In the past four chapters we studied the interactions of electric charges at rest;

Transcription

1 25 LERNNG GL By studying this chpte, you will len: The mening of electic cuent, nd how chges move in conducto. Wht is ment y the esistivity nd conductivity of sustnce. How to clculte the esistnce of conducto fom its dimensions nd its esistivity. How n electomotive foce (emf) mkes it possile fo cuent to flow in cicuit. How to do clcultions involving enegy nd powe in cicuits. CURRENT, RETNCE, ND ELECTRMTE FRCE?n flshlight, is the mount of cuent tht flows out of the ul less thn, gete thn, o equl to the mount of cuent tht flows into the ul? n the pst fou chptes we studied the intections of electic chges t est; now we e edy to study chges in motion. n electic cuent consists of chges in motion fom one egion to nothe. When this motion tkes plce within conducting pth tht foms closed loop, the pth is clled n electic cicuit. Fundmentlly, electic cicuits e mens fo conveying enegy fom one plce to nothe. s chged pticles move within cicuit, electic potentil enegy is tnsfeed fom souce (such s ttey o geneto) to device in which tht enegy is eithe stoed o conveted to nothe fom: into sound in steeo system o into het nd light in toste o light ul. Fom technologicl stndpoint, electic cicuits e useful ecuse they llow enegy to e tnspoted without ny moving pts (othe thn the moving chged pticles themselves). Electic cicuits e t the het of flshlights, CD plyes, computes, dio nd television tnsmittes nd eceives, nd household nd industil powe distiution systems. The nevous systems of nimls nd humns e specilized electic cicuits tht cy vitl signls fom one pt of the ody to nothe. n Chpte 26 we will see how to nlyze electic cicuits nd will exmine some pcticl pplictions of cicuits. Befoe we cn do so, howeve, you must undestnd the sic popeties of electic cuents. These popeties e the suject of this chpte. We ll egin y desciing the ntue of electic conductos nd consideing how they e ffected y tempetue. We ll len why shot, ft, cold coppe wie is ette conducto thn long, skinny, hot steel wie. We ll study the popeties of tteies nd see how they cuse cuent nd enegy tnsfe in cicuit. n this nlysis we will use the concepts of cuent, potentil diffeence (o voltge), esistnce, nd electomotive foce. Finlly, we ll look t electic cuent in mteil fom micoscopic viewpoint Cuent cuent is ny motion of chge fom one egion to nothe. n this section we ll discuss cuents in conducting mteils. The vst mjoity of technologicl pplictions of chges in motion involve cuents of this kind. n electosttic situtions (discussed in Chptes 21 though 24) the electic field is zeo eveywhee within the conducto, nd thee is no cuent. Howeve, this does not men tht ll chges within the conducto e t est. n n odiny metl such s coppe o lumium, some of the electons e fee to move within the conducting mteil. These fee electons move ndomly in ll diections, somewht like the molecules of gs ut with much gete speeds, of the ode of 10 6 m/s. The electons nonetheless do not escpe fom the conducting mteil, ecuse they e ttcted to the positive ions of the mteil. The motion of the electons is ndom, so thee is no net flow of chge in ny diection nd hence no cuent. Now conside wht hppens if constnt, stedy electic field E is estlished inside conducto. (We ll see lte how this cn e done.) chged pticle (such s fee electon) inside the conducting mteil is then sujected to stedy foce F 5 qe. f the chged pticle wee moving in vcuum, this stedy foce would cuse stedy cceletion in the diection of F, nd fte time the chged pticle would e moving in tht diection t high speed. But chged pticle moving in conducto undegoes fequent collisions with the mssive, nely sttiony ions of the mteil. n ech such collision the pticle s diection of motion undegoes ndom chnge. The net effect of the electic field E is tht in ddition to the ndom motion of the chged pticles within the conducto, thee is lso vey slow net motion o dift of the moving chged pticles s goup in the diection of the electic foce F 5 qe (Fig. 25.1). This motion is descied in tems of the dift velocity vd of the pticles. s esult, thee is net cuent in the conducto. While the ndom motion of the electons hs vey fst vege speed of out 10 6 m/s, the dift speed is vey slow, often on the ode of m/s. Given tht the electons move so slowly, you my wonde why the light comes on immeditely when you tun on the switch of flshlight. The eson is tht the electic field is set up in the wie with speed ppoching the speed of light, nd electons stt to move ll long the wie t vey nely the sme time. The time tht it tkes ny individul electon to get fom the switch to the light ul isn t elly elevnt. good nlogy is goup of soldies stnding t ttention when the segent odes them to stt mching; the ode eches the soldies es t the speed of sound, which is much fste thn thei mching speed, so ll the soldies stt to mch essentilly in unison. The Diection of Cuent Flow The dift of moving chges though conducto cn e intepeted in tems of wok nd enegy. The electic field E does wok on the moving chges. The esulting kinetic enegy is tnsfeed to the mteil of the conducto y mens of collisions with the ions, which vite out thei equiliium positions in the cystlline stuctue of the conducto. This enegy tnsfe inceses the vege vitionl enegy of the ions nd theefoe the tempetue of the mteil. Thus much of the wok done y the electic field goes into heting the conducto, not into mking the moving chges move eve fste nd fste. This heting is sometimes useful, s in n electic toste, ut in mny situtions is simply n unvoidle y-poduct of cuent flow. n diffeent cuent-cying mteils, the chges of the moving pticles my e positive o negtive. n metls the moving chges e lwys (negtive) electons, while in n ionized gs (plsm) o n ionic solution the moving E 25.1 Cuent f thee is no electic field inside conducto, n electon moves ndomly fom point P 1 to point P 2 in time Dt. f n electic field E is pesent, the electic foce F 5 qe imposes smll dift (getly exggeted hee) tht tkes the electon to point P 2, distnce Dt fom P 2 in the diection of the foce. Conducto without intenl E field Pth of electon without E field. Electon moves ndomly. Pth of electon with E field. The motion is mostly P 1 ndom, ut... P 2 P 2 Dt... the E field esults in net displcement long the wie. Conducto with intenl E field F 5 qe n electon hs negtive chge q, so the foce on it due to the E field is in the diection opposite to E. E 846

