ELECTRIC CURRENT. charge carrier. electric field MISN ELECTRIC CURRENT. by F. Reif, George Brackett and Jill Larkin CONTENTS

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1 MISN0423 ELECTRIC CURRENT ELECTRIC CURRENT y F. Rif, Gorg Brktt n Jill Lrkin hrg rrir ltri fil CONTENTS A. Dfinition n Proprtis of Currnt B. Currnt n Motion of Chrg Prtils C. Sty Stt Currnts D. TwoTrminl Systms n Ciruits E. Prinipls of Ciruit Anlysis F. Chrtristis of TwoTrminl Systm G. Summry H. Prolms Projt PHYSNET Physis Blg. Mihign Stt Univrsity Est Lnsing, MI 1 2

2 ID Sht: MISN0423 Titl: Eltri Currnt Author: F. Rif n J. Lrkin, Univ. of Cliforni, Brkly. Vrsion: 4/30/2002 Evlution: Stg 0 Lngth: 1 hr; 76 pgs Input Skills: 1. Voulry: mss urrnt, sty stt (MISN0418); ltri potntil, potntil rop, volt (MISN0420). 2. Stt th prinipl of onsrvtion of hrg (MISN0411). Output Skills (Knowlg): K1. Voulry: mpr, iruit igrm, fftiv urrnt, ltri urrnt, ltri rsistn, mf, ohm, prlll onntion, sris onntion, twotrminl systm. K2. Stt th rltionship mong urrnts into n out of rgion t sty stt. K3. Stt th rltionship mong potntil rops long ll pths twn two points in iruit. K4. Stt th hrtristi rltionship twn urrnt n potntil rop for twotrminl systm. K5. Intrhngly sri iruit: () with pitur or igrm; () with vrious quivlnt iruit igrms; () with wor sription; () in trms of sris or prlll onntions. Output Skills (Rul Applition): R1. Givn th vrg motion of th hrg prtils in systm, trmin th th mgnitu n sns of th urrnt in th systm. R2. Givn th numr of moving hrg prtils pr unit lngth of wir, th vrg vloity of ths prtils, n th sign of thir hrgs, trmin th mgnitu n sns of th urrnt in th wir. Output Skills (Prolm Solving): S1. Givn th urrnts n potntil rops in on prt of iruit, us th two prinipls of iruit nlysis to trmin th urrnts n potntil rops in othr prts of th iruit. THIS IS A DEVELOPMENTALSTAGE PUBLICATION OF PROJECT PHYSNET Th gol of our projt is to ssist ntwork of utors n sintists in trnsfrring physis from on prson to nothr. W support mnusript prossing n istriution, long with ommunition n informtion systms. W lso work with mployrs to intify si sintifi skills s wll s physis topis tht r n in sin n thnology. A numr of our pulitions r im t ssisting usrs in quiring suh skills. Our pulitions r sign: (i) to upt quikly in rspons to fil tsts n nw sintifi vlopmnts; (ii) to us in oth lssroom n profssionl sttings; (iii) to show th prrquisit pnnis xisting mong th vrious hunks of physis knowlg n skill, s gui oth to mntl orgniztion n to us of th mtrils; n (iv) to pt quikly to spifi usr ns rnging from singlskill instrution to omplt ustom txtooks. Nw uthors, rviwrs n fil tstrs r wlom. PROJECT STAFF Anrw Shnpp Eugn Kls Ptr Signll Wmstr Grphis Projt Dirtor ADVISORY COMMITTEE D. Aln Bromly Yl Univrsity E. Lonr Jossm Th Ohio Stt Univrsity A. A. Strssnurg S. U. N. Y., Stony Brook Viws xprss in moul r thos of th moul uthor(s) n r not nssrily thos of othr projt prtiipnts. 2002, Ptr Signll for Projt PHYSNET, PhysisAstronomy Blg., Mih. Stt Univ., E. Lnsing, MI 48824; (517) For our lirl us poliis s: 3 4

3 MISN0423 A1 SECT. A DEFINITION AND PROPERTIES OF CURRENT MISN0423 ELECTRIC CURRENT A. Dfinition n Proprtis of Currnt B. Currnt n Motion of Chrg Prtils C. Sty Stt Currnts D. TwoTrminl Systms n Ciruits E. Prinipls of Ciruit Anlysis F. Chrtristis of TwoTrminl Systm G. Summry H. Prolms Astrt: Up to now w hv isuss only hrg prtils whih r t rst. In th nxt thr units w shll stuy th mor importnt n intrsting sitution whr hrg prtils r moving so tht thy giv ris to ltri urrnts. Suh urrnts r th sis of most ltril vis n instrumnts us in vryy lif in th lortory. Whn hrg prtils r t rst, th prtils in ny rgion rmin in tht rgion. But whn hrg prtils r moving, som of th prtils in rgion orinrily ntr or lv this rgion