Take a length of copper wire and wrap it around a pencil to form a coil. If

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1 30 ANING GOA By sudyng hs chper, you wll lern: How me-vryng curren n one col cn nduce n emf n second, unconneced col. How o rele he nduced emf n crcu o he re of chnge of curren n he sme crcu. How o clcule he energy sored n mgnec feld. How o nlyze crcus h nclude oh ressor nd n nducor (col). Why elecrcl oscllons occur n crcus h nclude oh n nducor nd cpcor. Why oscllons decy n crcus wh n nducor, ressor, nd cpcor. INDUCTANC eween wo crcus when here s chngng curren n one of he crcus. Consder wo neghorng cols of wre, s n Fg A curren flowng n col produces mgnec feld B nd hence mgnec flux hrough col 2. If he curren n col chnges, he flux hrough col 2 chnges s well; ccordng o Frdy s lw, hs nduces n emf n col 2. In hs wy, chnge n he curren n one crcu cn nduce curren n second crcu. e s nlyze he suon shown n Fg. 30. n more del. We wll use lowercse leers o represen qunes h vry wh me; for exmple, mevryng curren s, ofen wh suscrp o denfy he crcu. In Fg. 30. curren n col ses up mgnec feld (s ndced y he lue lnes), nd some of hese feld lnes pss hrough col 2. We denoe he mgnec flux hrough ech urn of col 2, cused y he curren n col, s F B2. (If he flux s dfferen hrough dfferen urns of he col, hen F B2 denoes he verge flux.) The mgnec feld s proporonl o, so F B2 s lso proporonl o. When chnges, F B2 chnges; hs chngng flux nduces n emf n col 2, gven y Muul Inducnce A curren n col gves rse o mgnec flux hrough col 2. Muul nducnce: If he curren n col s chngng, he chngng flux hrough col 2 nduces n emf n col 2. Col N urns B Col 2 N 2 urns F B2?Mny rffc lghs chnge when cr rolls up o he nersecon. How does he lgh sense he presence of he cr? Tke lengh of copper wre nd wrp round pencl o form col. If you pu hs col n crcu, does ehve ny dfferenly hn srgh pece of wre? emrkly, he nswer s yes. In n ordnry gsolnepowered cr, col of hs knd mkes possle for he 2-vol cr ery o provde housnds of vols o he sprk plugs, whch n urn mkes possle for he plugs o fre nd mke he engne run. Oher cols of hs ype re used o keep fluorescen lgh fxures shnng. rger cols plced under cy srees re used o conrol he operon of rffc sgnls. All of hese pplcons, nd mny ohers, nvolve he nducon effecs h we suded n Chper 29. A chngng curren n col nduces n emf n n djcen col. The couplng eween he cols s descred y her muul nducnce. A chngng curren n col lso nduces n emf n h sme col. uch col s clled n nducor, nd he relonshp of curren o emf s descred y he nducnce (lso clled selfnducnce) of he col. If col s nlly crryng curren, energy s relesed when he curren decreses; hs prncple s used n uomove gnon sysems. We ll fnd h hs relesed energy ws sored n he mgnec feld cused y he curren h ws nlly n he col, nd we ll look some of he prccl pplcons of mgnec-feld energy. We ll lso ke frs look wh hppens when n nducor s pr of crcu. In Chper 3 we ll go on o sudy how nducors ehve n lernng-curren crcus; n h chper we ll lern why nducors ply n essenl role n modern elecroncs, ncludng communcon sysems, power supples, nd mny oher devces. 30. Muul Inducnce In econ 28.4 we consdered he mgnec nercon eween wo wres crryng sedy currens; he curren n one wre cuses mgnec feld, whch exers force on he curren n he second wre. Bu n ddonl nercon rses (30.) We could represen he proporonly of F B2 nd n he form F B2 5 (consn), u nsed s more convenen o nclude he numer of urns N 2 n he relonshp. Inroducng proporonly consn M 2, clled he muul nducnce of he wo cols, we wre N 2 F B2 5 M 2 where F B2 s he flux hrough sngle urn of col 2. From hs, nd we cn rewre q. (30.) s df B2 d N 2 5 M 2 df B2 2 52N 2 d 2 52M 2 (30.2) (30.3) Th s, chnge n he curren n col nduces n emf n col 2 h s drecly proporonl o he re of chnge of (Fg. 30.2). We my lso wre he defnon of muul nducnce, q. (30.2), s M 2 5 N 2 F B2 F B2 If he cols re n vcuum, he flux hrough ech urn of col 2 s drecly proporonl o he curren. Then he muul nducnce M 2 s consn h depends only on he geomery of he wo cols (he sze, shpe, numer of urns, nd orenon of ech col nd he sepron eween he cols). If mgnec merl s presen, M 2 lso depends on he mgnec properes of he merl. If he merl hs nonlner mgnec properes, h s, f he relve permely K m (defned n econ 28.8) s no consn nd mgnezon s no proporonl o mgnec feld, hen F B2 s no longer drecly proporonl o. In h cse he muul nducnce lso depends on he vlue of. In hs dscusson we wll ssume h ny mgnec merl presen hs consn K m so h flux s drecly proporonl o curren nd M 2 depends on geomery only. We cn repe our dscusson for he oppose cse n whch chngng curren 2 n col 2 cuses chngng flux F B nd n emf n col. We mgh expec h he correspondng consn M 2 would e dfferen from M 2 ecuse n generl he wo cols re no dencl nd he flux hrough hem s no he sme. I urns ou, however, h M 2 s lwys equl o M 2, even when he wo cols re no symmerc. We cll hs common vlue smply he muul nducnce, 30.2 Ths elecrc oohrush mkes use of muul nducnce. The se conns col h s suppled wh lernng curren from wll socke. Ths vryng curren nduces n emf n col whn he oohrush self, whch s used o rechrge he oohrush ery. Toohrush wh col conneced o ery Bse wh rechrgng col conneced o wll socke 030

2 032 CHAPT 30 Inducnce 30. Muul Inducnce 033 denoed y he symol M whou suscrps; chrcerzes compleely he nduced-emf nercon of wo cols. Then we cn wre d 2 52M where he muul nducnce M s M 5 N 2 F B2 d 2 nd 52M 5 N F B 2 (muully nduced emfs) (30.4) (muul nducnce) (30.5) The negve sgns n q. (30.4) re reflecon of enz s lw. The frs equon sys h chnge n curren n col cuses chnge n flux hrough col 2, nducng n emf n col 2 h opposes he flux chnge; n he second equon he roles of he wo cols re nerchnged. CAUTION Only me-vryng curren nduces n emf Noe h only mevryng curren n col cn nduce n emf nd hence curren n second col. quons (30.4) show h he nduced emf n ech col s drecly proporonl o he re of chnge of he curren n he oher col, no o he vlue of he curren. A sedy curren n one col, no mer how srong, cnno nduce curren n neghorng col. The I un of muul nducnce s clled he henry ( H), n honor of he Amercn physcs Joseph Henry ( ), one of he dscoverers of elecromgnec nducon. From q. (30.5), one henry s equl o one weer per mpere. Oher equvlen uns, oned y usng q. (30.4), re one vol-second per mpere, one ohm-second, or one joule per mpere squred: H 5 W/A 5 V # s/a 5 V # s 5 J/A 2 Jus s he frd s rher lrge un of cpcnce (see econ 24.), he henry s rher lrge un of muul nducnce. As xmple 30. shows, ypcl vlues of muul nducnce cn e n he mllhenry (mh) or mcrohenry mh 2 rnge. Drwcks