2 848 CHPTER 25 Cuent, Resistnce, nd Electomotive Foce 25.1 Cuent The sme cuent cn e poduced y () positive chges moving in the diection of the electic field E o () the sme nume of negtive chges moving t the sme speed in the diection opposite to E. () conventionl cuent is teted s flow of positive chges, egdless of whethe the fee chges in the conducto e positive, negtive, o oth. () n metllic conducto, the moving chges e electons ut the cuent still points in the diection positive chges would flow The cuent is the time te of chge tnsfe though the coss-sectionl e. The ndom component of ech moving chged pticle s motion veges to zeo, nd the cuent is in the sme diection s E whethe the moving chges e positive (s shown hee) o negtive (see Fig. 25.2). dt Cuent 5 dq dt E E chges my include oth electons nd positively chged ions. n semiconducto mteil such s gemnium o silicon, conduction is ptly y electons nd ptly y motion of vcncies, lso known s holes; these e sites of missing electons nd ct like positive chges. Fig shows segments of two diffeent cuent-cying mteils. n Fig the moving chges e positive, the electic foce is in the sme diection s E, nd the dift velocity vd is fom left to ight. n Fig the chges e negtive, the electic foce is opposite to E, nd the dift velocity vd is fom ight to left. n oth cses thee is net flow of positive chge fom left to ight, nd positive chges end up to the ight of negtive ones. We define the cuent, denoted y, to e in the diection in which thee is flow of positive chge. Thus we descie cuents s though they consisted entiely of positive chge flow, even in cses in which we know tht the ctul cuent is due to electons. Hence the cuent is to the ight in oth Figs nd This choice o convention fo the diection of cuent flow is clled conventionl cuent. While the diection of the conventionl cuent is not necessily the sme s the diection in which chged pticles e ctully moving, we ll find tht the sign of the moving chges is of little impotnce in nlyzing electic cicuits. Fig shows segment of conducto in which cuent is flowing. We conside the moving chges to e positive, so they e moving in the sme diection s the cuent. We define the cuent though the coss-sectionl e to e the net chge flowing though the e pe unit time. Thus, if net chge dq flows though n e in time dt, the cuent though the e is 5 dq dt (definition of cuent) (25.1) CUTN Cuent is not vecto lthough we efe to the diection of cuent, cuent s defined y Eq. (25.1) is not vecto quntity. n cuent-cying wie, the cuent is lwys long the length of the wie, egdless of whethe the wie is stight o cuved. No single vecto could descie motion long cuved pth, which is why cuent is not vecto. We ll usully descie the diection of cuent eithe in wods (s in the cuent flows clockwise ound the cicuit ) o y choosing cuent to e positive if it flows in one diection long conducto nd negtive if it flows in the othe diection. The unit of cuent is the mpee; one mpee is defined to e one coulom pe second C/s 2. This unit is nmed in hono of the Fench scientist ndé Mie mpèe ( ). When n odiny flshlight (D-cell size) is tuned on, the cuent in the flshlight is out 0.5 to 1 ; the cuent in the wies of c engine s stte moto is ound 200. Cuents in dio nd television cicuits e usully expessed in millimpees 1 1 m o micompees 1 1 m , nd cuents in compute cicuits e expessed in nnompees 1 1 n o picompees 1 1 p Cuent, Dift elocity, nd Cuent Density We cn expess cuent in tems of the dift velocity of the moving chges. Let s conside gin the sitution of Fig. 25.3, conducto with coss-sectionl e nd n electic field E diected fom left to ight. To egin with, we ll ssume tht the fee chges in the conducto e positive; then the dift velocity is in the sme diection s the field. uppose thee e n moving chged pticles pe unit volume. We cll n the concenttion of pticles; its unit is m 23. ssume tht ll the pticles move with the sme dift velocity with mgnitude. n time intevl dt, ech pticle moves distnce dt. The pticles tht flow out of the ight end of the shded cylinde with length dt duing dt e the pticles tht wee within this cylinde t the eginning of the intevl dt. The volume of the cylinde is dt, nd the nume of pticles within it is n dt. f ech pticle hs chge q, the chge dq tht flows out of the end of the cylinde duing time dt is nd the cuent is The cuent pe unit coss-sectionl e is clled the cuent density J: The units of cuent density e mpees pe sque mete 1 /m 2 2. f the moving chges e negtive the thn positive, s in Fig. 25.2, the dift velocity is opposite to E. But the cuent is still in the sme diection s E t ech point in the conducto. Hence the cuent nd cuent density J don t depend on the sign of the chge, nd so in the ove expessions fo nd J we eplce the chge q y its solute vlue 0 q 0 : 5 dq dt J 5 5 n 0 q 0 dq 5 q 1 n dt 2 5 nq dt 5 n 0 q 0 (genel expession fo cuent) (25.2) (genel expession fo cuent density) (25.3) The cuent in conducto is the poduct of the concenttion of moving chged pticles, the mgnitude of chge of ech such pticle, the mgnitude of the dift velocity, nd the coss-sectionl e of the conducto. We cn lso define vecto cuent density J tht includes the diection of the dift velocity: J 5 nq 5 dq dt 5 nq J 5 5 nq (vecto cuent density) (25.4) Thee e no solute vlue signs in Eq. (25.4). f q is positive, v is in the sme diection s E if is negtive, is opposite to E d ; q v n eithe cse, is in the sme diection s E d. J. Eqution (25.3) gives the mgnitude J of the vecto cuent density J. CUTN Cuent density vs. cuent Note tht cuent density J is vecto, ut cuent is not. The diffeence is tht the cuent density J descies how chges flow t cetin point, nd the vecto s diection tells you out the diection of the flow t tht point. By contst, the cuent descies how chges flow though n extended oject such s wie. Fo exmple, hs the sme vlue t ll points in the cicuit of Fig. 25.3, ut J does not: the cuent density is diected downwd in the left-hnd side of the loop nd upwd in the ight-hnd side. The mgnitude of J cn lso vy ound cicuit. n Fig the cuent density mgnitude J 5 / is less in the ttey (which hs lge coss-sectionl e ) thn in the wies (which hve smll coss-sectionl e). n genel, conducto my contin sevel diffeent kinds of moving chged pticles hving chges q 1, q 2, c, concenttions n 1, n 2, c, nd dift velocities with mgnitudes 1, 2, c. n exmple is cuent flow in n ionic solution (Fig. 25.4). n sodium chloide solution, cuent cn e cied y oth positive sodium ions nd negtive chloine ions; the totl cuent is found y dding up the cuents due to ech kind of chged pticle, using Eq. (25.2). Likewise, the totl vecto cuent density J is found y using Eq. (25.4) fo ech kind of chged pticle nd dding the esults. We will see in ection 25.4 tht it is possile to hve cuent tht is stedy (tht is, one tht is constnt in time) only if the conducting mteil foms 25.4 Pt of the electic cicuit tht includes this light ul psses though eke with solution of sodium chloide. The cuent in the solution is cied y oth positive chges 1 N 1 ions) nd negtive chges ( Cl 2 ions).