y moving through its ounry surf. (Th prtils, n thir ssoit hrg, r thn si to flow through th surf into, or out of, th rgion.) How thn n on sri th rt spifying how rpily hrg flows through ny surf? Dfinition of urrnt Any surf hs two sis n thus sprts th jnt sp into th two rgions on ithr si of th surf. To istinguish twn ths two rgions, w n spify on of thm y hoosing sns init y short rrow pointing from th surf into this rgion. (S Fig. A1.) * * Thr r thus only two possil snss, pning on whih of th two rgions on wishs to spify. Lt Q th hrg in th rgion spifi y th sns rrow. During smll nough tim t, this hrg thn hngs y smll mount Q s rsult of hrg prtils moving through th surf. Th rt t whih hrg psss through th surf n thn sri y th rtio Q/t whih is ll th ltri urrnt (or th hrg urrnt ) flowing through th surf. This urrnt, onvntionlly not y th symol I, is thus fin s follows: Df. Eltri urrnt: I = Q t PROPERTIES OF ELECTRIC CURRENT (A1) Currnt is numr Aoring to Df. (A1), th ltri urrnt I is numr whos mgnitu inits th mgnitu of th hrg pssing through surf pr unit tim. This numr my positiv, ngtiv, or zro (pning on whthr th hrg in th rgion spifi y th sns rrow inrss, rss, or rmins th sm s rsult of th hrg pr 5 6

4 MISN0423 A2 MISN0423 A3 surf rgion 1 rgion 2 (spifi y sns rrow) Fig. A1: Surf sprting two rgions. tils pssing through th surf). Not tht th ltri urrnt is not vtor, spit th ft tht w hv us sns rrow to spify on of two possil rgions. (In, th pris irtion of this rrow is irrlvnt s long s it points into th rgion to spifi.) Unit of I Aoring to Df. (A1), th unit of urrnt is unit of I = oulom son = mpr (A2) whr th unit mpr (rvit A ) mrly mns oulom pr son. [Th nm mpr ommmorts th Frnh physiist Anr Mri Ampr ( ), fmous for his stuis of th mgnti ffts of ltri urrnts.] Typil mgnitu of I Th mgnitu of th ltri urrnt through n orinry 100 wtt light ul is out 1 mpr. Dpning on thir mgnitus, ltri urrnts through iologil tissus n hv pronoun ffts. For xmpl, urrnt of lss thn mpr pssing through th humn oy is not prptil, ut urrnt 0.01 mpr through th oy is pinful. A urrnt of 0.1 mpr through th oy n us th y ltroution sin it is nough to prou firilltion of th hrt, i.., isruption of th rgulr hrt rhythm. (Only smll prt of this urrnt psss through th hrt, ut 10 5 mpr pssing irtly through th hrt is nough to prou firilltion.) Finlly, urrnts of svrl mprs pssing through th oy prou urns n rspirtory prlysis. PARTICLE MOTION AND SIGN OF CURRENT I without nt hrg Whn som of th hrg prtils in mtril mov, thy n prou n ltri urrnt through surf lthough th mtril hs no nt hrg. This sitution is illustrt in Fig. A2 whih shows mtl 1 tim t ltr tim t' Fig. A2: Motion of ltrons through mtl wir with no nt hrg. Th igrm illustrts vry shmtilly th positions of th ltrons n sttionry ions t two slightly iffrnt tims. wir whih is ltrilly nutrl us it ontins s mny ngtivly hrg ltrons s ions with opposit positiv hrgs. Nvrthlss, th motion of th ltrons through rossstionl surf of this wir prous urrnt through this surf. Sign of I Aoring to Df. (A1), th sign of th urrnt I = Q/t through surf, rltiv to som hosn sns, pns on how th hrg Q in th rgion spifi y th sns rrow hngs with tim s rsult of th hrg prtils moving through th surf. Thus th urrnt I is positiv if th hrg Q in th spifi rgion inrss with tim (so tht Q is positiv). This n hppn ithr if positivly hrg prtils flow into this rgion s shown in Fig. A3, or if ngtivly hrg prtils flow out of this rgion s shown in Fig. A3. Convrsly, th urrnt I is ngtiv if th hrg Q in th spifi rgion rss with tim (so tht Q is ngtiv). This n hppn ithr if ngtivly hrg prtils flow into this rgion s shown in Fig. A3, or if positivly hrg prtils flow out of this rgion s shown in Fig. A3. To summriz, th urrnt rltiv to hosn sns is positiv if positivly hrg prtils mov long this sns, or if ngtivly hrg prtils mov opposit to this sns. * * W sy tht prtils mov (or flow) through surf long hosn sns if thy pss through th surf towr th rgion spifi y this sns. Convrsly, th urrnt is ngtiv if ngtivly hrg prtils mov long this sns,or if positivly hrg prtils mov opposit to 7 8