nd Uses of Muul Inducnce Muul nducnce cn e nusnce n elecrc crcus, snce vrons n curren n one crcu cn nduce unwned emfs n oher nery crcus. To mnmze hese effecs, mulple-crcu sysems mus e desgned so h M s s smll s possle; for exmple, wo cols would e plced fr pr or wh her plnes perpendculr. Hpply, muul nducnce lso hs mny useful pplcons. A rnsformer, used n lernng-curren crcus o rse or lower volges, s fundmenlly no dfferen from he wo cols shown n Fg A me-vryng lernng curren n one col of he rnsformer produces n lernng emf n he oher col; he vlue of M, whch depends on he geomery of he cols, deermnes he mplude of he nduced emf n he second col nd hence he mplude of he oupu volge. (We ll descre rnsformers n more del n Chper 3 fer we ve dscussed lernng curren n greer deph.) 30.3 A long solenod wh cross-seconl re A nd N urns (shown n lck) s surrounded s cener y col wh N 2 urns (shown n lue). T UP: We use q. (30.5) o deermne he muul nducnce M. Accordng o h equon, we need o know eher () he flux F B2 hrough ech urn of he ouer col due o curren n he solenod or () he flux F B hrough ech urn of he solenod due o curren 2 n he ouer col. We choose opon () snce from xmple 28.9 (econ 28.7) we hve smple expresson for he feld he cener of long curren-crryng solenod, gven y q. (28.23). Noe h we re no gven vlue for he curren n he solenod. Ths omsson s no cuse for lrm, however: The vlue of he muul nducnce doesn depend on he vlue of he curren, so he quny should cncel ou when we clcule M. xmple 30.2 Cross-seconl re A mf due o muul nducnce In xmple 30., suppose he curren 2 n he ouer, surroundng col s gven y A/s 2 (currens n wres cn ndeed ncrese hs rpdly for ref perods). () A me ms, wh verge mgnec flux hrough ech urn of he solenod s cused y he curren n he ouer, surroundng col? () Wh s he nduced emf n he solenod? IDNTIFY: In xmple 30. we found he muul nducnce y relng he curren n he solenod o he flux produced n he ouer col. In hs exmple we re gven he curren n he ouer col nd wn o fnd he resulng flux n he solenod. The key pon s h he muul nducnce s he sme n eher cse. T UP: Gven he vlue of he muul nducnce M 5 25 mh from xmple 30., we use q. (30.5) o deermne he flux F B hrough ech urn of he solenod cused y gven curren 2 n he ouer col. We hen use q. (30.4) o deermne he emf nduced n he solenod y he me vron of he ouer col s curren. XCUT: () A me ms s, he curren n he ouer col (col 2) s A/s s 2 5 l Blue col: N 2 urns Blck col: N urns XCUT: From xmple 28.9, long solenod crryng curren produces mgnec feld B h pons long he xs of he solenod. The feld mgnude B s proporonl o nd o n, he numer of urns per un lengh: The flux hrough cross secon of he solenod equls B A. nce very long solenod produces no mgnec feld ousde of s col, hs s lso equl o he flux F B2 hrough ech urn of he ouer, surroundng col, no mer wh he cross-seconl re of he ouer col. From q. (30.5) he muul nducnce M s M 5 N 2 F B2 B 5m 0 n 5 m 0 N l 5 N 2 B A 5 N 2 VAUAT: The muul nducnce of ny wo cols s lwys proporonl o he produc N N 2 of her numers of urns. Noce h he muul nducnce M depends only on he geomery of he wo cols, no on he curren. Here s numercl exmple o gve you n de of mgnudes. uppose l m, A 5 0 cm m 2, N urns, nd N urns. Then M 5 4p 3027 W/A # m m m W/A H 5 25 mh 6.0 A. To fnd he verge flux hrough ech urn of he solenod (col ), we solve q. (30.5) for F B : F B 5 M H 26.0 A 2 N 000 Noe h hs s n verge vlue; he flux cn vry consderly eween he cener nd he ends of he solenod. () The nduced emf s gven y q. (30.4): m 0 N l A 5 m 0 AN N 2 l W d 2 52M H 2 d A/s H A/s V VAUAT: Ths s susnl nduced emf n response o very rpd re of chnge of curren. In n operng Tesl col, here s hgh-frequency lernng curren rher hn connuously ncresng curren s n hs exmple; oh d 2/ nd lerne s well, wh mpludes h cn e housnds of mes lrger hn n hs exmple. xmple 30. Clculng muul nducnce In one form of Tesl col ( hgh-volge generor h you my hve seen n scence museum), long solenod wh lengh l nd cross-seconl re A s closely wound wh N urns of wre. A col wh N 2 urns surrounds s cener (Fg. 30.3). Fnd he muul nducnce. IDNTIFY: Muul nducnce occurs n hs suon ecuse curren n one of he cols ses up mgnec feld h cuses flux hrough he oher col. Tes Your Undersndng of econ 30. Consder he Tesl col descred n xmple 30.. If you mke he solenod ou of wce s much wre, so h hs wce s mny urns nd s wce s long, how much lrger s he muul nducnce? () M s four mes greer; () M s wce s gre; () M s unchnged; (v) M s 2 s gre; (v) M s 4 s gre.

3 034 CHAPT 30 Inducnce 30.2 elf-inducnce nd Inducors The curren n he crcu cuses mgnec feld B n he col nd hence flux hrough he col. elf-nducnce: If he curren n he col s chngng, he chngng flux hrough he col nduces n emf n he col. B 30.2 elf-inducnce nd Inducors In our dscusson of muul nducnce we consdered wo sepre, ndependen crcus: A curren n one crcu crees mgnec feld nd hs feld gves rse o flux hrough he second crcu. If he curren n he frs crcu chnges, he flux hrough he second crcu chnges nd n emf s nduced n he second crcu. An mporn reled effec occurs even f we consder only sngle soled crcu. When curren s presen n crcu, ses up mgnec feld h cuses mgnec flux hrough he sme crcu; hs flux chnges when he curren chnges. Thus ny crcu h crres vryng curren hs n emf nduced n y he vron n s own mgnec feld. uch n emf s clled selfnduced emf. By enz s lw, self-nduced emf lwys opposes he chnge n he curren h cused he emf nd so ends o mke more dffcul for vrons n curren o occur. For hs reson, self-nduced emfs cn e of gre mpornce whenever here s vryng curren. elf-nduced emfs cn occur n ny crcu, snce here s lwys some mgnec flux hrough he closed loop of curren-crryng crcu. Bu he effec s grely enhnced f he crcu ncludes col wh N urns of wre (Fg. 30.4). As resul of he curren, here s n verge mgnec flux F B hrough ech urn of he col. In nlogy o q. (30.5) we defne he self-nducnce of he crcu s 5 NF B (self-nducnce) (30.6) When here s no dnger of confuson wh muul nducnce, he self-nducnce s clled smply he nducnce. Comprng qs. (30.5) nd (30.6), we see h he uns of self-nducnce re he sme s hose of muul nducnce; he I un of self-nducnce s one henry. If he curren n he crcu chnges, so does he flux F B ; from rerrngng q. (30.6) nd kng he dervve wh respec o me, he res of chnge re reled y From Frdy s lw for col wh N urns, q. (29.4), he self-nduced emf s 52N df B/, so follows h d 52 df B N d 5 (self-nduced emf) (30.7) ke ressors nd cpcors, nducors re mong he ndspensle crcu elemens of modern elecroncs. Ther purpose s o oppose ny vrons n he curren hrough he crcu. An nducor n drec-curren crcu helps o mnn sedy curren despe