3 850 CHPTER 25 Cuent, Resistnce, nd Electomotive Foce 25.2 Resistivity 851 closed loop, clled complete cicuit. n such stedy sitution, the totl chge in evey segment of the conducto is constnt. Hence the te of flow of chge out t one end of segment t ny instnt equls the te of flow of chge in t the othe end of the segment, nd the cuent is the sme t ll coss sections of the cicuit. We ll mke use of this osevtion when we nlyze electic cicuits lte in this chpte. n mny simple cicuits, such s flshlights o codless electic dills, the diection of the cuent is lwys the sme; this is clled diect cuent. But home pplinces such s tostes, efigetos, nd televisions use ltenting cuent, in which the cuent continuously chnges diection. n this chpte we ll conside diect cuent only. ltenting cuent hs mny specil fetues wothy of detiled study, which we ll exmine in Chpte Tle 25.1 Resistivities t Room Tempetue 1 20 C 2 ustnce ( # m ) ustnce ( # m ) Conductos emiconductos Metls ilve Pue con (gphite) Coppe Pue gemnium 0.60 Gold Pue silicon 2300 luminum nsultos Tungsten me teel Glss Led Lucite Mecuy Mic lloys Mngnin (Cu 84%, Mn 12%, Ni 4%) Qutz (fused) Constntn (Cu 60%, Ni 40%) ulfu Nichome Teflon Wood Exmple 25.1 Cuent density nd dift velocity in wie n 18-guge coppe wie (the size usully used fo lmp cods) hs nominl dimete of 1.02 mm. This wie cies constnt cuent of 1.67 to 200-wtt lmp. The density of fee electons is electons pe cuic mete. Find the mgnitudes of () the cuent density nd () the dift velocity. LUTN DENTFY: This polem uses the eltionships mong cuent, cuent density, nd dift velocity. ET UP: We e given the cuent nd the dimensions of the wie, so we use Eq. (25.3) to find the mgnitude J of the cuent density. We then use Eq. (25.3) gin to find the dift speed fom J nd the concenttion of electons. EXECUTE: () The coss-sectionl e is 5 pd p m m 2 4 The mgnitude of the cuent density is J m /m 2 () olving Eq. (25.3) fo the dift velocity mgnitude, we find 5 J n 0 q /m m C m/s mm/s ELUTE: t this speed n electon would equie 6700 s, o out 1 h 50 min, to tvel the length of wie 1 m long. The speeds of ndom motion of the electons e of the ode of 10 6 m/s. o in this exmple the dift speed is ound times slowe thn the speed of ndom motion. Pictue the electons s ouncing ound fnticlly, with vey slow nd sluggish dift! Test You Undestnding of ection 25.1 uppose we eplced the wie in Exmple 25.1 with 12-guge coppe wie, which hs twice the dimete of 18-guge wie. f the cuent emins the sme, wht effect would this hve on the mgnitude of the dift velocity? (i) none would e unchnged; (ii) would e twice s get; (iii) would e fou times gete; (iv) would e hlf s get; (v) would e one-fouth s get Resistivity The cuent density J in conducto depends on the electic field nd on the popeties of the mteil. n genel, this dependence cn e quite complex. But fo some mteils, especilly metls, t given tempetue, J is nely diectly popotionl to E, nd the tio of the mgnitudes of E nd J is constnt. This eltionship, clled hm s lw, ws discoveed in 1826 y the Gemn physicist Geog imon hm ( ). The wod lw should ctully e in quottion mks, since hm s lw, like the idel-gs eqution nd Hooke s lw,isnidelized model tht descies the ehvio of some mteils quite well ut is not genel desciption of ll mtte. n the following discussion we ll ssume tht hm s lw is vlid, even though thee e mny situtions in which it is not. The sitution is comple to ou epesenttion of the ehvio of the sttic nd kinetic fiction foces; we teted these fiction foces s eing diectly popotionl to the noml foce, even though we knew tht this ws t est n ppoximte desciption. E We define the esistivity of mteil s the tio of the mgnitudes of electic field nd cuent density: 5 E J (definition of esistivity) (25.5) The gete the esistivity, the gete the field needed to cuse given cuent density, o the smlle the cuent density cused y given field. Fom Eq. (25.5) the units of e 1 /m 2 / 1 /m # m/. s we will discuss in the next section, 1 / is clled one ohm ( 1 ; we use the Geek lette, o omeg, which is llitetive with ohm ). o the units fo e # m (ohm-metes). Tle 25.1 lists some epesenttive vlues of esistivity. pefect conducto would hve zeo esistivity, nd pefect insulto would hve n infinite esistivity. Metls nd lloys hve the smllest esistivities nd e the est conductos. The esistivities of insultos e gete thn those of the metls y n enomous fcto, on the ode of The ecipocl of esistivity is conductivity. ts units e 1 # m Good conductos of electicity hve lge conductivity thn insultos. Conductivity is the diect electicl nlog of theml conductivity. Comping Tle 25.1 with Tle 17.5 (Theml Conductivities), we note tht good electicl conductos, such s metls, e usully lso good conductos of het. Poo electicl conductos, such s cemic nd plstic mteils, e lso poo theml conductos. n metl the fee electons tht cy chge in electicl conduction lso povide the pincipl mechnism fo het conduction, so we should expect coeltion etween electicl nd theml conductivity. Becuse of the enomous diffeence in conductivity etween electicl conductos nd insultos, it is esy to confine electic cuents to well-defined pths o cicuits (Fig. 25.5). The vition in theml conductivity is much less, only fcto of 10 3 o so, nd it is usully impossile to confine het cuents to tht extent. emiconductos hve esistivities intemedite etween those of metls nd those of insultos. These mteils e impotnt ecuse of the wy thei esistivities e ffected y tempetue nd y smll mounts of impuities. mteil tht oeys hm s lw esonly well is clled n ohmic conducto o line conducto. Fo such mteils, t given tempetue, is constnt tht does not depend on the vlue of E. Mny mteils show sustntil deptues fom hm s-lw ehvio; they e nonohmic, o nonline. n these mteils, J depends on E in moe complicted mnne. nlogies with fluid flow cn e ig help in developing intuition out electic cuent nd cicuits. Fo exmple, in the mking of wine o mple syup, the poduct is sometimes filteed to emove sediments. pump foces the fluid though the filte unde pessue; if the flow te (nlogous to J ) is popotionl to the pessue diffeence etween the upstem nd downstem sides (nlogous to E), the ehvio is nlogous to hm s lw The coppe wies, o tces, on this cicuit od e pinted diectly onto the sufce of the dk-coloed insulting od. Even though the tces e vey close to ech othe (only out millimete pt), the od hs such high esistivity (nd low conductivity) comped to the coppe tht no cuent cn flow etween the tces. Conducting pths (tces)