5 MISN0423 A4 MISN0423 A5 () () () () sns of I Fig. A3: Prtils with iffrnt hrgs moving in iffrnt irtions through surf. Th short rrow though th surf inits sns hosn so s to spify th rgion on th right si of th surf. Th tul sns of th urrnt is init y th rrows t th ottom of th igrm. this sns. Thus ngtivly hrg prtils moving long hosn sns prou urrnt of th sm sign s positivly hrg prtils moving opposit to this sns. Sns of I Th sns of th urrnt through surf is th prtiulr sns spifying th rgion in whih hrg inrss s rsult of th urrnt. * * Th sns of urrnt is somtims lso ll th irtion of th urrnt, lthough this woring is onfusing us th urrnt is not vtor n thus hs no sptil irtion, ut only on of two possil snss. If th urrnt long hosn sns is positiv (s in Fig. A3 n Fig. A3), th tul sns of th urrnt is thus th sm s this hosn sns. But if th urrnt long hosn sns is ngtiv (s in Fig. A3 n Fig. A3), th tul sns of th urrnt is opposit to this hosn sns. Not tht th sns of th ltri urrnt is th sm s th sns of motion of th prtils if thir hrg is positiv (s in Fig. A3 n Fig. A3), ut is opposit to th sns of motion of ths prtils if thir hrg is ngtiv (s in Fig. A3 n Fig. A3). Exmpl A1: Eltri urrnt in n ioni solution Figur A4 illustrts shmtilly solution of soium hlori whih ontins positivly hrg soium (N ) ions n ngtivly hrg hlorin (Cl ) ions. Suppos tht th solution is pl in n xtrnlly prou ltri fil E pointing to th right. Wht thn is ` E ` v surf N ` N v ` Cl v Cl sns of I ` v ` E Fig. A4: Extrnl ltri fil E using motion of N n Cl ions in solution of soium hlori. th urrnt through th init surf prpniulr to this fil? Th ltri fil E xrts on th positivly hrg N ions n ltri for to th right n thus uss ths ions to mov through th surf into th rgion on th right. Aoringly, ths positivly hrg N ions moving to th right prou n ltri urrnt hving sns towr th right. On th othr hn, th ltri fil E xrts on th ngtivly hrg Cl ions n ltri for to th lft n thus uss ths ions to mov through th surf into th rgion on th lft. Aoringly, ths ngtivly hrg Cl ions moving to th lft prou n ltri urrnt whih hs lso sns towr th right (sin th hrg in th right rgion inrss s rsult of th motion of ngtivly hrg ions out of this rgion). Although th xtrnl ltri fil E uss th positivly n th ngtivly hrg ions to mov in opposit irtions, th ltri urrnts u to ths moving prtils hv thus th sm sns (towr th rgion towr whih th ltri fil points). Unrstning th Dfinition of Currnt (Cp. 1) Sttmnt n xmpl: () Stt th finition of urrnt in A1 trms of th usul lgri symols for th quntitis involv. () During tim intrvl of 4.0 ms ( s), soium ions, h hving hrg of C, pss into nrv xon through its mmrn. During this tim intrvl, wht is th hng in th hrg of th xon u to ions pssing through its mmrn? Wht is th orrsponing urrnt through th mmrn? (Answr: 7) Proprtis: () Answr th following qustions for hrg n for A2 urrnt. Wht is th usul lgri symol for h quntity? Is it numr or vtor? If it is numr, wht signs n it hv (,, 0)? Wht is its SI unit? () Is mpr, 1 mpr, or 100 mpr 9 10