ny flucuons n he ppled emf; n n lernng-curren crcu, n nducor ends o suppress vrons of he curren h re more rpd hn desred. In hs chper nd he nex we wll explore he ehvor nd pplcons of nducors n crcus n more del. To undersnd he ehvor of crcus connng nducors, we need o develop generl prncple nlogous o Krchhoff s loop rule (dscussed n econ 26.2). To pply h rule, we go round conducng loop, mesurng poenl dfferences cross successve crcu elemens s we go. The lgerc sum of hese dfferences round ny closed loop mus e zero ecuse he elecrc feld produced y chrges dsrued round he crcu s conservve. In econ 29.7 we denoed such conservve feld s c. When n nducor s ncluded n he crcu, he suon chnges. The mgneclly nduced elecrc feld whn he cols of he nducor s no conservve; s n econ 29.7, we ll denoe y n. We need o hnk very crefully ou he roles of he vrous felds. e s ssume we re delng wh n nducor whose cols hve neglgle ressnce. Then neglgly smll elecrc feld s requred o mke chrge move hrough he cols, so he ol elecrc feld c n whn he cols mus e zero, even hough neher feld s ndvdully zero. Becuse c s nonzero, we know here hve o e ccumulons of chrge on he ermnls of he nducor nd he surfces of s conducors, o produce hs feld. Consder he crcu shown n Fg. 30.5; he ox conns some comnon of eres nd vrle ressors h enles us o conrol he curren n he crcu. Accordng o Frdy s lw, q. (29.0), he lne negrl of round he crcu s he negve of he re of chnge of flux hrough he crcu, whch n urn s gven y q. (30.7). Comnng hese wo relonshps, we ge where we negre clockwse round he loop (he drecon of he ssumed curren). Bu s dfferen from zero only whn he nducor. Therefore he negrl of n n round he whole loop cn e replced y s negrl only from o hrough he nducor; h s, Nex, ecuse c n 5 0 rewre hs s # C n d d l 52 # 3 n d d l 52 n ech pon whn he nducor cols, we cn 30.5 A crcu connng source of emf nd n nducor. The source s vrle, so he curren nd s re of chnge d/ cn e vred. Vrle source of emf The mnus sgn n q. (30.7) s reflecon of enz s lw; sys h he selfnduced emf n crcu opposes ny chnge n he curren n h crcu. (er n hs secon we ll explore n greer deph he sgnfcnce of hs mnus sgn.) quon (30.7) lso ses h he self-nducnce of crcu s he mgnude of he self-nduced emf per un re of chnge of curren. Ths relonshp mkes possle o mesure n unknown self-nducnce n relvely smple wy: Chnge he curren n he crcu known re d/, mesure he nduced emf, nd ke he ro o deermne. Bu hs negrl s jus he poenl fnlly on # 3 c d d l 5 V of pon wh respec o pon, so we d V 5 V 2 V 5 (30.8) Inducors As Crcu lemens A crcu devce h s desgned o hve prculr nducnce s clled n nducor, or choke. The usul crcu symol for n nducor s We conclude h here s genune poenl dfference eween he ermnls of he nducor, ssoced wh conservve, elecrosc forces, despe he fc h he elecrc feld ssoced wh he mgnec nducon effec s nonconservve. Thus we re jusfed n usng Krchhoff s loop rule o nlyze crcus h nclude nducors. quon (30.8) gves he poenl dfference cross n nducor n crcu.

4 036 CHAPT 30 Inducnce 30.2 elf-inducnce nd Inducors () The poenl dfference cross ressor depends on he curren. (), (c), (d) The poenl dfference cross n nducor depends on he re of chnge of he curren. () essor wh curren flowng from o : poenl drops from o. V 5. 0 () Inducor wh consn curren flowng from o : no poenl dfference. consn: d/ 5 0 V 5 d (c) Inducor wh ncresng curren flowng from o : poenl drops from o. ncresng: d/. 0 V 5 d. 0 (d) Inducor wh decresng curren flowng from o : poenl ncreses from o. decresng: d/, 0 V 5 d, These fluorescen lgh ues re wred n seres wh n nducor, or lls, h helps o susn he curren flowng hrough he ues. CAUTION elf-nduced emf opposes chnges n curren Noe h he selfnduced emf does no oppose he curren self; rher, opposes ny chnge d/ 2 n he curren. Thus he crcu ehvor of n nducor s que dfferen from h of ressor. Fgure 30.6 compres he ehvors of ressor nd n nducor nd summrzes he sgn relonshps. Applcons of Inducors Becuse n nducor opposes chnges n curren, plys n mporn role n fluorescen lgh fxures (Fg. 30.7). In such fxures, curren flows from he wrng no he gs h flls he ue, onzng he gs nd cusng o glow. However, n onzed gs or plsm s hghly nonohmc conducor: The greer he curren, he more hghly onzed he plsm ecomes nd he lower s ressnce. If suffcenly lrge volge s ppled o he plsm, he curren cn grow so much h dmges he crcury ousde he fluorescen ue. To preven hs prolem, n nducor or mgnec lls s pu n seres wh he fluorescen ue o keep he curren from growng ou of ounds. The lls lso mkes possle for he fluorescen ue o work wh he lernng volge provded y household wrng. Ths volge osclles snusodlly wh frequency of 60 Hz, so h goes momenrly o zero 20 mes per second. If here were no lls, he plsm n he fluorescen ue would rpdly deonze when he volge wen o zero nd he ue would shu off. Wh lls presen, self-nduced emf susns he curren nd keeps he ue l. Mgnec llss re lso used for hs purpose n sreelghs (whch on her lgh from glowng vpor of mercury or sodum oms) nd n neon lghs. (In compc fluorescen lmps, he mgnec lls s replced y more complced scheme for regulng curren. Ths scheme ulzes rnssors, dscussed n Chper 42.) The self-nducnce of crcu depends on s sze, shpe, nd numer of urns. For N urns close ogeher, s lwys proporonl o N 2. I lso depends on he mgnec properes of he merl enclosed y he crcu. In he followng exmples we wll ssume h he crcu encloses only vcuum (or r, whch from he sndpon of mgnesm s essenlly vcuum). If, however, he flux s concenred n regon connng mgnec merl wh permely m, hen n he expresson for B we mus replce m 0 (he permely of vcuum) y m5k m m 0, s dscussed n econ If he merl s dmgnec or prmgnec, hs replcemen mkes very lle dfference, snce K m s very close o. If he merl s ferromgnec, however, he dfference s of crucl mpornce. A solenod wound on sof ron core hvng K m cn hve n nducnce pproxmely 5000 mes s gre s h of he sme solenod wh n r core. Ferromgnec-core nducors re very wdely used n vrey of elecronc nd elecrc-power pplcons. An dded complcon s h wh ferromgnec merls he mgnezon s n generl no lner funcon of mgnezng curren, especlly s suron s pproched. As resul, he nducnce s no consn u cn depend on curren n frly complced wy. In our dscusson we wll gnore hs complcon nd ssume lwys h he nducnce s consn. Ths s resonle ssumpon even for ferromgnec merl f he mgnezon remns well elow he suron level. Becuse uomoles conn seel, ferromgnec merl, drvng n? uomole over col cuses n pprecle ncrese n he col s nducnce. Ths effec s used n rffc lgh sensors, whch use lrge, curren-crryng col emedded under he rod surfce ner n nersecon. The crcury conneced o he