ifference for given chrge i reuce y fctor K. The ielectric contnt K i pure numer. Becue C i lwy greter thn C 0, K i lwy greter thn unity. ome repreenttive vlue of K re given in Tle 24.. For vcuum, K y efinition. For ir t orinry temperture n preure, K i out.0006; thi i o nerly equl to tht for mot purpoe n ir cpcitor i equivlent to one in vcuum. Note tht while wter h very lrge vlue of K, it i uully not very prcticl ielectric for ue in cpcitor. The reon i tht while pure wter i very poor conuctor, it i lo n excellent ionic olvent. ny ion tht re iolve in the wter will cue chrge to flow etween the cpcitor plte, o the cpcitor ichrge. Tle 24. Vlue of Dielectric Contnt K t 20 C Mteril K Mteril Vcuum Polyvinyl chlorie 3.8 ir ( tm).0009 Plexigl 3.40 ir (00 tm).048 Gl 0 Teflon 2. Neoprene 6.70 Polyethylene 2.2 Germnium 6 Benzene 2.28 Glycerin 42. Mic 36 Wter 80.4 Mylr 3. trontium titnte 30 No rel ielectric i perfect inultor. Hence there i lwy ome lekge current etween the chrge plte of cpcitor with ielectric. We tcitly ignore thi effect in ection 24.2 when we erive expreion for the equivlent cpcitnce of cpcitor in erie, Eq. (24.), n in prllel, Eq. (24.7). But if lekge current flow for long enough time to utntilly chnge the chrge from the vlue we ue to erive Eq. (24.) n (24.7), thoe eqution my no longer e ccurte. Inuce Chrge n Polriztion When ielectric mteril i inerte etween the plte while the chrge i kept contnt, the potentil ifference etween the plte ecree y fctor K. Therefore the electric fiel etween the plte mut ecree y the me fctor. If i the vcuum vlue n E i the vlue with the ielectric, then E 0 E E 0 K (when i contnt) K (24.4) ince the electric-fiel mgnitue i mller when the ielectric i preent, the urfce chrge enity (which cue the fiel) mut e mller well. The urfce chrge on the conucting plte oe not chnge, ut n inuce chrge of the oppoite ign pper on ech urfce of the ielectric (Fig. 24.). The ielectric w originlly electriclly neutrl n i till neutrl; the inuce urfce chrge rie reult of reitriution of poitive n negtive chrge within the ielectric mteril, phenomenon clle polriztion. We firt encountere polriztion in ection 2.2, n we ugget tht you rere the icuion of Fig. 2.8. We will ume tht the inuce urfce chrge i irectly proportionl to the electric-fiel mgnitue E in the mteril; thi i inee the ce for mny common ielectric. (Thi irect proportionlity i nlogou to 24.4 Effect of ielectric etween the plte of prllel-plte cpcitor. () With given chrge, the potentil ifference i V 0. () With the me chrge ut with ielectric etween the plte, the potentil ifference V i mller thn V 0. () Vcuum () Dielectric 2 V 0 V () Vcuum E 0 2 Electrometer (meure potentil ifference cro plte) ing the ielectric reuce the potentil ifference cro the cpcitor. 24. Electric fiel line with () vcuum etween the plte n () ielectric etween the plte. () Dielectric i i E Inuce chrge i i For given chrge enity, the inuce chrge on the ielectric urfce reuce the electric fiel etween the plte.
830 CHPTER 24 Cpcitnce n Dielectric Hooke lw for pring.) In tht ce, K i contnt for ny prticulr mteril. When the electric fiel i very trong or if the ielectric i me of certin crytlline mteril, the reltionhip etween inuce chrge n the electric fiel cn e more complex; we won t conier uch ce here. We cn erive reltionhip etween thi inuce urfce chrge n the chrge on the plte. Let enote the mgnitue of the chrge per unit re inuce on the urfce of the ielectric (the inuce urfce chrge enity) y i. The mgnitue of the urfce chrge enity on the cpcitor plte i, uul. Then the net urfce chrge on ech ie of the cpcitor h mgnitue i 2, hown in Fig. 24.. we foun in Exmple 2.3 (ection 2.) n in Exmple 22.8 (ection 22.4), the fiel etween the plte i relte to the net urfce chrge enity y E net/p 0. Without n with the ielectric, repectively, we hve E 0 E i (24.) P 0 P 0 Uing thee expreion in Eq. (24.4) n rerrnging the reult, we fin i 2 (24.6) K 2 (inuce urfce chrge enity) Thi eqution