6 MISN0423 A6 MISN0423 A7 S 1 S 2 S 3 Fig. A5. rsonl mgnitu for th urrnt flowing through light ul in ring lmp? (Answr: 4) Intrprttion of Q: Th rgion shown in Fig. A5 hs thr A3 surfs through whih hrg prtils mov. During tim intrvl of 0.5 s, prtils with totl hrg 0.3 C flow into th rgion through th surf S 1, whil prtils with totl hrgs of 0.1 C n 0.2 C flow out of th rgion through th surfs S 2 n S 3. () Wht r th urrnts I 1, I 2, n I 3 whih flow into this rgion through h of th surfs S 1, S 2, n S 3? () Wht is th hng Q in th hrg of th rgion uring th 0.5 s intrvl sri? (Answr: 1) Rlting Currnt Sign n Sns to Prtil Motion (Cp. 2) () NH 4 n wtr ron ro MnO pst 2 Fig. A6. wir () zin ontinr () ron ro 1 2 MnO2 pst 1 2 Zn zin ontinr ltrons lving th ron ro n pssing into th solution). () Whih rrow (1 or 2) inits th sns of th urrnt through th surf init in Fig. A6? () Is this sns th sm or opposit to th sns sriing th vrg motion of th ltron? (Answr: 8) Whn urrnt is u to th motion of positivly hrg prtils, is th sns of th urrnt long or opposit to th sns of A6 prtil motion? Answr th pring qustion for urrnt u to th motion of ngtivly hrg prtils. (Answr: 5) (Prti: [p1]) Figur A6 shows simplifi igrm of ryll ttry. Figur A6 inits th motion of hrg prtils ross th sur A4 fs of th zin ontinr n of th ron ro in its ntr. Positivly hrg zin ions (Zn ) lv th zin ontinr n om fr ions in th wtr solution. Eltrons ( ) lv th ron ro (n omin with NH 4 ions n with MnO 2 n wtr moluls to form nw ompouns). () Whih of th rrows ll y 1 n 2 inits th snss of th urrnts ross th ron n ross th zin surfs shown? () Rltiv to hosn sns init y th rrow ll 1, wht is th sign of th urrnt ross h surf? () Rltiv to hosn sns init y th rrow ll 2, wht is th sign of th urrnt ross h surf? (Answr: 10) (Suggstion: [s11]) Figur A6 inits th motion of ngtivly hrg ltrons A5 in th wir, shown onnt to th ry ll in Fig. A6. Eltrons (rmining in th zin ftr th prtur of th positivly hrg zin ions) mov into th wir t its right n. At th lft n, ltrons lv th wir, moving into th ron ro, (n ultimtly rpling th 11 12

7 MISN0423 B1 MISN0423 B2 SECT. B CURRENT AND MOTION OF CHARGED PARTI CLES () () In th pring stion w xmin how th sign, or sns, of urrnt pns on th motion of th hrg prtils rsponsil for this urrnt. Lt us now isuss how th mgnitu of th urrnt is rlt to th motion of th hrg prtils. Avrg vloity Consir moil hrg prtils in mtril (for xmpl, positiv or ngtiv ions in solution, or ltrons in mtl). Suh prtils mov roun with lrg sps in rnom irtions, so tht thir vrg vloity long ny prtiulr irtion is orinrily zro. (S Fig. B1.) But, in th prsn of n ltri fil long som prtiulr irtion, th prtils tn to mov prfrntilly in th irtion of th ltri for prou y this fil n thus quir smll vrg vloity in th irtion of this for. (S Fig. B1.) Thus ltrons in mtl wir quir n vrg vloity long th wir if n xtrnl vi (suh s ttry) mintins n ltri fil insi th wir. Motion prpniulr to surf Consir prtils, h hving hrg q, moving with som vrg vloity v in irtion ˆx prpniulr to som pln surf, s init in Fig. B2. (For xmpl, th prtils might ltrons in mtl wir n th surf might ross stion of this wir.) How is th ltri urrnt I through suh surf rlt to th motion of th prtils? Chrtristis of I Th sign of th urrnt (rltiv to th sns init y th vtor ˆx long whih th prtils mov) is positiv if th prtils r positivly hrg n is ngtiv if th prtils r ngtivly hrg. Th mgnitu of th urrnt (i.., th mgnitu of th rt t whih hrg psss through th surf) shoul lrgr if thr r mor prtils pssing through th surf, if h of ths prtils hs lrgr hrg, or if ths prtils mov with lrgr vloity. CALCULATION OF CURRENT To mk th pring qulittiv ommnts mor quntittiv, lt us ssum for simpliity tht vry prtil hs th sm hrg q n movs ` E = 0 Fig. B1: Pth of moving hrg prtil insi mtril whr it ollis rptly with th toms in th mtril. with th sm vloity v (qul to its vrg vloity) in th irtion long ˆx. As rsult of thir motion, ths prtils ross th surf in Fig. B2 from lft to right. During som smll tim t, ths prtils thn hng th hrg in th rgion on th right si of th surf y som mount Q, thus prouing n ltri urrnt I = Q/t through th surf (rltiv to th sns init y th vtor ˆx pointing into th rgion on th right). To fin this urrnt, onsir th N moving hrg prtils ontin t som instnt on th lft si of th surf in ylinril rgion