col deecs he nducnce chnge s cr drves over. When preprogrmmed numer of crs hve pssed over he col, he lgh chnges o green o llow he crs hrough he nersecon. xmple 30.3 Clculng self-nducnce A orodl solenod wh cross-seconl re A nd men rdus r curren d/. We re no gven ny nformon ou he emf, so s closely wound wh N urns of wre (Fg. 30.8). The orod s we mus use he frs pproch. We use he resuls of xmple 28.0 (econ 28.7), n whch we found he mgnec feld n wound on nonmgnec core. Deermne s self-nducnce. Assume h B s unform cross cross secon (h s, neglec he he neror of orodl solenod. vron of B wh dsnce from he orod xs). XCUT: From q. (30.6), he self-nducnce s 5 NF B/. From xmple 28.0, he feld mgnude dsnce r from he IDNTIFY: Our rge vrle s he self-nducnce of he orod xs s B 5m 0 N/2pr. If we ssume h he feld hs hs orodl solenod. mgnude over he enre cross-seconl re A, hen he mgnec flux hrough he cross secon s T UP: We cn deermne n one of wo wys: eher wh q. (30.6), whch requres knowng he flux F B hrough ech urn F nd he curren n he col, or from q. (30.7), whch requres B 5 BA 5 m 0 NA 2pr knowng he self-nduced emf due o gven re of chnge of The flux F B s he sme hrough ech urn, nd he self-nducnce s 30.8 Deermnng he self-nducnce of closely wound orodl solenod. For clry, only few urns of he wndng re shown. Pr of he orod hs een cu wy o show he cross-seconl re A nd rdus r. xmple 30.4 Numer of urns 5 N (only few re shown) Clculng self-nduced emf If he curren n he orodl solenod n xmple 30.3 ncreses unformly from 0 o 6.0 A n 3.0 ms, fnd he mgnude nd drecon of he self-nduced emf. IDNTIFY: We re gven, he self-nducnce, nd d/, he re of chnge of he curren. Our rge vrle s he selfnduced emf. T UP: We clcule he emf usng q. (30.7). XCUT: The re of chnge of he solenod curren s d/ A2 / s A/s. From q. (30.7), he mgnude of he nduced emf s A r 5 NF B 5 m 0 N 2 A 2pr Tes Your Undersndng of econ 30.2 nk he followng nducors n order of he poenl dfference V, from mos posve o mos negve. In ech cse he nducor hs zero ressnce nd he curren flows from pon hrough he nducor o pon. () The curren hrough 2.0-mH nducor ncreses from.0 A o 2.0 A n 0.50 s; () he curren hrough 4.0-mH nducor decreses from 3.0 A o 0 n 2.0 s; () he curren hrough.0-mh nducor remns consn 4.0 A; (v) he curren hrough.0-mh nducor ncreses from 0 o 4.0 A n 0.25 s. VAUAT: uppose N urns, 0 24 m 2, nd r m; hen H 5 40 mh (self-nducnce of orodl solenod) A cm p 3027 W / A # m m 2 2 2p 0.0 m 2 er n hs chper we wll use he expresson 5m 0 N 2 A/2pr for he nducnce of orodl solenod o help develop n expresson for he energy sored n mgnec feld. d P P H A/s V The curren s ncresng, so ccordng o enz s lw he drecon of he emf s oppose o h of he curren. Ths corresponds o he suon n Fg. 30.6c; he emf s n he drecon from o, lke ery wh s he ermnl nd he 2 ermnl, endng o oppose he curren ncrese from he exernl crcu. VAUAT: Ths exmple shows h even smll nducnce cn gve rse o susnl nduced emf f he curren chnges rpdly.

5 038 CHAPT 30 Inducnce 30.3 Mgnec-Feld nergy A ressor s devce n whch energy s rrecoverly dssped. By conrs, energy sored n curren-crryng nducor cn e recovered when he curren decreses o zero. essor wh curren : energy s dssped. Inducor wh curren : energy s sored Mgnec-Feld nergy slshng curren n n nducor requres n npu of energy, nd n nducor crryng curren hs energy sored n. e s see how hs comes ou. In Fg. 30.5, n ncresng curren n he nducor cuses n emf eween s ermnls, nd correspondng poenl dfference V eween he ermnls of he source, wh pon hgher poenl hn pon. Thus he source mus e ddng energy o he nducor, nd he nsnneous power P (re of rnsfer of energy no he nducor) s P 5 V. nergy ored n n Inducor We cn clcule he ol energy npu U needed o eslsh fnl curren I n n nducor wh nducnce f he nl curren s zero. We ssume h he nducor hs zero ressnce, so no energy s dssped whn he nducor. e he curren some nsn e nd le s re of chnge e d/; he curren s ncresng, so d/. 0. The volge eween he ermnls nd of he nducor hs nsn s V 5 d/, nd he re P whch energy s eng delvered o he nducor (equl o he nsnneous power suppled y he exernl source) s The energy du suppled o he nducor durng n nfnesml me nervl s du 5 P, so The ol energy U suppled whle he curren ncreses from zero o fnl vlue I s I U 5 3 d 5 2 I 2 0 d P 5 V 5 du 5 d (energy sored n n nducor) (30.9) Afer he curren hs reched s fnl sedy vlue I, d/ 5 0 nd no more energy s npu o he nducor. When here s no curren, he sored energy U s zero; when he curren s I, he energy s 2 I 2. When he curren decreses from I o zero, he nducor cs s source h supples ol moun of energy 2 I 2 o he exernl crcu. If we nerrup he crcu suddenly y openng swch or ynkng plug from wll socke, he curren decreses very rpdly, he nduced emf s very lrge, nd he energy my e dssped n n rc cross he swch concs. Ths lrge emf s he elecrcl nlog of he lrge force exered y cr runnng no rck wll nd soppng very suddenly. CAUTION nergy, ressors, nd nducors I s mporn no o confuse he ehvor of ressors nd nducors where energy s concerned (Fg. 30.9). nergy flows no ressor whenever curren psses hrough, wheher he curren s sedy or vryng; hs energy s dssped n he form of he. By conrs, energy flows no n del, zeroressnce nducor only when he curren n he nducor ncreses. Ths energy s no dssped; s sored n he nducor nd relesed when he curren decreses. When sedy curren flows hrough n nducor, here s no energy flow n or ou. Mgnec nergy Densy The energy n n nducor s cully sored n he mgnec feld whn he col, jus s he energy of cpcor s sored n he elecrc feld eween s ples. We cn develop relonshps for mgnec-feld energy nlogous o hose we oned for elecrc-feld energy n econ 24.3 [qs. (24.9) nd (24.)]. We wll concenre on one smple cse, he del orodl solenod. Ths sysem hs he dvnge h s mgnec feld s confned compleely o fne regon of spce whn s core. As n xmple 30.3, we ssume h he cross-seconl re A s smll enough h we cn preend h he mgnec feld s unform over he re. The volume V enclosed y he orodl solenod s pproxmely equl o he crcumference 2pr mulpled y he re A: V 5 2prA. From xmple 30.3, he self-nducnce of he orodl solenod wh vcuum whn s cols s From q. (30.9), he energy U sored n he orodl solenod when he curren s I s The mgnec feld nd herefore hs energy re loclzed n he volume V 5 2prA enclosed y he wndngs. The energy per un volume, or mgnec energy densy, s u 5 U/V: We cn express hs n erms of he mgnude B of he mgnec feld nsde he orodl solenod. From q. (28.24) n xmple 28.0 (econ 28.7), hs s nd so When we susue hs no he ove equon for u, we fnlly fnd he expresson for mgnec energy densy n vcuum: u 5 B2 2m 0 (mgnec energy densy n vcuum) (30.0) Ths s he mgnec nlog of he energy per un volume n n elecrc feld n vcuum, u 5 2 P 0 2, whch we derved n econ When he merl nsde he orod s no vcuum u merl wh (consn) mgnec permely m5k m m 0, we replce m 0 y m n q. (30.0). The energy per un volume n he mgnec feld s hen u 5 B2 2m U 5 2 I 2 5 m 0 N 2 A 2 2pr u 5 5 m 0 N 2 A 2pr U 2prA 5 2 m 0 N 2 I 2 2pr 2 2 B 5 m 0 NI 2pr N 2 I 2 2pr 2 5 B2 2 2 m 0 (mgnec energy densy n merl) (30.) Alhough we hve derved q. (30.) only for one specl suon, urns ou o e he correc expresson for he energy per un volume ssoced wh ny mgnec-feld confguron n merl wh consn permely. For vcuum, q. (30.) reduces o q. (30.0). We wll use he expressons for elecrc-feld nd mgnec-feld energy n Chper 32 when we sudy he energy ssoced wh elecromgnec wves. I 2