4 852 CHPTER 25 Cuent, Resistnce, nd Electomotive Foce 25.3 Resistnce ition of esistivity with solute tempetue T fo () noml metl, () semiconducto, nd (c) supeconducto. n () the line ppoximtion to s function of T is shown s geen line; the ppoximtion gees exctly t T 5 T 0, whee 5 0. () 0 () (c) Metl: Resistivity inceses with incesing tempetue. T 0 lope 5 0 emiconducto: Resistivity deceses with incesing tempetue. T T upeconducto: t tempetues elow T c, the esistivity is zeo. T c T Resistivity nd Tempetue The esistivity of metllic conducto nely lwys inceses with incesing tempetue, s shown in Fig s tempetue inceses, the ions of the conducto vite with gete mplitude, mking it moe likely tht moving electon will collide with n ion s in Fig. 25.1; this impedes the dift of electons though the conducto nd hence educes the cuent. ve smll tempetue nge (up to 100 C o so), the esistivity of metl cn e epesented ppoximtely y the eqution 1 T T 2 T 0 24 (tempetue dependence of esistivity) (25.6) whee 0 is the esistivity t efeence tempetue T 0 (often tken s 0 C o 20 C 2 nd 1 T 2 is the esistivity t tempetue T, which my e highe o lowe thn T 0. The fcto is clled the tempetue coefficient of esistivity. ome epesenttive vlues e given in Tle The esistivity of the lloy mngnin is pcticlly independent of tempetue. Tle 25.2 Tempetue Coefficients of Resistivity (ppoximte lues Ne Room Tempetue) Mteil 3 ( 8C ) 21 4 Mteil 3 ( 8C ) 21 4 luminum Led Bss Mngnin Con (gphite) Mecuy Constntn Nichome Coppe ilve on Tungsten The esistivity of gphite ( nonmetl) deceses with incesing tempetue, since t highe tempetues, moe electons e shken loose fom the toms nd ecome moile; hence the tempetue coefficient of esistivity of gphite is negtive. This sme ehvio occus fo semiconductos (Fig. 25.6). Mesuing the esistivity of smll semiconducto cystl is theefoe sensitive mesue of tempetue; this is the pinciple of type of themomete clled themisto. ome mteils, including sevel metllic lloys nd oxides, show phenomenon clled supeconductivity. s the tempetue deceses, the esistivity t fist deceses smoothly, like tht of ny metl. But then t cetin citicl tempetue T c phse tnsition occus nd the esistivity suddenly dops to zeo, s shown in Fig. 25.6c. nce cuent hs een estlished in supeconducting ing, it continues indefinitely without the pesence of ny diving field. upeconductivity ws discoveed in 1911 y the Dutch physicist Heike Kmelingh nnes ( ). He discoveed tht t vey low tempetues, elow 4.2 K, the esistivity of mecuy suddenly dopped to zeo. Fo the next 75 yes, the highest T c ttined ws out 20 K. This ment tht supeconductivity occued only when the mteil ws cooled using expensive liquid helium, with oiling-point tempetue of 4.2 K, o explosive liquid hydogen, with oiling point of 20.3 K. But in 1986 Kl Mülle nd Johnnes Bednoz discoveed n oxide of ium, lnthnum, nd coppe with T c of nely 40 K, nd the ce ws on to develop high-tempetue supeconducting mteils. By 1987 complex oxide of yttium, coppe, nd ium hd een found tht hs vlue of T c well ove the 77 K oiling tempetue of liquid nitogen, efigent tht is oth inexpensive nd sfe. The cuent (2006) ecod fo T c t tmospheic pessue is 138 K, nd mteils tht e supeconductos t oom tempetue my ecome elity. The implictions of these discoveies fo powe-distiution systems, compute design, nd tnspottion e enomous. Menwhile, supeconducting electomgnets cooled y liquid helium e used in pticle cceletos nd some expeimentl mgnetic-levittion ilods. upeconductos hve othe exotic popeties tht equie n undestnding of mgnetism to exploe; we will discuss these futhe in Chpte 29. Test You Undestnding of ection 25.2 You mintin constnt electic field inside piece of semiconducto while loweing the semiconducto s tempetue. Wht hppens to the cuent density in the semiconducto? (i) t inceses; (ii) it deceses; (iii) it emins the sme Resistnce Fo conducto with esistivity, the cuent density t point whee the electic field is is given y Eq. (25.5), which we cn wite s E (25.7) When hm s lw is oeyed, is constnt nd independent of the mgnitude of the electic field, so E is diectly popotionl to J. ften, howeve, we e moe inteested in the totl cuent in conducto thn in J nd moe inteested in the potentil diffeence etween the ends of the conducto thn in E. This is so lgely ecuse cuent nd potentil diffeence e much esie to mesue thn e J nd E. uppose ou conducto is wie with unifom coss-sectionl e nd length L, s shown in Fig Let e the potentil diffeence etween the highe-potentil nd lowe-potentil ends of the conducto, so tht is positive. The diection of the cuent is lwys fom the highe-potentil end to the lowepotentil end. Tht s ecuse cuent in conducto flows in the diection of E, no mtte wht the sign of the moving chges (Fig. 25.2), nd ecuse E points in the diection of decesing electic potentil (see ection 23.2). s the cuent flows though the potentil diffeence, electic potentil enegy is lost; this enegy is tnsfeed to the ions of the conducting mteil duing collisions. We cn lso elte the vlue of the cuent to the potentil diffeence etween the ends of the conducto. f the mgnitudes of the cuent density J nd the electic field E e unifom thoughout the conducto, the totl cuent is given y 5 J, nd the potentil diffeence etween the ends is 5 EL. When we solve these equtions fo J nd E, espectively, nd sustitute the esults in Eq. (25.7), we otin (25.8) L 5 o 5 L This shows tht when is constnt, the totl cuent is popotionl to the potentil diffeence. The tio of to fo pticul conducto is clled its esistnce R: R 5 (25.9) Comping this definition of R to Eq. (25.8), we see tht the esistnce R of pticul conducto is elted to the esistivity of its mteil y R 5 L (eltionship etween esistnce nd esistivity) (25.10) f is constnt, s is the cse fo ohmic mteils, then so is R. The eqution 5 R E 5 J (eltionship mong voltge, cuent, nd esistnce) (25.11) is often clled hm s lw, ut it is impotnt to undestnd tht the el content of hm s lw is the diect popotionlity (fo some mteils) of to o of J to E. Eqution (25.9) o (25.11) defines esistnce R fo ny conducto, whethe o not it oeys hm s lw, ut only when R is constnt cn we coectly cll this eltionship hm s lw. J 25.7 conducto with unifom coss section. The cuent density is unifom ove ny coss section, nd the electic field is constnt long the length. Cuent flows fom highe to lowe electic potentil. Highe potentil L J E Lowe potentil 5 potentil diffeence etween ends