of smll lngth l n of rossstionl r qul to tht of th surf. (S Fig. B2.) Ths prtils, moving long ˆx with vloity of mgnitu v, rquir tim t = l/v for ll of thm pss through th surf. * * This is th tim ftr whih prtil t th frthst istn l from th surf rrivs t th surf s rsult of moving with sp v. Sin th totl hrg of ths N prtils is Nq, ths prtils pssing through th surf ontriut hrg Q = Nq to th rgion on th right si of th surf. Aoringly, th ltri urrnt I prou y ths prtils pssing through th surf is simply qul to N prtils q ` v l x^ Fig. B2: Moving hrg prtils pssing through surf. È 13 14

8 MISN0423 B3 I = Q t = ( Nq ) (B1) l v Simplifying this frtion y multiplying oth its nomintor n numrtor y v, w thn gt ( ) N I = qv (B2) l Hr N/l is th numr of moving hrg prtils pr unit lngth (in ylinril rgion of r qul to tht of th surf). Th rltion (B2) llows us to fin th urrnt I through surf from knowlg of th numr N/l of moving hrg prtils pr unit lngth, th hrg q of h of ths prtils, n th mgnitu v of th vrg vloity of ths prtils moving in irtion prpniulr to th surf. Not tht th urrnt I inrss proportiontly to ll of ths quntitis, in orn with our prvious qulittiv xpttions. Rlting Currnt Mgnitu to Prtil Motion (Cp. 2) () Whn urrntrrying mtl wir is ht, th numr of B1 moving ltrons ontin in lngth l of wir rmins (vry nrly) fix. Howvr, thir vrg vloity rss (u to inrs intrtion with th mor rpily virting toms of th mtl). Dos hting urrntrrying wir inrs or rs th mgnitu of th urrnt through it? () Th numr of moil ltrons in smionutor inrss whn mor impurity toms r. If two othrwis intil smionutors r onnt twn th trminls of intil ttris, is th urrnt through th smionutor with mor impurity toms lrgr or smllr in mgnitu thn th urrnt through th smionutor with fwr suh toms? (Answr: 12) (Suggstion: [p2]) Illustrtion: Sp of Eltrons In Wirs B2 Stnr lmp or ontins out toms of oppr in h mtr of its lngth. Eh tom of oppr hs 1 moil ltron with hrg of oulom. If urrnt of mgnitu 2.0 mpr flows through suh wir, wht is th mgnitu of th vrg vloity of th moving ltrons in th wir? (Answr: 2) MISN0423 C1 SECT. C STEADY STATE CURRENTS Dfinition of sty stt flow Suppos tht mtl wir, or som othr systm, is onnt to som xtrnl vi (suh s ttry) whih n mintin n ltri urrnt through th wir. Immitly ftr th wir is onnt, th ltrons rrrng thmslvs throughout th wir so tht th motion of ths ltrons is quit omplx. But, ftr vry short tim, th motion of th ltrons usully sttls own to simpl stystt sitution hrtriz s follows: Df. Sty stt: A sitution in whih th totl hrg in ny rgion rmins unhng. (C1) Anlogous stystt situtions our ommonly in th flow of liquis (s isuss in txt stion B of Unit 418). For xmpl, whn hos is first onnt to wtr fut, th flow of th wtr through th hos is initilly quit omplx s th wtr grully fills th hos. But, ftr short tim, th hos oms fill n th sitution sttls own to sty stt whr th wtr is flowing through th hos whil th mount of wtr in ny rgion of th hos rmins unhng. Currnts in sty stt Bus of th onsrvtion of hrg, th totl hrg in ny rgion n only hng us of hrgs pssing through its ounry surf. In sty stt, whr th totl hrg in ny rgion rmins unhng, th totl ltri urrnt I in flowing into th rgion (through vrious prts of its ounry) must thn qul to th ltri urrnt I out flowing out of th rgion (through vrious othr prts of its ounry). This funmntl onition rlting th urrnts in sty stt n thus summriz: I in = I out. (C2) This rltion n sily visuliz y thinking of th nlogous flow of liqui in sty stt (s isuss in txt stion B of Unit 418). During ny tim, th flow of liqui into ny rgion must qul to th mount of flui flowing out of this rgion