6 040 CHAPT 30 Inducnce 30.4 The - Crcu The energy requred o fre n uomole sprk plug s derved from mgnec-feld energy sored n he gnon col. Mgnec-feld energy plys n mporn role n he gnon sysems of gsolne-powered uomoles. A prmry col of ou 250 urns s conneced o he cr s ery nd produces srong mgnec feld. Ths col s surrounded y secondry col wh some 25,000 urns of very fne wre. When s me for sprk plug o fre (see Fg n econ 20.3), he curren o he prmry col s nerruped, he mgnec feld quckly drops o zero, nd n emf of ens of housnds of vols s nduced n he secondry col. The energy sored n he mgnec feld hus goes no powerful pulse of curren h rvels hrough he secondry col o he sprk plug, generng he sprk h gnes he fuel r mxure n he engne s cylnders (Fg. 30.0) The - Crcu e s look some exmples of he crcu ehvor of n nducor. One hng s cler lredy; n nducor n crcu mkes dffcul for rpd chnges n curren o occur, hnks o he effecs of self-nduced emf. quon (30.7) shows h he greer he re of chnge of curren d/, he greer he self-nduced emf nd he greer he poenl dfference eween he nducor ermnls. Ths equon, ogeher wh Krchhoff s rules (see econ 26.2), gves us he prncples we need o nlyze crcus connng nducors. ONIN 4. The Crcu Prolem-olvng regy 30. Inducors n Crcus xmple 30.5 orng energy n n nducor The elecrc-power ndusry would lke o fnd effcen wys o sore surplus energy genered durng low-demnd hours o help mee cusomer requremens durng hgh-demnd hours. Perhps lrge nducor cn e used. Wh nducnce would e needed o sore.00 kw # h of energy n col crryng 200-A curren? IDNTIFY: We re gven he requred moun of sored energy U nd he curren I. Our rge vrle s he self-nducnce. T UP: We solve for usng q. (30.9) XCUT: We hve I A nd U 5.00 kw # h W s J. olvng q. (30.9) for, we fnd 5 2U I J H A 2 2 xmple 30.6 Mgnec energy densy In proon cceleror used n elemenry prcle physcs expermens, he rjecores of proons re conrolled y endng mgnes h produce mgnec feld of 6.6 T. Wh s he energy densy n hs feld n he vcuum eween he poles of such mgne? IDNTIFY: Our rge vrle s he mgnec energy densy u. we re gven he mgnude B of he mgnec feld. T UP: In vcuum, m5m 0 nd he energy densy s gven y q. (30.0). Ths s more hn mllon mes greer hn he self-nducnce of he orodl solenod of xmple 30.3 (econ 30.2). VAUAT: Convenonl wres h re o crry 200 A would hve o e of lrge dmeer o keep he ressnce low nd vod uncceple energy losses due o I 2 heng. As resul, 80-H nducor usng convenonl wre would e very lrge (room-sze). A superconducng nducor could e much smller, snce he ressnce of superconducor s zero nd much hnner wres could e used; one drwck s h he wres would hve o e kep low emperure o remn superconducng, nd energy would hve o e used o mnn hs low emperure. As resul, hs scheme s mprccl wh presen echnology. XCUT: The energy densy n he mgnec feld s u 5 B2 2m T p T # m/a J/m 3 VAUAT: As n neresng comprson, he he of comuson of nurl gs, expressed on n energy per un volume ss, s ou J/m 3. Tes Your Undersndng of econ 30.3 The curren n solenod s reversed n drecon whle keepng he sme mgnude. () Does hs chnge he mgnec feld whn he solenod? () Does hs chnge he mgnec energy densy n he solenod? IDNTIFY he relevn conceps: An nducor s jus noher crcu elemen, lke source of emf, ressor, or cpcor. One key dfference s h when n nducor s ncluded n crcu, ll he volges, currens, nd cpcor chrges re n generl funcons of me, no consns s hey hve een n mos of our prevous crcu nlyss. Bu Krchhoff s rules, whch we suded n econ 26.2, re sll vld. When he volges nd currens vry wh me, Krchhoff s rules hold ech nsn of me. T UP he prolem usng he followng seps:. Follow he sme procedure descred n Prolem-olvng regy 26.2 n econ (Now would e n excellen me o revew h sregy.) Drw lrge crcu dgrm nd lel ll qunes, known nd unknown. Apply he juncon rule mmedely ny juncon. 2. Deermne whch qunes re he rge vrles. XCUT he soluon s follows:. As n Prolem-olvng regy 26.2, pply Krchhoff s loop rule o ech loop n he crcu. Curren Growh n n - Crcu We cn lern severl sc hngs ou nducor ehvor y nlyzng he crcu of Fg A crcu h ncludes oh ressor nd n nducor, nd possly source of emf, s clled n - crcu. The nducor helps o preven rpd chnges n curren, whch cn e useful f sedy curren s requred u he exernl source hs flucung emf. The ressor my e sepre crcu elemen, or my e he ressnce of he nducor wndngs; every rel-lfe nducor hs some ressnce unless s mde of superconducng wre. By closng swch, we cn connec he - comnon o source wh consn emf. (We ssume h he source hs zero nernl ressnce, so he ermnl volge equls he emf.) uppose oh swches re open o egn wh, nd hen some nl me 5 0 we close swch. The curren cnno chnge suddenly from zero o some fnl vlue, snce d/ nd he nduced emf n he nducor would oh e nfne. Insed, he curren egns o grow re h depends only on he vlue of n he crcu. e e he curren some me fer swch v s closed, nd le d/ e s re of chnge h me. The poenl dfference cross he ressor h me s nd he poenl dfference v c v 5 cross he nducor s d v c 5 2. As n ll crcu nlyss, geng he correc sgn for ech poenl dfference s essenl. (You should revew he rules gven n Prolem-olvng regy 26.2.) To ge he correc sgn for he poenl dfference eween he ermnls of n nducor, rememer enz s lw nd he sgn rule descred n econ 30.2 n conjuncon wh q. (30.7) nd Fg In Krchhoff s loop rule, when we go hrough n nducor n he sme drecon s he ssumed curren, we encouner volge drop equl o d/, so he correspondng erm n he loop equon s 2 d/. When we go hrough n nducor n he oppose drecon from he ssumed curren, he poenl dfference s reversed nd he erm o use n he loop equon s d/. 3. As lwys, solve for he rge vrles. VAUAT your nswer: Check wheher your nswer s conssen wh he wy h nducors ehve. If he curren hrough n nducor s chngng, your resul should ndce h he poenl dfference cross he nducor opposes he chnge. If no, you proly used n ncorrec sgn somewhere n your clculon. 30. An - crcu. Closng swch connecs he - comnon n seres wh source of emf. Closng swch 2 whle openng swch dsconnnecs he comnon from he source. c 2