5 854 CHPTER 25 Cuent, Resistnce, nd Electomotive Foce 25.3 Resistnce long fie hose offes sustntil esistnce to wte flow. To mke wte pss though the hose pidly, the upstem end of the hose must e t much highe pessue thn the end whee the wte emeges. n n nlogous wy, thee must e lge potentil diffeence etween the ends of long wie in ode to cuse sustntil electic cuent though the wie. Tle 25.3 Colo Codes fo Resistos lue s lue s Colo Digit Multiplie Blck 0 1 Bown 1 10 Red nge Yellow Geen Blue iolet Gy White This esisto hs esistnce of 5.7 k with pecision (tolence) of 610%. econd digit Fist digit Multiplie Tolence ntepeting Resistnce Eqution (25.10) shows tht the esistnce of wie o othe conducto of unifom coss section is diectly popotionl to its length nd invesely popotionl to its coss-sectionl e. t is lso popotionl to the esistivity of the mteil of which the conducto is mde. The flowing-fluid nlogy is gin useful. n nlogy to Eq. (25.10), now wte hose offes moe esistnce to flow thn ft one, nd long hose hs moe esistnce thn shot one (Fig. 25.8). We cn incese the esistnce to flow y stuffing the hose with cotton o snd; this coesponds to incesing the esistivity. The flow te is ppoximtely popotionl to the pessue diffeence etween the ends. Flow te is nlogous to cuent, nd pessue diffeence is nlogous to potentil diffeence ( voltge ). Let s not stetch this nlogy too f, though; the wte flow te in pipe is usully not popotionl to its coss-sectionl e (see ection 14.6). The unit of esistnce is the ohm, equl to one volt pe mpee / 2. The kilohm 1 1 k nd the megohm 1 1 M e lso in common use. 100-m length of 12-guge coppe wie, the size usully used in household wiing, hs esistnce t oom tempetue of out W, 120- light ul hs esistnce (t opeting tempetue) of 140. f the sme cuent flows in oth the coppe wie nd the light ul, the potentil diffeence 5 R is much gete coss the light ul, nd much moe potentil enegy is lost pe chge in the light ul. This lost enegy is conveted y the light ul filment into light nd het. You don t wnt you household wiing to glow white-hot, so its esistnce is kept low y using wie of low esistivity nd lge coss-sectionl e. Becuse the esistivity of mteil vies with tempetue, the esistnce of specific conducto lso vies with tempetue. Fo tempetue nges tht e not too get, this vition is ppoximtely line eltionship, nlogous to Eq. (25.6): R 1 T 2 5 R T 2 T 0 24 (25.12) n this eqution, R 1 T 2 is the esistnce t tempetue T nd R 0 is the esistnce t tempetue T 0, often tken to e 0 C o 20 C. The tempetue coefficient of esistnce is the sme constnt tht ppes in Eq. (25.6) if the dimensions L nd in Eq. (25.10) do not chnge ppecily with tempetue; this is indeed the cse fo most conducting mteils (see Polem 25.67). Within the limits of vlidity of Eq. (25.12), the chnge in esistnce esulting fom tempetue chnge T 2 T 0 is given y R 0 1 T 2 T 0 2. cicuit device mde to hve specific vlue of esistnce etween its ends is clled esisto. Resistos in the nge 0.01 to 10 7 cn e ought off the shelf. ndividul esistos used in electonic cicuity e often cylindicl, few millimetes in dimete nd length, with wies coming out of the ends. The esistnce my e mked with stndd code using thee o fou colo nds ne one end (Fig. 25.9), ccoding to the scheme shown in Tle The fist two nds (stting with the nd neest n end) e digits, nd the thid is poweof-10 multiplie, s shown in Fig Fo exmple, geen violet ed mens , o 5.7 k. The fouth nd, if pesent, indictes the pecision (tolence) of the vlue; no nd mens 620%, silve nd 610%, nd gold nd 65%. nothe impotnt chcteistic of esisto is the mximum powe it cn dissipte without dmge. We ll etun to this point in ection Fo esisto tht oeys hm s lw, gph of cuent s function of potentil diffeence (voltge) is stight line (Fig ). The slope of the line is 1/R. f the sign of the potentil diffeence chnges, so does the sign of the cuent poduced; in Fig this coesponds to intechnging the highe- nd lowepotentil ends of the conducto, so the electic field, cuent density, nd cuent Cuent voltge eltionships fo two devices. nly fo esisto tht oeys hm s lw s in () is cuent popotionl to voltge. () hmic esisto (e.g., typicl metl wie): t given tempetue, cuent is popotionl to voltge. lope 5 1 R emiconducto diode: nonohmic esisto n the diection of positive cuent nd voltge, inceses nonlinely with. n the diection of negtive cuent nd voltge, little cuent flows. ll evese diection. n devices tht do not oey hm s lw, the eltionship of voltge to cuent my not e diect popotion, nd it my e diffeent fo the two diections of cuent. Figue shows the ehvio of semiconducto diode, device used to convet ltenting cuent to diect cuent nd to pefom wide viety of logic functions in compute cicuity. Fo positive potentils of the node (one of two teminls of the diode) with espect to the cthode (the othe teminl), inceses exponentilly with incesing ; fo negtive potentils the cuent is extemely smll. Thus positive potentil diffeence cuses cuent to flow in the positive diection, ut potentil diffeence of the othe sign cuses little o no cuent. Hence diode cts like one-wy vlve in cicuit. Exmple 25.2 () Electic field, potentil diffeence, nd esistnce in wie The 18-guge coppe wie in Exmple 25.1 (ection 25.1) hs dimete of 1.02 mm nd coss-sectionl e of m 2. t cies cuent of Find () the electic-field mgnitude in the wie; () the potentil diffeence etween two points in the wie 50.0 m pt; (c) the esistnce of 50.0-m length of this wie. LUTN DENTFY: We e given the vlues of coss-sectionl e nd cuent. u tget viles e the electic-field mgnitude E, potentil diffeence, nd esistnce R. ET UP: The mgnitude of the cuent density is J 5 / nd the esistivity is given in Tle We find the electic-field mgnitude y using Eq. (25.5), E 5J. nce we hve found E, the potentil diffeence is simply the poduct of E nd the length of the wie. We find the esistnce y using Eq. (25.11). EXECUTE: () Fom Tle 25.1, the esistivity of coppe is # m. Hence, using Eq. (25.5), E 5J # m m /m () The potentil diffeence is given y 5 EL /m m (c) Fom Eq. (25.11) the esistnce of 50.0-m length of this wie is R ELUTE: To check ou esult in pt (c), we clculte the esistnce using Eq. (25.10): R 5 L # m m m We emphsize tht the esistnce of the wie is defined to e the tio of voltge to cuent. f the wie is mde of nonohmic mteil, then R is diffeent fo diffeent vlues of ut is lwys given y R 5 /. Resistnce is lso lwys given y R 5L/; if the mteil is nonohmic, is not constnt ut depends on E (o, equivlently, on 5 EL 2.