9 MISN0423 C2 MISN0423 C3 () () I 1 I 1 I 2 I 2 I 1 I 2 Fig. C1: Currnts flowing through wir. I 3 Fig. C2: Currnts in wirs join togthr. Th juntion rgion is shown sh. CURRENT THROUGH A WIRE Consir ny mtl wir (not nssrily of uniform rossstion), suh s th stright wir in Fig. C1 or th urv wir in Fig. C1. In sty stt, wht n w sy out th urrnt t ny point in th wir (i.., out th urrnt through rossstionl surf of th wir t ny suh point)? Equlity of urrnts Lt us hoos snss t vrious points long th wir in suh wy tht th tip of th rrow initing on sns lwys points towr th til of th rrow initing th nxt sns. (S Fig. C1.) Lt us not y I 1 n I 2 th urrnts, rltiv to ths hosn snss, through th rossstionl surfs t ny two points of th wir. If ths urrnts r positiv, th urrnt into th rgion twn th rossstionl surfs in Fig. C1 is thn I 1, whil th urrnt out of this rgion is I 2. In sty stt, th rltion I in = I out thus implis simply tht * I 1 = I 2 (C3) * If th urrnts I 1 n I 2 in th pring isussion wr ngtiv, I 1 woul th urrnt out of th rgion n I 2 woul th urrnt into th rgion. Thus th qulity I 1 = I 2 of Eq. (C3) woul rmin vli. In sty stt, th sm urrnt flows thus through ny rossstionl surf of th wir. Aoringly w n simply tlk out th urrnt I flowing through th wir, without ning to spify ny prtiulr rossstion of th wir. Unonnt wir If on n of wir is unonnt to nything ls, th urrnt flowing through this n of th wir must zro. In sty stt, th urrnt through ny othr rossstion of th wir must thn, y Eq. (C 3), lso zro (irrsptiv of wht th othr n of th wir might onnt to). Currnt n prtil flow Suppos tht th urrnt I in wir flows in th sns init y th rrows in Fig. C1. As isuss in S. A, this urrnt n ithr th rsult of positivly hrg prtils moving through th wir long this sns, or of ngtivly hrg prtils moving through th wir opposit to this sns. But, irrsptiv of th tul sitution, it is simplst to imgin tht th urrnt rsults from th flow of positivly hrg prtils, sin th sns of th ltri urrnt is thn th sm s tht of th motion of th prtils. CURRENTS AT A JUNCTION Consir svrl wirs (suh s th thr wirs in Fig. C2) whih mt t juntion. In sty stt, how r th urrnts in ths wirs rlt? Lt I 1, I 2, n I 3 not th urrnts through ths wirs rltiv to snss hosn s init y th rrows in Fig. C2. In sty stt, th totl urrnt I in into th juntion rgion must qul to th totl urrnt I out out of this rgion. But in Fig. C2 th urrnt into th juntion rgion is I in = I 1 whil th urrnt out of this rgion is I out = I 2 I 3. Hn th rltion I in = I out implis tht I 1 = I 2 I 3 (C4) This rltion llows us to fin th urrnt in on of th wirs from knowlg of th urrnts in th othr wirs. * 17 18

10 MISN0423 C4 MISN0423 C5 * Any of th urrnts in Eq. (C4) might ngtiv. This mrly mns tht th tul sns of suh urrnt is opposit to th hosn sns init in Fig. C2. 15 A 15 A Th sitution in Fig. C2 is gin nlogous to tht of th sty flow of liqui through svrl pips join togthr. During nytim, th mount of liqui flowing into th juntion rgion through on pip is simply qul to th mount of flui flowing out of this rgion through th othr pips. REMARK ON ALTERNATING CURRENTS As will isuss mor fully in txt stion D of Unit 425, th urrnts suppli to our homs through wll outlts r ltrnting, i.., thy vry rptitivly in tim (t rt of 60 rptitions pr son in th Unit Stts) so s to s oftn positiv s ngtiv th mgnitu of suh n ltrnting urrnt n sri y n fftiv vlu (whih is fix frtion of its mximum vlu). Suh ltrnting urrnts ommonly lso flow in sty stt whr th totl hrg in ny rgion rmins unhng. Hn th rltion I in = I out lso hols for ltrnting urrnts t ny instnt, or for thir fftiv vlus. Unrstning I In = I out (Cp. 1) Exmpl: Figur C3 shows floor pln of kithn in whih C1 urrnt flows through th stov, rfrigrtor, n through wll outlt to whih is onnt tostr n off mkr. Arrows n numrs in Fig. C3 init th mgnitus n snss of th urrnts flowing through th kithn wlls t on instnt of tim. Wht r th urrnts I in n I out for this kithn? (Answr: 9) Appliility: () Th initilly unhrg plts of pitor r C2 onnt to th trminls of ttry. Consir on pitor plt uring th tim this plt is quiring positiv hrg from th ttry. For this plt, I in hs som positiv vlu, ut I out = 0. Why os th rltion I in = I out not pply to this sitution? () Now onsir ltr tim whn th sm plt is fully hrg. Now no urrnt flows in or out of th plt so tht I in = I out = 0. Why is I in = I out pplil in this sitution, lthough it ws not pplil to th sitution sri in prt ()? (Answr: 11) (Suggstion: [s8]) (Prti: [p3]) 6 A 6 A Fig. C3. 