7 042 CHAPT 30 Inducnce 30.4 The - Crcu Grph of versus for growh of curren n n - crcu wh n emf n seres. The fnl curren s I 5 /; fer one me consn, he curren s 2 /e of hs vlue. I 5 O wch s closed ( ) I 2 e The fnl curren I does no depend on he nducnce ; s he sme s would e f he ressnce lone were conneced o he source wh emf. Fgure 30.2 shows he ehvor of he curren s funcon of me. To derve he equon for hs curve (h s, n expresson for curren s funcon of me), we proceed jus s we dd for he chrgng cpcor n econ Frs we rerrnge q. (30.3) o he form Ths sepres he vrles, wh on he lef sde nd on he rgh. Then we negre oh sdes, renmng he negron vrles r nd r so h we cn use nd s he upper lms. (The lower lm for ech negrl s zero, correspondng o zero curren he nl me 5 0.) We ge Now we ke exponenls of oh sdes nd solve for. We leve he dels for you o work ou; he fnl resul s 5 2 e 2 / 2 2 d 2 / 2 52 dr 3 0 ln 2 / 2 2/ r 2 / (curren n n - crcu wh emf) (30.4) Ths s he equon of he curve n Fg Tkng he dervve of q. (30.4), we fnd d 5 e 2 / 2 r (30.5) Noe h f he curren s n he drecon shown n Fg. 30. nd s ncresng, A me 5 0, 5 0 nd d/ 5 /. As `, / nd d/ 0, s we hen oh v nd v c re posve; s hgher poenl hn, whch n urn s predced. hgher poenl hn c. (Compre o Fgs nd c.) We pply Krchhoff s As Fg shows, he nsnneous curren frs rses rpdly, hen loop rule, srng he negve ermnl nd proceedng counerclockwse ncreses more slowly nd pproches he fnl vlue I 5 / sympoclly. A round he loop: me equl o / he curren hs rsen o 2 /e 2, or ou 63%, of s fnl d vlue. The quny / s herefore mesure of how quckly he curren ulds 2 r 2 (30.2) 5 0 owrd s fnl vlue; hs quny s clled he me consn for he crcu, denoed y : olvng hs for d/, we fnd h he re of ncrese of curren s d (30.3) (me consn for n - crcu) (30.6) In me equl o 2, he curren reches 86% of s fnl vlue; n 5, 99.3%; nd n 0, %. (Compre he dscusson n econ 26.4 of chrgng cpcor A he nsn h swch s frs closed, 5 0 nd he poenl drop cross of cpcnce C h ws n seres wh ressor of ressnce ; he me consn for h suon ws he produc C.) s zero. The nl re of chnge of curren s d 2 5 The grphs of versus hve he sme generl shpe for ll vlues of. For nl gven vlue of, he me consn s greer for greer vlues of. When s smll, he curren rses rpdly o s fnl vlue; when s lrge, rses more As we would expec, he greer he nducnce, he more slowly he curren slowly. For exmple, f 5 00 V nd 5 0 H, ncreses. As he curren ncreses, he erm / 2 n q. (30.3) lso ncreses, nd 5 he re of ncrese of curren gven y q. (30.3) ecomes smller nd smller. 5 0 H 00 V s Ths mens h he curren s pprochng fnl, sedy-se vlue I. When he nd he curren ncreses o ou 63% of s fnl vlue n 0.0 s. (ecll h curren reches hs vlue, s re of ncrese s zero. Then q. (30.3) ecomes H 5 V # s.) Bu f H, s ms, nd he rse s d fnl 2 much more rpd. I nd nergy consderons offer us ddonl nsgh no he ehvor of n - I 5 crcu. The nsnneous re whch he source delvers energy o he crcu s P 5. The nsnneous re whch energy s dssped n he ressor s 2, nd he re whch energy s sored n he nducor s v c 5 d/ [or, equvlenly, d/ d/]. When we mulply q. (30.2) y nd rerrnge, we fnd 5 2 d (30.7) Of he power suppled y he source, pr 2 2 s dssped n he ressor nd pr d/ 2 goes o sore energy n he nducor. Ths dscusson s compleely nlogous o our power nlyss for chrgng cpcor, gven he end of econ xmple 30.7 Anlyzng n - crcu A sensve elecronc devce of ressnce 75 V s o e conneced o source of emf y swch. The devce s desgned o h he curren e no more hn ma 5 58 ms; o s- men h he fnl curren s o e 36 ma. The oher requremen s opere wh curren of 36 ma, u o vod dmge o he sfy hs, we use q. (30.4) for he curren s funcon of me devce, he curren cn rse o no more hn 4.9 ma n he frs nd solve for he nducnce, whch s he only unknown quny. 58 ms fer he swch s closed. To proec he devce, s conneced n seres wh n nducor s n Fg. 30.; he swch n quon (30.6) hen ells us he me consn. queson s. () Wh emf mus he source hve? Assume neglgle nernl ressnce. () Wh nducnce s requred? (c) Wh expresson I 5 / for he fnl curren nd solvng for he emf, XCUT: () Usng I 5 36 ma A nd 5 75 V n he s he me consn? we fnd 5 I A 275 V V IDNTIFY: Ths prolem concerns curren growh n n - crcu, so we cn use he des of hs secon. Frs we mulply hrough y 2/ 2 nd hen dd o oh sdes () To fnd he requred nducnce, we solve q. (30.4) for. o on T UP: Fgure 30.2 shows h he fnl curren s I 5 /. nce he ressnce s gven, he emf s deermned y he requre- 2 5 e 2 / 2 Connued

8 044 CHAPT 30 Inducnce 30.5 The -C Crcu 045 Then we ke nurl logs of oh sdes, solve for, nd nser he (c) From q. (30.6), numers: 2 5 ln 2 / V s mh ln A 275 V 2 / 6.3 V H 75 V s ms VAUAT: We noe h 58 ms s much less hn he me consn. In 58 ms he curren ulds up only from zero o 4.9 ma, smll frcon of s fnl vlue of 36 ma; fer 390 ms he curren equls 2 /e 2 of s fnl vlue, or ou ma ma. Th s, only 0.00 or.0% of he energy nlly sored n he nducor remns, so 99.0% hs een dssped n he ressor. VAUAT: To ge sense of wh hs resul mens, consder he - crcu we nlyzed n xmple 30.7, for whch he me consn s 390 ms. Wh 5 69 mh H nd n nl curren I ma A, he moun of energy n he nducor nlly s U I H A J. Of hs, 99.0% or J s dssped n ms s ms. In oher words, hs crcu cn e powered off lmos compleely n 0.90 ms, nd cn e powered on n he sme moun of me. The mnmum me for complee on-off cycle s herefore.8 ms. For mny purposes, such s n fs swchng neworks for elecommuncon, n even shorer cycle me s requred. In such cses smller me consn 5/s needed Grph of versus for decy of curren n n - crcu. Afer one me consn, he curren s /e of s nl vlue. I 0 O I 0e 5 xmple wch 2 s closed 5 0. nergy n n - crcu Curren Decy n n - Crcu Now suppose swch n he crcu of Fg. 30. hs een closed for whle nd he curren hs reched he vlue I 0. eseng our sopwch o redefne he nl me, we close swch 2 me 5 0, ypssng he ery. (A he sme me we should open o sve he ery from run.) The curren hrough nd does no nsnneously go o zero u decys smoohly, s shown n Fg The Krchhoff s-rule loop equon s oned from q. (30.2) y smply omng he erm. We chllenge you o rerce he seps n he ove nlyss nd show h he curren vres wh me ccordng o 5 I 0 e 2 / 2 (30.8) where I 0 s he nl curren me 5 0. The me consn, 5/, s he me for curren o decrese o /e, or ou 37%, of s orgnl vlue. In me 2 hs dropped o 3.5%, n me 5 o 0.67%, nd n 0 o %. The energy h s needed o mnn he curren durng hs decy s provded y he energy sored n he mgnec feld of he nducor. The deled energy nlyss s smpler hs me. In