6 856 CHPTER 25 Cuent, Resistnce, nd Electomotive Foce 25.4 Electomotive Foce nd Cicuits 857 Exmple 25.3 Tempetue dependence of esistnce uppose the esistnce of the wie in Exmple 25.2 is 1.05 t t T C, tempetue of 20 C. Find the esistnce t 0 C nd t 100 C. R C C 2 20 C46 LUTN DENTFY: This exmple concens how esistnce (the tget vile) depends on tempetue. s Tle 25.2 shows, this tempe- ELUTE: The esistnce t 100 C is gete thn tht t 0 C y fcto of tue dependence diffes fo diffeent sustnces. / n othe wods, ising the tempetue of odiny coppe wie fom 0 C to 100 C inceses ET UP: u tget viles e the vlues of the wie esistnce its esistnce y 42%. Fom Eq. (25.11), 5 R, this mens tht R t two tempetues, T 5 0 C nd T C. To find these vlues we use Eq. (25.12). Note tht we e given the esistnce R C thn t 0 C. This is sustntil effect tht must e tken 42% moe voltge is equied to poduce the sme cuent t 1.05 t efeence tempetue T C, nd we know fom into ccount in designing electic cicuits tht e to opete ove Exmple 25.2 tht the wie is mde of coppe. wide nge of tempetues. EXECUTE: Fom Tle 25.2 the tempetue coefficient of esistivity of coppe is C Fom Eq. (25.12), the esistnce t T 5 0 C is R 5 R T 2 T C C 2 20 C Exmple 25.4 Clculting esistnce The hollow cylinde shown in Fig hs length L nd inne nd oute dii nd. t is mde of mteil with esistivity. potentil diffeence is set up etween the inne nd oute sufces of the cylinde (ech of which is n equipotentil sufce) so tht cuent flows dilly though the cylinde. Wht is the esistnce to this dil cuent flow? LUTN DENTFY: Figue shows tht the cuent flows dilly fom the inside of the conducto towd the outside, not long the length of the conducto s in Fig Hence we must use the ides of this section to deive new fomul fo esistnce (ou tget vile) ppopite fo dil cuent flow. ET UP: We cn t use Eq. (25.10) diectly ecuse the coss section though which the chge tvels is not constnt; it vies fom 2pL t the inne sufce to 2pL t the oute sufce. nsted, we clculte the esistnce to dil cuent flow though thin cylindicl shell of inne dius nd thickness d. We then comine the esistnces fo ll such shells etween the inne nd oute dii of the cylinde. EXECUTE: The e fo the shell is 2pL, the sufce e tht the cuent encountes s it flows outwd. The length of the cuent pth though the shell is d. The esistnce dr of this shell, etween inne nd oute sufces, is tht of conducto with length d nd e 2pL: dr 5 d 2pL The cuent hs to pss successively though ll such shells etween the inne nd oute dii nd. Fom Eq. (25.11) the potentil diffeence coss one shell is d 5 dr, nd the totl potentil diffeence etween the inne nd oute sufces is the sum of the potentil diffeences fo ll shells. The totl cuent is the sme though ech shell, so the totl esistnce is the sum of the esistnces of ll the shells. f the e 2pL wee constnt, we could just integte d fom 5 to 5 to get the totl length of the cuent pth. But the e inceses s the cuent psses though shells of gete dius, so we hve to integte the ove expession fo dr. The totl esistnce is thus given y R 5 3 dr 5 d 2pL 3 5 2pL ln ELUTE: The conducto geomety shown in Fig plys n impotnt ole in you ody s nevous system. Ech neuon, o neve cell, hs long extension clled neve fie o xon. n xon hs cylindicl memne shped much like the esisto in Fig , with one conducting fluid inside the memne nd nothe outside it. dinily ll of the inne fluid is t the sme potentil, so no cuent tends to flow long the length of the xon. f the xon is stimulted t cetin point long its length, howeve, chged ions flow dilly coss the cylindicl memne t tht point, s in Fig This flow cuses potentil diffeence etween tht point nd othe points long the length of the xon, which mkes neve signl flow long tht length Finding the esistnce fo dil cuent flow. L d J J J Coss section J Test You Undestnding of ection 25.3 uppose you incese the voltge coss the coppe wie in Exmples 25.2 nd The incesed voltge cuses moe cuent to flow, which mkes the tempetue of the wie incese. (The sme thing hppens to the coils of n electic oven o toste when voltge is pplied to them. We ll exploe this issue in moe depth in ection 25.5.) f you doule the voltge coss the wie, the cuent in the wie inceses. By wht fcto does it incese? (i) 2; (ii) gete thn 2; (iii) less thn Electomotive Foce nd Cicuits Fo conducto to hve stedy cuent, it must e pt of pth tht foms closed loop o complete cicuit. Hee s why. f you estlish n electic field E 1 inside n isolted conducto with esistivity tht is not pt of complete cicuit, cuent egins to flow with cuent density J 5 E 1/ (Fig ). s esult net positive chge quickly ccumultes t one end of the conducto nd net negtive chge ccumultes t the othe end (Fig ). These chges themselves poduce n electic field E in the diection opposite to E 2 1, cusing the totl electic field nd hence the cuent to decese. Within vey smll fction of second, enough chge uilds up on the conducto ends tht the totl electic field E 5 E 1 1 E inside the conducto. Then J 5 0 s well, nd the cuent stops ltogethe (Fig c). o thee cn e no stedy motion of chge in such n incomplete cicuit. To see how to mintin stedy cuent in complete cicuit, we ecll sic fct out electic potentil enegy: f chge q goes ound complete cicuit nd etuns to its stting point, the potentil enegy must e the sme t the end of the ound tip s t the eginning. s descied in ection 25.3, thee is lwys decese in potentil enegy when chges move though n odiny conducting mteil with esistnce. o thee must e some pt of the cicuit in which the potentil enegy inceses. The polem is nlogous to n onmentl wte fountin tht ecycles its wte. The wte pous out of openings t the top, cscdes down ove the teces nd spouts (moving in the diection of decesing gvittionl potentil enegy), nd collects in sin in the ottom. pump then lifts it ck to the top (incesing the potentil enegy) fo nothe tip. Without the pump, the wte would just fll to the ottom nd sty thee. Electomotive Foce n n electic cicuit thee must e device somewhee in the loop tht cts like the wte pump in wte fountin (Fig ). n this device chge tvels uphill, fom lowe to highe potentil enegy, even though the electosttic foce is tying to push it fom highe to lowe potentil enegy. The diection of cuent in such device is fom lowe to highe potentil, just the opposite of wht hppens in n odiny conducto. The influence tht mkes cuent flow fom lowe to highe potentil is clled electomotive foce (evited emf nd ponounced ee-em-eff ). This is poo tem ecuse emf is not foce ut n enegy-pe-unit-chge quntity, like potentil. The unit of emf is the sme s tht fo potentil, the volt J/C 2. typicl flshlight ttey hs n emf of 1.5 ; this mens tht the ttey does 1.5 J of wok on evey coulom of chge tht psses though it. We ll use the symol E ( scipt cpitl E) fo emf. Evey complete cicuit with stedy cuent must include some device tht povides emf. uch device is clled souce of emf. Btteies, electic genetos, sol cells, themocouples, nd fuel cells e ll exmples of souces of emf. ll such devices convet enegy of some fom (mechnicl, chemicl, theml, nd so on) into electic potentil enegy nd tnsfe it into the cicuit to which the device is connected. n idel souce of emf mintins constnt potentil f n electic field is poduced inside conducto tht is not pt of complete cicuit, cuent flows fo only vey shot time. () n electic field E 1 poduced inside n isolted conducto cuses cuent. J J E 1 () The cuent cuses chge to uild up t the ends. E 1 (c) fte vey shot time E 2 hs the sme mgnitude s E 1 ; then the totl field is E totl 5 0 nd the cuent stops completely. E 2 E totl The chge uildup poduces n opposing field E 2, thus educing the cuent. 0 E 1 E 2 J 0 E totl Just s wte fountin equies pump, n electic cicuit equies souce of electomotive foce to sustin stedy cuent.