how tht when K i very lrge, i i nerly lrge. In thi ce, i nerly cncel, n the fiel n potentil ifference re much mller thn their vlue in vcuum. The prouct KP 0 i clle the permittivity of the ielectric, enote y P: home repir worker to locte metl tu hien ehin wll urfce. It conit of metl plte with ocite circuitry. The plte ct one hlf of cpcitor, with the wll cting the other hlf. If the tu finer move over metl tu, the effective ielectric contnt for the cpcitor chnge, chnging the cpcitnce n triggering ignl. Prolem-olving trtegy 24.2 Dielectric IDENTIFY the relevnt concept: The reltionhip in thi ection re ueful whenever there i n electric fiel in ielectric, uch ielectric etween chrge cpcitor plte. Typiclly you will e ke to relte the potentil ifference etween the plte, the electric fiel in the cpcitor, the chrge enity on the cpcitor plte, n the inuce chrge enity on the urfce of the cpcitor. ET UP the prolem uing the following tep:. Mke rwing of the itution. 2. Ientify the trget vrile, n chooe which of the key eqution of thi ection will help you fin thoe vrile. EXECUTE the olution follow:. In prolem uch the next exmple, it i ey to get lot in lizzr of formul. k yourelf t ech tep wht kin of quntity ech ymol repreent. For exmple, itinguih 24.4 Dielectric 83 clerly etween chrge n chrge enitie, n etween electric fiel n electric potentil ifference. 2. you clculte, continully check for conitency of unit. Thi effort i it more complex with electricl quntitie thn it w in mechnic. Ditnce mut lwy e in meter. Rememer tht microfr i 0 26 fr, n o on. Don t confue the numericl vlue of P 0 with the vlue of /4pP 0. There re everl lterntive et of unit for electric-fiel mgnitue, incluing N/C n V/m. The unit of re C 2 /N # m 2 or F/m. P 0 EVLUTE your nwer: When you check numericl vlue, rememer tht with ielectric preent, () the cpcitnce i lwy greter thn without ielectric; () for given mount of chrge on the cpcitor, the electric fiel n potentil ifference re le thn without ielectric; n (c) the inuce urfce chrge enity i on the ielectric i lwy le in mgnitue thn the chrge enity on the cpcitor plte. P KP 0 (efinition of permittivity) (24.7) In term of P we cn expre the electric fiel within the ielectric Exmple 24.0 cpcitor with n without ielectric The cpcitnce when the ielectric i preent i given y C KC 0 KP 0 P E P (prllel-plte cpcitor, ielectric etween plte) (24.8) (24.9) We cn repet the erivtion of Eq. (24.) for the energy enity u in n electric fiel for the ce in which ielectric i preent. The reult i u 2 KP 0 E 2 2 PE 2 (electric energy enity in ielectric) (24.20) In empty pce, where K, P P 0 n Eq. (24.9) n (24.20) reuce to Eq. (24.2) n (24.), repectively, for prllel-plte cpcitor in vcuum. For thi reon, P 0 i ometime clle the permittivity of free pce or the permittivity of vcuum. Becue K i pure numer, P n P 0 hve the me unit, C 2 /N # m 2 or F/m. Eqution (24.9) how tht extremely high cpcitnce cn e otine with plte tht hve lrge urfce re n re eprte y mll itnce y ielectric with lrge vlue of K. In n electrolytic oule-lyer cpcitor, tiny cron grnule here to ech plte: The vlue of i the comine urfce re of the grnule, which cn e tremenou. The plte with grnule ttche re eprte y very thin ielectric heet. cpcitor of thi kin cn hve cpcitnce of 000 fr yet fit in the plm of your hn (compre Exmple 24. in ection 24.). everl prcticl evice mke ue of the wy in which cpcitor repon to chnge in ielectric contnt. One exmple i n electric tu finer, ue y uppoe the prllel plte in Fig. 24. ech hve n re of 2000 cm 2 2.00 3 0 2 m 2 2 n re.00 cm.00 3 0 22 m 2 prt. The cpcitor i connecte to power upply n chrge to potentil ifference V 0 3000 V 3.00 kv. It i then iconnecte from the power upply, n heet of inulting pltic mteril i inerte etween the plte, completely filling the pce etween them. We fin tht the potentil ifference ecree to 000 V while the chrge on ech cpcitor plte remin contnt. Compute () the originl cpcitnce C 0 ; () the mgnitue