plce of q. (30.7) we hve d (30.9) In hs cse, d/ s negve; q. (30.9) shows h he energy sored n he nducor decreses re equl o he re of dsspon of energy 2 n he ressor. Ths enre dscusson should look fmlr; he suon s very smlr o h of chrgng nd dschrgng cpcor, nlyzed n econ I would e good de o compre h secon wh our dscusson of he - crcu. Tes Your Undersndng of econ 30.4 () In Fg. 30., wh re he lgerc sgns of he poenl dfferences v nd v c when swch s closed nd swch 2 s open? () v. 0, v c. 0; () v. 0, v c, 0; () v, 0, v c. 0; (v) v, 0, v c, 0. () Wh re he sgns of v nd v c when s open, 2 s closed, nd curren s flowng n he drecon shown? () v. 0, v c. 0; () v. 0, v c, 0; () v, 0, v c. 0; (v) v, 0, v c, The -C Crcu A crcu connng n nducor nd cpcor shows n enrely new mode of ehvor, chrcerzed y oscllng curren nd chrge. Ths s n shrp conrs o he exponenl pproch o sedy-se suon h we hve seen wh oh -C nd - crcus. In he -C crcu n Fg we chrge he Cpcor fully chrged; zero curren V m Cpcor polry reverses. Cpcor fully dschrged; curren mxml Cpcor fully chrged; zero curren V m Q m Q m ONIN 4.2 AC Crcus: The C Oscllor (Quesons 6) 30.4 In n oscllng -C crcu, he chrge on he cpcor nd he curren hrough he nducor oh vry snusodlly wh me. nergy s rnsferred eween mgnec energy n he nducor U B 2 nd elecrc energy n he cpcor U 2. As n smple hrmonc moon, he ol energy remns consn. (Compre Fg. 3.4 n econ 3.3.) Q m Q m m C Cpcor dschrgng; I ncresng I m B m I m Cpcor chrgng; I decresng m Curren drecon reverses. Cpcor dschrgng; I ncresng I m Cpcor fully dschrged; curren mxml B m I m When he curren n n - crcu s decyng, wh frcon of he orgnl energy sored n he nducor hs een dssped fer 2.3 me consns? IDNTIFY: Ths prolem concerns curren decy n n - crcu s well s he relonshp eween he curren n n nducor nd he moun of sored energy. T UP: The curren ny me for hs suon s gven y q. (30.8). The sored energy ssoced wh hs curren s gven y q. (30.9), U XCUT: From q. (30.8), he curren ny me s 5 I 0 e 2 / 2 The energy U n he nducor ny me s oned y susung hs expresson no U We on U 5 2 I 0 2 e 22 / 2 5 U 0 e 22 / 2 where U I 2 0 s he energy he nl me 5 0. When /, we hve U 5 U 0 e U 0 e U 0 zero 5 U B U Crcu s energy ll sored n elecrc feld () 5 0 nd 5 T (close swch 5 0) 5 U B U Crcu s energy ll sored n mgnec feld () 5 4 zero T Cpcor chrgng; I decresng zero 5 U B U Crcu s energy ll sored n elecrc feld (c) 5 2 T 5 U B U Crcu s energy ll sored n mgnec feld (d) zero T

9 046 CHAPT 30 Inducnce 30.5 The -C Crcu Applyng Krchhoff s loop rule o he -C crcu. The drecon of rvel round he loop n he loop equon s shown. Jus fer he crcu s compleed nd he cpcor frs egns o dschrge, s n Fg. 30.4, he curren s negve (oppose o he drecon shown). C q q Trvel cpcor o poenl dfference V m nd nl chrge Q 5 CV m on s lef-hnd ple nd hen close he swch. Wh hppens? The cpcor egns o dschrge hrough he nducor. Becuse of he nduced emf n he nducor, he curren cnno chnge nsnneously; srs zero nd evenully ulds up o mxmum vlue I m. Durng hs uldup he cpcor s dschrgng. A ech nsn he cpcor poenl equls he nduced emf, so s he cpcor dschrges, he re of chnge of curren decreses. When he cpcor poenl ecomes zero, he nduced emf s lso zero, nd he curren hs leveled off s mxmum vlue I m. Fgure 30.4 shows hs suon; he cpcor hs compleely dschrged. The poenl dfference eween s ermnls (nd hose of he nducor) hs decresed o zero, nd he curren hs reched s mxmum vlue I m. Durng he dschrge of he cpcor, he ncresng curren n he nducor hs eslshed mgnec feld n he spce round, nd he energy h ws nlly sored n he cpcor s elecrc feld s now sored n he nducor s mgnec feld. Alhough he cpcor s compleely dschrged n Fg. 30.4, he curren persss ( cnno chnge nsnneously), nd he cpcor egns o chrge wh polry oppose o h n he nl se. As he curren decreses, he mgnec feld lso decreses, nducng n emf n he nducor n he sme drecon s he curren; hs slows down he decrese of he curren. venully, he curren nd he mgnec feld rech zero, nd he cpcor hs een chrged n he sense oppose o s nl polry (Fg. 30.4c), wh poenl dfference 2V m nd chrge 2Q on s lef-hnd ple. The process now repes n he reverse drecon; lle ler, he cpcor hs gn dschrged, nd here s curren n he nducor n he oppose drecon (Fg. 30.4d). ll ler, he cpcor chrge reurns o s orgnl vlue (Fg. 30.4), nd he whole process repes. If here re no energy losses, he chrges on he cpcor connue o osclle ck nd forh ndefnely. Ths process s clled n elecrcl oscllon. From n energy sndpon he oscllons of n elecrcl crcu rnsfer energy from he cpcor s elecrc feld o he nducor s mgnec feld nd ck. The ol energy ssoced wh he crcu s consn. Ths s nlogous o he rnsfer of energy n n oscllng mechncl sysem from poenl energy o knec energy nd ck, wh consn ol energy. As we wll see, hs nlogy goes much furher. lecrcl Oscllons n n -C Crcu To sudy he flow of chrge n del, we proceed jus s we dd for he - crcu. Fgure 30.5 shows our defnons of q nd. CAUTION Posve curren n n -C crcu Afer exmnng Fg. 30.4, he posve drecon for curren n Fg my seem ckwrd o you. In fc we ve chosen hs drecon o smplfy he relonshp eween curren nd cpcor chrge. We defne he curren ech nsn o e 5 dq/, he re of chnge of he chrge on he lef-hnd cpcor ple. Hence f he cpcor s nlly chrged nd egns o dschrge s n Fgs nd 30.4, hen dq/, 0 nd he nl curren s negve; he drecon of he curren s hen oppose o he (posve) drecon shown n Fg We pply Krchhoff s loop rule o he crcu n Fg rng he lower-rgh corner of he crcu nd ddng volges s we go clockwse round he loop, we on d 2 2 q C 5 0 nce 5 dq/, follows h d/ 5 d 2 q/ 2. We susue hs expresson no he ove equon nd dvde y 2 o on (30.20) quon (30.20) hs excly he sme form s he equon we derved for smple hrmonc moon n econ 3.2, q. (3.4). Th equon s d 2 x/ k/m 2 x, or (You should revew econ 3.2 efore gong on wh hs dscusson.) In he -C crcu he cpcor chrge q plys he role of he dsplcemen x, nd he curren 5 dq/ s nlogous o he prcle s velocy v x 5 dx/. The nducnce s nlogous o he mss m, nd he recprocl of he cpcnce, /C, s nlogous o he force consn k. Pursung hs nlogy, we recll h he ngulr frequency v52pf of he hrmonc oscllor s equl o k/m 2 / 2, nd he poson s gven s funcon of me y q. (3.3), where he mplude A nd he phse ngle f depend on he nl condons. In he nlogous elecrcl suon he cpcor chrge q s gven y nd he ngulr frequency v of oscllon s gven y (ngulr frequency of oscllon n n -C crcu) (30.2) (30.22) You should verfy h q. (30.2) ssfes he loop equon, q. (30.20), when v hs he vlue gven y q. (30.22). In dong hs, you wll fnd h he nsnneous curren 5 dq/ s gven y (30.23) Thus he chrge nd curren n n -C crcu osclle snusodlly wh me, wh n ngulr frequency deermned y he vlues of nd C. The ordnry frequency f, he numer of cycles per second, s equl o v/2p s lwys. The consns Q nd f n qs. (30.2) nd (30.23) re deermned y he nl condons. If me 5 0 he lef-hnd cpcor ple n Fg hs s mxmum chrge Q nd he curren s zero, hen f50. If q 5 0 me 5 0, hen f56p/2 rd. nergy n n -C Crcu We cn lso nlyze he -C crcu usng n energy pproch. The nlogy o smple hrmonc moon s eqully useful here. In he mechncl prolem ody wh mss m s ched o sprng wh force consn k. uppose we dsplce he ody dsnce A from s equlrum poson nd relese from res me 5 0. The knec energy of he sysem ny ler me s 2 mv x 2, nd s elsc poenl energy s 2 kx2. Becuse he sysem s conservve, he sum of hese energes equls he nl energy of he sysem, 2 ka2. We fnd he velocy ny poson x jus s we dd n econ 3.3, q. (3.22): v x v5 Å C d 2 q q 5 0 (-C crcu) 2 C d 2 x 2 k m x 5 0 x 5 A cos v f2 q 5 Q cos v f2 52vQ sn v f2 k v x 56 Å m "A2 2 x 2 (30.24)