7 858 CHPTER 25 Cuent, Resistnce, nd Electomotive Foce 25.4 Electomotive Foce nd Cicuits chemtic digm of souce of emf in n open-cicuit sitution. The electic-field foce F nd the nonelectosttic foce F e 5 qe n e shown fo positive chge q. 5 E del emf souce E F n q Teminl t highe potentil F e 5 qe Nonelectosttic foce tending to move chge to highe potentil Foce due to electic field Teminl t lowe potentil When the emf souce is not pt of closed cicuit, F n 5 F e nd thee is no net motion of chge etween the teminls chemtic digm of n idel souce of emf in complete cicuit. The electic-field foce F nd the nonelectosttic foce F e 5 qe n e shown fo positive chge q. The cuent is in the diection fom to in the extenl cicuit nd fom to within the souce. Potentil coss teminls cetes electic field in cicuit, cusing chges to move. 5 E NLNE 12.1 DC eies Cicuits (Qulittive) del emf souce E F n F e When el (s opposed to idel) emf souce is connected to cicuit, nd thus F e fll, so tht F n. F e nd F n does wok on the chges. E E E diffeence etween its teminls, independent of the cuent though it. We define electomotive foce quntittively s the mgnitude of this potentil diffeence. s we will see, such n idel souce is mythicl est, like the fictionless plne nd the mssless ope. We will discuss lte how el-life souces of emf diffe in thei ehvio fom this idelized model. Fig is schemtic digm of n idel souce of emf tht mintins potentil diffeence etween conductos nd, clled the teminls of the device. Teminl, mked 1, is mintined t highe potentil thn teminl, mked 2. ssocited with this potentil diffeence is n electic field E in the egion ound the teminls, oth inside nd outside the souce. The electic field inside the device is diected fom to, s shown. chge q within the souce expeiences n electic foce F e 5 qe. But the souce lso povides n dditionl influence, which we epesent s nonelectosttic foce F n. This foce, opeting inside the device, pushes chge fom to in n uphill diection ginst the electic foce F Thus F mintins the potentil diffeence etween the teminls. f F e. n n wee not pesent, chge would flow etween the teminls until the potentil diffeence ws zeo. The oigin of the dditionl influence F n depends on the kind of souce. n geneto it esults fom mgnetic-field foces on moving chges. n ttey o fuel cell it is ssocited with diffusion pocesses nd vying electolyte concenttions esulting fom chemicl ections. n n electosttic mchine such s n de Gff geneto (see Fig ), n ctul mechnicl foce is pplied y moving elt o wheel. f positive chge q is moved fom to inside the souce, the nonelectosttic foce F n does positive mount of wok W n 5 qe on the chge. This displcement is opposite to the electosttic foce F e, so the potentil enegy ssocited with the chge inceses y n mount equl to q, whee 5 2 is the (positive) potentil of point with espect to point. Fo the idel souce of emf tht we ve descied, F nd F e n e equl in mgnitude ut opposite in diection, so the totl wok done on the chge q is zeo; thee is n incese in potentil enegy ut no chnge in the kinetic enegy of the chge. t s like lifting ook fom the floo to high shelf t constnt speed. The incese in potentil enegy is just equl to the non-electosttic wok W n, so qe 5 q, o 5 E (idel souce of emf) (25.13) Now let s mke complete cicuit y connecting wie with esistnce R to the teminls of souce (Fig ). The potentil diffeence etween teminls nd sets up n electic field within the wie; this cuses cuent to flow ound the loop fom towd, fom highe to lowe potentil. Whee the wie ends, equl mounts of positive nd negtive chge pesist on the inside nd outside of the end. These chges exet the foces tht cuse the cuent to follow the ends in the wie. Fom Eq. (25.11) the potentil diffeence etween the ends of the wie in Fig is given y 5 R. Comining with Eq. (25.13), we hve E 5 5 R (idel souce of emf) (25.14) Tht is, when positive chge q flows ound the cicuit, the potentil ise E s it psses though the idel souce is numeiclly equl to the potentil dop 5 R s it psses though the eminde of the cicuit. nce E nd R e known, this eltionship detemines the cuent in the cicuit. CUTN Cuent is not used up in cicuit t s common misconception tht in closed cicuit, cuent is something tht squits out of the positive teminl of ttey nd is consumed o used up y the time it eches the negtive teminl. n fct the cuent is the sme t evey point in simple loop cicuit like tht in Fig , even if the thickness of the wies is diffeent t diffeent points in the cicuit. This hppens ecuse chge is conseved (tht is, it cn e neithe ceted no destoyed) nd ecuse chge cnnot ccumulte in the cicuit devices we hve descied. f chge did ccumulte, the? potentil diffeences would chnge with time. t s like the flow of wte in n onmentl fountin; wte flows out of the top of the fountin t the sme te t which it eches the ottom, no mtte wht the dimensions of the fountin. None of the wte is used up long the wy! ntenl Resistnce Rel souces of emf in cicuit don t ehve in exctly the wy we hve descied; the potentil diffeence coss el souce in cicuit is not equl to the emf s in Eq. (25.14). The eson is tht chge moving though the mteil of ny el souce encountes esistnce. We cll this the intenl esistnce of the souce, denoted y. f this esistnce ehves ccoding to hm s lw, is constnt nd independent of the cuent. s the cuent moves though, it expeiences n ssocited dop in potentil equl to. Thus, when cuent is flowing though souce fom the negtive teminl to the positive teminl, the potentil diffeence etween the teminls is (teminl voltge, souce with intenl esistnce) (25.15) The potentil, clled the teminl voltge, is less thn the emf E ecuse of the tem epesenting the potentil dop coss the intenl esistnce. Expessed nothe wy, the incese in potentil enegy q s chge q moves fom to within the souce is now less thn the wok qe done y the nonelectosttic foce F n, since some potentil enegy is lost in tvesing the intenl esistnce ttey hs n emf of 1.5, ut the teminl voltge of the ttey is equl to 1.5 only if no cuent is flowing though it so tht 5 0 in Eq. (25.15). f the ttey is pt of complete cicuit though which cuent is flowing, the teminl voltge will e less thn 1.5. Fo el souce of emf, the teminl voltge equls the emf only if no cuent is flowing though the souce (Fig ). Thus we cn descie the ehvio of souce in tems of two popeties: n emf E, which supplies constnt potentil diffeence independent of cuent, in seies with n intenl esistnce. The cuent in the extenl cicuit connected to the souce teminls nd is still detemined y 5 R. Comining this with Eq. (25.15), we find E 2 5 R 5 E 2 (cuent, souce with o 5 E (25.16) R 1 intenl esistnce) Tht is, the cuent equls the souce emf divided y the totl cicuit esistnce 1 R 1 2. CUTN ttey is not cuent souce You might hve thought tht ttey o othe souce of emf lwys poduces the sme cuent, no mtte wht cicuit it s used in. But s Eq. (25.16) shows, the cuent tht souce of emf poduces in given cicuit depends on the esistnce R of the extenl cicuit (s well s on the intenl esistnce of the souce). The gete the esistnce, the less cuent the souce will poduce. t s nlogous to pushing n oject though thick, viscous liquid such s oil o molsses; if you exet cetin stedy push (emf), you cn move smll oject t high speed (smll R, lge ) o lge oject t low speed (lge R, smll ). ymols fo Cicuit Digms n impotnt pt of nlyzing ny electic cicuit is dwing schemtic cicuit digm. Tle 25.4 shows the usul symols used in cicuit digms. We will use these symols extensively in this chpte nd the next. We usully ssume tht the wies tht connect the vious elements of the cicuit hve negligile esistnce; fom Eq. (25.11), 5 R, the potentil diffeence etween the ends of such wie is zeo The emf of this ttey tht is, the teminl voltge when it s not connected to nything is 12. But ecuse the ttey hs intenl esistnce, the teminl voltge of the ttey is less thn 12 when it is supplying cuent to light ul.