50 A 50 A Fig. C4. Rlting Currnt Sign n Sns to Prtil Motion (Cp. 2) Figur C4 shows th vloitis of positivly hrg non ions C3 n ngtivly hrg ltrons long th long tu of non light. () Whih rrow (1 or 2) inits th sns of th urrnt u to th ltrons n u to th ions? () Rltiv to hosn sns init y rrow 1, wht is th sign of h of ths urrnts? (Answr: 6) (Suggstion: [s5]) Using I In = I out to Rlt Currnts (Cp. 3) Now: Go to tutoril stion C. Rognizing qul mgnitus: Th igrms in Fig. C5 h C4 show st of onnt wirs with rrows initing th snss of th sty urrnts through thm. () In Fig. C5, urrnts flow through surfs t th points A, B, C, D, n E. Through whih of ths surfs must th urrnt qul in mgnitu to th urrnt through th surf t A? () In Fig. C5, through whih of th light uls A, B, C, n D must th urrnt qul in mgnitu to th urrnt through th ttry? (Answr: 3) (Suggstion: [2]) Rlting snss: Figur C6 shows svrl sts of onnt wirs. C5 In h st, singl rrows init th known sns of th urrnt in on or mor wirs. A pir of rrows (ll 1 n 2) init th two possil snss of n unknown urrnt in nothr wir. From h pir of rrows, hoos th on whih inits th sns of th urrnt (or stt tht this urrnt oul hv ithr sns). (Answr: 15) (Suggstion: [3])

11 MISN0423 C6 MISN0423 C7 () () A B B A T T E R Y E A D C B C () 1 () 2 () () () 1 2 Fig. C5. D Fig. C Rlting mgnitus n snss: Figur C7 shows igrm C6 of th urrnts flowing in hous trilr. Fin th mgnitu n sns of th urrnt I l through th living room lights y pplying I in = I out to th init rgion. (Answr: 18) Rlting mgnitus: Figur C8 shows th snss of vrious urrnts flowing in onnt wirs in kithn. If th urrnt into C7 (n out of) th rfrigrtor, tostr, n off mkr hv th mgnitus 6.6 A, 8.3 A, n 12.2 A, wht is th mgnitu of th urrnt I flowing out of th kithn? (Answr: 13) (Suggstion: [s7]) (Prti: [p4]) 5 A B A T T E R Y 2 A room lights rgion I l living room lights rfrigrtor tostr S off mkr Swiths n zro urrnts: A swith is simply vi whih C8 n ithr los so s to onnt two wirs or opn so s to isonnt ths wirs from h othr. In Fig. C8, th wir through whih urrnt flows wy from th off mkr hs swith ll y S. Suppos this swith is opn. Whih of ths urrnts r thn zro? () Currnt out of th off mkr. () Currnt into th off mkr. () Currnt into th tostr. (Answr: 16) (Suggstion: [s6]) Fig. C7. Fig. C

12 MISN0423 D1 MISN0423 D2 SECT. D TWOTERMINAL SYSTEMS AND CIRCUITS Twotrminl systms Fig. D1: Two trminl systm with trminls n. In mny importnt prtil situtions, ltri urrnts flow through vrious systms suh s lmps, tprorrs, iologil orgnisms, n so forth. To isuss suh systms, w shll introu this finition: Df. Twotrminl systm: Any systm with two points (ll its trminls ) suh tht ltri urrnt n flow into th systm through smll surf t on trminl n n flow out of th systm through smll surf t th othr trminl. (D1) Suh twotrminl systm is shown shmtilly in Fig. D1. Not tht twotrminl systm n of ny gr of omplxity. For xmpl, suh systm might simply oppr wir whos two ns r th trminls of th systm. Or it might vuum lnr whos two trminls r th two prongs on th plug. Or it might mn whos two trminls r th thums of his hns. Dsription of systm Th ltri proprtis of twotrminl systm n sri quit usfully without ning to know ny tils out th intrnl strutur of th systm. Thus w my sri suh systm mrly in trms of quntitis whih n msur t its trminls. Ths