of chrge on ech plte; (c) the cpcitnce C fter the ielectric i inerte; () the ielectric contnt K of the ielectric; (e) the permittivity P of the ielectric; (f) the mgnitue of the inuce chrge i on ech fce of the ielectric; (g) the originl electric fiel E 0 etween the plte; n (h) the electric fiel E fter the ielectric i inerte. IDENTIFY: Thi prolem ue mot of the reltionhip we hve icue for cpcitor n ielectric. ET UP: Mot of the trget vrile cn e otine in everl ifferent wy. The metho ue elow re repreenttive mple; we encourge you to think of other n compre your reult. EXECUTE: () With vcuum etween the plte, we ue Eq. (24.9) with K : C 0 P 0 8.8 3 022 F/m 2 2.00 3 02 m 2.00 3 0 22 m.77 3 0 20 F 77 pf () Uing the efinition of cpcitnce, Eq. (24.), C 0 V 0.77 3 0 20 F 23.00 3 0 3 V 2.3 3 0 27 C 0.3 mc (c) When the ielectric i inerte, the chrge remin the me ut the potentil ecree to V 000 V. Hence from Eq. (24.), the new cpcitnce i C V.3 3 027 C.00 3 0 3 V.3 3 020 F 3 pf () From Eq. (24.2), the ielectric contnt i K C.3 3 020 F 3 pf C 0.77 3 0 20 F 77 pf 3.00 lterntively, from Eq. (24.3), K V 0 V 3000 V 000 V 3.00 (e) Uing K from prt () in Eq. (24.7), the permittivity i P KP 0 3.00 28.8 3 0 22 C 2 /N # m 2 2 2.66 3 0 2 C 2 /N # m 2 (f) Multiplying Eq. (24.) y the re of ech plte give the inuce chrge i i in term of the chrge on ech plte: i 2 K 2.3 3 027 C 2 2 3.00 2 3.4 3 0 27 C Continue
832 CHPTER 24 Cpcitnce n Dielectric *24. Moleculr Moel of Inuce Chrge 833 (g) ince the electric fiel etween the plte i uniform, it mgnitue i the potentil ifference ivie y the plte eprtion: E 0 V 0 3000 V.00 3 0 22 m 3.00 3 0 V/m (h) With the new potentil ifference fter the ielectric i inerte, E V or, from Eq. (24.7), E P P.00 3 0 V/m Exmple 24. Energy torge with n without ielectric Fin the totl energy tore in the electric fiel of the cpcitor in Exmple 24.0 n the energy enity, oth efore n fter the ielectric heet i inerte. IDENTIFY: In thi prolem we hve to exten the nlyi of Exmple 24.0 to inclue the ie of energy tore in cpcitor n electric-fiel energy. ET UP: We ue Eq. (24.9) to fin the tore energy efore n fter the ielectric i inerte, n Eq. (24.20) to fin the energy enity. EXECUTE: Let the originl energy e U 0 n let the energy with the ielectric in plce e U. From Eq. (24.9), U 0 2 C 0 V 0 2 2.77 3 020 F 23000 V 2 2 7.97 3 0 24 J U 2 CV 2 2.3 3 020 F 2000 V 2 2 2.66 3 0 24 J The finl energy i one-thir of the originl energy. The energy enity without the ielectric i given y Eq. (24.20) with K : u 0 2 P 0 E 2 0 2 8.8 3 022 C 2 /N # m 2 23.0 3 0 N/C 2 2 0.398 J/m 3 With the ielectric in plce, u 2 PE 2 2 2.66 3 02 C 2 /N # m 2 2.00 3 0 N/C 2 2 0.33 J/m 3 000 V.00 3 0 22 m.00 3 0 V/m.3 3 0 27 C 2.66 3 0 2 C 2 /N # m 2 22.00 3 0 2 m 2 2 The energy enity with the ielectric i one-thir of the originl energy enity. or, from Eq. (24.), E i 2 i P 0 P 0.3 2 3.4 2 3 0 27 C 8.8 3 0 22 C 2 /N # m 2 22.00 3 0 2 m 2 2.00 3 0 V/m or, from Eq. (24.4), EVLUTE: It lwy ueful to check the reult y fining them in more thn one wy, we i in prt () n (h). Our reult how tht inerting the ielectric incree the cpcitnce y fctor of K 3.00 n reuce the electric fiel etween the plte y fctor of /K /3.00. It i o y eveloping inuce chrge on the fce of the ielectric of mgnitue 2 /K 2 2 /3.00 2 0.667. EVLUTE: We cn check our nwer for u 0 y noting tht the volume etween the plte i V 0.200 m 2 2 0.000 m 2 0.00200 m 3. ince the electric fiel i uniform etween the plte, u 0 i uniform well n the energy enity i jut the tore energy ivie y the volume: u 0 U 0 V 7.97 3 024 J 0.00200 m 3 0.398 J/m 3 which gree with our erlier nwer. You houl ue the me pproch to check the vlue for U, the energy enity with the ielectric. We cn generlize the reult of thi exmple. When ielectric i inerte into cpcitor while the chrge on ech plte remin the me, the permittivity P incree y fctor of K (the ielectric contnt), the electric fiel ecree y fctor of /K, n the energy enity u 2PE 2 ecree y fctor of /K. Where i the energy go? The nwer lie in the fringing fiel t the ege of rel prllel-plte cpcitor. Fig. 24.6 how, tht fiel ten to pull the ielectric into the pce etween the plte, oing work on it it oe o. We coul ttch pring to the left en of the ielectric in Fig. 24.6 n ue thi force to tretch the pring. Becue work i one y the fiel, the fiel energy enity ecree. 