10 048 CHAPT 30 Inducnce 30.6 The --C eres Crcu 049 Tle 30. Oscllon of Mss- prng ysem Compred wh lecrcl Oscllon n n -C Crcu Mss-prng ysem The -C crcu s lso conservve sysem. Agn le Q e he mxmum cpcor chrge. The mgnec-feld energy 2 2 n he nducor ny me corresponds o he knec energy of he oscllng ody, nd he elecrcfeld energy q 2 2 mv2 /2C n he cpcor corresponds o he elsc poenl energy 2 kx2 of he sprng. The sum of hese energes equls he ol energy Q 2 /2C of he sysem: 2 2 q2 2C 5 Q2 2C (30.25) The ol energy n he -C crcu s consn; osclles eween he mgnec nd he elecrc forms, jus s he consn ol mechncl energy n smple hrmonc moon s consn nd osclles eween he knec nd poenl forms. olvng q. (30.25) for, we fnd h when he chrge on he cpcor s q, he curren s Fg T/4 2 nd Fg. 30.4c 5 T/2 2. Comprng hose fgures o Fg. 30.5, we expec he cpcor chrge q o e negve (h s, here wll e negve chrge on he lef-hnd ple of he cpcor) nd he curren o e negve s well (h s, curren wll e rvelng n counerclockwse drecon). To fnd he vlue of q, we use q. (30.2). The chrge s mxmum 5 0, so f50 nd Q 5 C F2300 V C. The chrge q ny me s A me s, xmple 30.0 q C 2 cos v nergy n n oscllng crcu v rd/s s rd q C 2 cos 2.4 rd C The curren ny me s A me s, 52vQ sn v rd/s C 2 sn 2.4 rd A VAUAT: Noe h he sgns of q nd re oh negve, s we predced. Knec energy 5 2 mv 2 x Poenl energy 5 2 kx2 2 mv x 2 2 kx2 5 2 ka2 v x 56 "k/m "A 2 2 x 2 v x 5 dx/ k v5 Å m x 5 A cos v f2 Inducor-Cpcor Crcu Mgnec energy lecrc energy 5 q 2 /2C 2 2 q 2 /2C 5 Q 2 /2C 56 "/C "Q 2 2 q 2 5 dq/ v5 Å C q 5 Q cos v f2 56 Å C "Q2 2 q 2 (30.26) You cn verfy hs equon y susung q from q. (30.2) nd from q. (30.23). Comprng qs. (30.24) nd (30.26), we see h curren 5 dq/ nd chrge q re reled n he sme wy s re velocy v x 5 dx/ nd poson x n he mechncl prolem. The nloges eween smple hrmonc moon nd -C crcu oscllons re summrzed n Tle 30.. The srkng prllel shown here eween mechncl nd elecrcl oscllons s one of mny such exmples n physcs. Ths prllel s so close h we cn solve complced mechncl nd couscl prolems y seng up nlogous elecrcl crcus nd mesurng he currens nd volges h correspond o he mechncl nd couscl qunes o e deermned. Ths s he sc prncple of mny nlog compuers. Ths nlogy cn e exended o dmped oscllons, whch we consder n he nex secon. In Chper 3 we wll exend he nlogy furher o nclude forced elecrcl oscllons, whch occur n ll lernng-curren crcus. Consder gn he -C crcu of xmple () Fnd he mgnec energy nd elecrc energy 5 0. () Fnd he mgnec energy nd elecrc energy 5.2 ms. IDNTIFY: Ths prolem sks for he mgnec energy (sored n he nducor) nd he elecrc energy (sored n he cpcor) wo dfferen mes durng he oscllon of he -C crcu. T UP: From xmple 30.9 we know he vlues of he cpcor chrge q nd crcu curren for oh of he mes of neres. We use hem o clcule he mgnec energy sored n he nducor, gven y U B 5 2 2, nd he elecrc energy sored n he cpcor, gven y U 5 q 2 /2C. XCUT: () A 5 0 here s no curren nd q 5 Q. Hence here s no mgnec energy, nd ll he energy n he crcu s n he form of elecrc energy n he cpcor: U B U 5 Q2 2C C J F 2 () As we menoned n xmple 30.9, 5.2 ms corresponds o suon nermede eween Fg T/4 2 nd Fg. 30.4c 5 T/2 2. o we expec he energy o e pr mgnec nd pr elecrc hs me. From xmple 30.9, 520 A nd q C, so U B H 220 A J U 5 q2 2C C J F 2 VAUAT: The mgnec nd elecrc energes re he sme 5 3T/ T, excly hlfwy eween he suons n Fgs nd 30.4c. The me we re consderng here s slghly ler nd U B s slghly less hn U, s we would expec. We emphsze h ll mes, he ol energy 5 U B U hs he sme vlue,. J. An -C crcu whou ressnce s conservve sysem; no energy s dssped. xmple 30.9 An oscllng crcu A 300-V dc power supply s used o chrge 25-mF cpcor. Afer he cpcor s fully chrged, s dsconneced from he power supply nd conneced cross 0-mH nducor. The ressnce n he crcu s neglgle. () Fnd he frequency nd perod of oscllon of he crcu. () Fnd he cpcor chrge nd he crcu curren.2 ms fer he nducor nd cpcor re conneced. IDNTIFY: Our rge vrles re he frequency f nd perod T, s well s he vlues of chrge q nd curren gven me. T UP: We re gven he cpcnce C nd he nducnce, from whch we cn clcule he frequency nd perod usng q. (30.22). We fnd he chrge nd curren usng qs. (30.2) nd (30.23). Inlly he cpcor s fully chrged nd he curren s zero, s n Fg. 30.4, so he phse ngle s f50[see he dscusson h follows q. (30.23)]. XCUT: () The nurl ngulr frequency s v5 Å C 5 Å H F rd/s The frequency f s /2p mes hs: f 5 v 2p rd/s Hz 2p rd/cycle The perod s he recprocl of he frequency: T 5 f Hz s 5 3. ms () nce he perod of he oscllon s T 5 3. ms, 5.2 ms equls 0.38T; hs corresponds o suon nermede eween Tes Your Undersndng of econ 30.5 One wy o hnk ou he energy sored n n -C crcu s o sy h he crcu elemens do posve or negve work on he chrges h move ck nd forh hrough he crcu. () Beween sges () nd () n Fg. 30.4, does he cpcor do posve work or negve work on he chrges? () Wh knd of force (elecrc or mgnec) does he cpcor exer on he chrges o do hs work? (c) Durng hs process, does he nducor do posve or negve work on he chrges? (d) Wh knd of force (elecrc or mgnec) does he nducor exer on he chrges? 30.6 The --C eres Crcu In our dscusson of he -C crcu we ssumed h here ws no ressnce n he crcu. Ths s n delzon, of course; every rel nducor hs ressnce n s wndngs, nd here my lso e ressnce n he connecng wres. Becuse of ressnce, he elecromgnec energy n he crcu s dssped nd convered o oher forms, such s nernl energy of he crcu merls. essnce n n elecrc crcu s nlogous o frcon n mechncl sysem. uppose n nducor wh nducnce nd ressor of ressnce re conneced n seres cross he ermnls of chrged cpcor, formng n -- C seres crcu. As efore, he cpcor srs o dschrge s soon s he crcu ONIN 4.2 AC Crcus: The C Oscllor (Quesons 7 0)