8 860 CHPTER 25 Cuent, Resistnce, nd Electomotive Foce Conceptul Exmple 25.5 Tle 25.4 includes two metes tht e used to mesue the popeties of cicuits. delized metes do not distu the cicuit in which they e connected. voltmete, intoduced in ection 23.2, mesues the potentil diffeence etween its teminls; n idelized voltmete hs infinitely lge esistnce nd mesues potentil diffeence without hving ny cuent diveted though it. n mmete mesues the cuent pssing though it; n idelized mmete hs zeo esistnce nd hs no potentil diffeence etween its teminls. Becuse metes ct s pt of the cicuit in which they e connected, these popeties e impotnt to ememe. Tle 25.4 ymols fo Cicuit Digms o R E E E souce in n open cicuit Fig shows souce ( ttey) with n emf E of 12 nd n intenl esistnce of 2. (Fo compison, the intenl esistnce of commecil 12- led stoge ttey is only few thousndths of n ohm.) The wies to the left of nd to the ight of the mmete e not connected to nything. Detemine the edings of the idelized voltmete nd the idelized mmete. LUTN Thee is no cuent ecuse thee is no complete cicuit. (Thee is no cuent though ou idelized voltmete, with its infinitely lge esistnce.) Hence the mmete eds 5 0. Becuse thee is no cuent though the ttey, thee is no potentil diffeence coss its intenl esistnce. Fom Eq. (25.15) with 5 0, the potentil Conducto with negligile esistnce Resisto ouce of emf (longe veticl line lwys epesents the positive teminl, usully the teminl with highe potentil) ouce of emf with intenl esistnce ( cn e plced on eithe side) oltmete (mesues potentil diffeence etween its teminls) mmete (mesues cuent though it) souce of emf in n open cicuit. 5 2, E 5 12 diffeence coss the ttey teminls is equl to the emf. o the voltmete eds 5 E The teminl voltge of el, nonidel souce equls the emf only if thee is no cuent flowing though the souce, s in this exmple. EXECUTE: The idel mmete hs zeo esistnce, so the esistnce extenl to the souce is R 5 4. Fom Eq. (25.16), the cuent though the cicuit is 5 E R The mmete eds 5 2. u idelized conducting wies hve zeo esistnce, nd the idelized mmete lso hs zeo esistnce. o thee is no potentil diffeence etween points nd o etween points nd ; tht is, 5. We cn find y consideing nd eithe s the teminls of the esisto o s the teminls of the souce. Con- Conceptul Exmple 25.7 Using voltmetes nd mmetes The voltmete nd mmete in Exmple 25.6 e moved to diffeent positions in the cicuit. Wht e the voltmete nd mmete edings in the situtions shown in () Fig nd () Fig ? Diffeent plcements of voltmete nd n mmete in complete cicuit. () LUTN 5 2, E 5 12 R 5 4 () 5 2, E 5 12 R 5 4 () The voltmete now mesues the potentil diffeence etween points nd. But s mentioned in Exmple 25.6, 5, so the voltmete eds the sme s in Exmple 25.6: 5 8. CUTN Cuent in simple loop You might e tempted to conclude tht the mmete in Fig , which is locted upstem of the esisto, would hve highe eding thn the 25.4 Electomotive Foce nd Cicuits 861 sideing them s teminls of the esisto, we use hm s lw 1 5 R 2 : 5 R Consideing them s the teminls of the souce, we hve 5 E Eithe wy, we conclude tht the voltmete eds 5 8. ELUTE: With cuent flowing though the souce, the teminl voltge is less thn the emf. The smlle the intenl esistnce, the less the diffeence etween nd E. one locted downstem of the esisto in Fig But this conclusion is sed on the misconception tht cuent is somehow used up s it moves though esisto. s chges move though esisto, thee is decese in electic potentil enegy, ut thee is no chnge in the cuent. The cuent in simple loop is the sme t evey point. n mmete plced s in Fig eds the sme s one plced s in Fig : 5 2. () Thee is no cuent though the voltmete ecuse it hs infinitely lge esistnce. ince the voltmete is now pt of the cicuit, thee is no cuent t ll in the cicuit, nd the mmete eds 5 0. The voltmete mesues the potentil diffeence etween points nd. ince 5 0, the potentil diffeence coss the esisto is 5 R 5 0, nd the potentil diffeence etween the ends nd of the idelized mmete is lso zeo. o is equl to, the teminl voltge of the souce. s in Conceptul Exmple 25.5, thee is no cuent flowing, so the teminl voltge equls the emf, nd the voltmete eding is 5 E This exmple shows tht mmetes nd voltmetes e cicuit elements, too. Moving the voltmete fom the position in Fig to tht in Fig chnges the cuent nd potentil diffeences in the cicuit in this cse the dmticlly. f you wnt to mesue the potentil diffeence etween two points in cicuit without distuing the cicuit, use voltmete s in Figs o 25.19, not s in Fig Exmple 25.8 souce with shot cicuit Exmple 25.6 souce in complete cicuit Using the ttey in Conceptul Exmple 25.5, we dd 4- esisto to fom the complete cicuit shown in Fig Wht e the voltmete nd mmete edings now? LUTN DENTFY u fist tget vile is the cuent though the cicuit (equl to the mmete eding). The second is the potentil diffeence (equl to the voltmete eding). ET UP: We find using Eq. (25.16). To find, we note tht we cn egd this eithe s the potentil diffeence coss the souce o s the potentil diffeence ound the cicuit though the extenl esisto souce of emf in complete cicuit , E 5 12 R 5 4 Using the sme ttey s in the peceding thee exmples, we now eplce the 4- esisto with zeo-esistnce conducto. Wht e the mete edings now? LUTN DENTFY: u tget viles e nd, the sme s in Exmple The only diffeence fom tht exmple is tht the extenl esistnce is now R 5 0. ET UP: Figue shows the new cicuit. Thee is now zeoesistnce pth etween points nd (though the lowe loop in Fig ). Hence the potentil diffeence etween these points must e zeo, which we cn use to help solve the polem u sketch fo this polem. Continued