24.6 The fringing fiel t the ege of the cpcitor exert force F n F 2i i on the negtive n poitive inuce urfce chrge of ielectric, pulling the ielectric into the cpcitor. Dielectric E E 0 K 3.00 3 0 V/m.00 3 0 V/m 3.00 F 2i F i E Dielectric Brekown We mentione erlier tht when ny ielectric mteril i ujecte to ufficiently trong electric fiel, ielectric rekown tke plce n the ielectric ecome conuctor (Fig. 24.7). Thi occur when the electric fiel i o trong tht electron re rippe looe from their molecule n crh into other molecule, lierting even more electron. Thi vlnche of moving chrge, forming prk or rc ichrge, often trt quite uenly. Becue of ielectric rekown, cpcitor lwy hve mximum voltge rting. When cpcitor i ujecte to exceive voltge, n rc my form through lyer of ielectric, urning or melting hole in it. Thi rc crete conucting pth ( hort circuit) etween the conuctor. If conucting pth remin fter the rc i extinguihe, the evice i renere permnently uele cpcitor. The mximum electric-fiel mgnitue tht mteril cn withtn without the occurrence of rekown i clle it ielectric trength. Thi quntity i ffecte ignificntly y temperture, trce impuritie, mll irregulritie in the metl electroe, n other fctor tht re ifficult to control. For thi reon we cn give only pproximte figure for ielectric trength. The ielectric trength of ry ir i out 3 3 0 6 V/m. Vlue of ielectric trength for few common inulting mteril re hown in Tle 24.2. Note tht the vlue re ll utntilly greter thn the vlue for ir. For exmple, lyer of polycronte 0.0 mm thick (out the mllet prcticl thickne) h 0 time the ielectric trength of ir n cn withtn mximum voltge of out 3 3 0 7 V/m 2 3 0 2 m 2 300 V. Tle 24.2 Dielectric Contnt n Dielectric trength of ome Inulting Mteril Mteril Dielectric Contnt, K Dielectric trength, E m ( V/m ) Polycronte 2.8 3 3 0 7 Polyeter 3.3 6 3 0 7 Polypropylene 2.2 7 3 0 7 Polytyrene 2.6 2 3 0 7 Pyrex gl 4.7 3 0 7 Tet Your Unertning of ection 24.4 The pce etween the plte of n iolte prllel-plte cpcitor i fille y l of ielectric with ielectric contnt K. The two plte of the cpcitor hve chrge n 2. You pull out the ielectric l. If the chrge o not chnge, how oe the energy in the cpcitor chnge when you remove the l? (i) It incree; (ii) it ecree; (iii) it remin the me. *24. Moleculr Moel of Inuce Chrge In ection 24.4 we icue inuce urfce chrge on ielectric in n electric fiel. Now let look t how thee urfce chrge cn rie. If the mteril were conuctor, the nwer woul e imple. Conuctor contin chrge tht i free to move, n when n electric fiel i preent, ome of the chrge reitriute itelf on the urfce o tht there i no electric fiel inie the conuctor. But n iel ielectric h no chrge tht re free to move, o how cn urfce chrge occur? To unertn thi, we hve to look gin t rerrngement of chrge t the moleculr level. ome molecule, uch H 2 O n N 2 O, hve equl mount of poitive n negtive chrge ut lopie itriution, with exce poitive chrge concentrte on one ie of the molecule n negtive chrge on the other. we ecrie in ection 2.7, uch n rrngement i clle n electric ipole, n the molecule i clle polr molecule. When no electric fiel i preent in g or liqui with polr molecule, the molecule re oriente rnomly (Fig. 24.8). When they re plce in n electric fiel, however, they ten 24.7 very trong electric fiel cue ielectric rekown in lock of Plexigl. The reulting flow of chrge etche thi pttern into the lock. 24.8 Polr molecule () without n () with n pplie electric fiel E. () () In the ence of n electric fiel, polr molecule orient rnomly. E When n electric fiel i pplie, the molecule ten to lign with it.