quntitis r th ltri urrnts flowing through ths trminls n th potntils t ths trminls. (As w shll s, knowlg of how ths quntitis r rlt n provi vry usful informtion out twotrminl systms n thir intrtion.) Sty stt In sty stt, th totl hrg in twotrminl systm rmins unhng. Hn th ltri urrnt flowing into th systm through on trminl must thn qul to th urrnt flowing out of th systm through its othr trminl. W shll ll this urrnt th urrnt through th systm. Nonissiptiv systms In som spil twotrminl ss (.g., in vuum or in supr onutor) hrg prtils ost long with onstnt vloity if thr is no ltri fil prsnt. Tht is us thr is no mhnism for th issiption of thir kinti nrgis. Th hrg prtils n m to lrt (or lrt) y pplying fil. Dissiptiv systms Orinrily, howvr, th hrg prtils moving through twotrminl systm intrt (or olli) with th mny toms in th mtrils within th systm. As rsult of suh intrtions, th nrgy of th moving hrg prtils is issipt, i.., is onvrt into th rnom intrnl nrgy of ll th tomi prtils in th mtril. (Suh twotrminl systm is thrfor ll issiptiv. ) Hn sty urrnt n flow through suh systm only if th nrgy lost y th prtils to rnom intrnl nrgy is rsuppli from othr sours. For xmpl, th nrgy lost to rnom intrnl nrgy my suppli t th xpns of rs in th oulom ltri potntil nrgy of th hrg prtils moving through th systm. Chrtristi rltion Th rs in th oulom potntil nrgy of th hrg prtils moving through systm is simply rlt to th potntil rop V twn its trminls. Hn w xpt tht th sty urrnt I flowing through th systm shoul rlt to th potntil rop V ross th systm (i.., from on trminl to th othr). Th xt rltion twn I n V is hrtristi of th ntur of th prtiulr systm unr onsirtion. W shll isuss in S. F how I is rlt to V in th s of th most ommon twotrminl systms. CIRCUITS Twotrminl systms n onnt togthr so tht th urrnt flowing out of trminl of on systm n flow into trminl of nothr systm. Any olltion of twotrminl systms onnt togthr is ll n ltri iruit (or ntwork )

13 MISN0423 D3 MISN0423 D4 () () g () wir ttry g h ttry rio ttry h g h wir rio rio lmp 1 f lmp 1 lmp 2 lmp 1 lmp 2 lmp 2 Fig. D2: Ciruit onsisting of storg ttry, two hlmps n rio: () physil rrngmnt; () n (), quivlnt iruit igrms rprsnting this iruit. (Th onnting wirs in th tul iruit r ssum short nough so th potntil rop long h of thm is ngligil.) Ciruit igrm A iruit igrm (suh s th on illustrt in Fig. D2) is rwing rprsnting symolilly how th vrious twotrminl systms in iruit r onnt togthr. In suh igrm, lin onnting ny two trminls is us to init tht urrnt n flow twn ths trminls n tht ths trminls hv th sm potntil. (For xmpl, suh onnting lin might rprsnt th ft tht th tul trminls touh h othr or r onnt y suffiintly short mtl wir.) Not tht iruit igrm inits only th onntions twn trminls, irrsptiv of th tul positions of th systms in th igrm. For xmpl, Fig. D2 shows rwing of n utomoil storg ttry onnt y insult wirs to two hlmps n r rio. * f f * For ny ttry, th trminl t th highr potntil is onvntionlly init y sign n th trminl t th lowr potntil y sign. This physil sitution is rprsnt y th two quivlnt iruit igrms shown in Fig. D2 n Fig. D2. Although ths two igrms look iffrnt, thy r ompltly quivlnt us thy show prisly th sm trminls onnt to h othr. SPECIAL CIRCUIT CONNECTIONS Two or mor twotrminl systms n onnt togthr in som spil simpl wys so tht th rsulting iruit is nothr twotrminl systm. Sris onntion Two or mor systms n onnt lik hin, s shown in Fig. D3, so tht on trminl of h systm is onnt to on trminl of th nxt systm. Th systms r thn si to onnt in sris. On trminl of th first systm n on trminl of th lst systm r lft unonnt (i.., thy r th ns of th hin) n r th trminls of th omposit twotrminl systm onsisting of th iniviul systms onnt in sris. Prlll onntion Two or mor systms n lso onnt, s shown in Fig. D3, y onnting togthr on trminl of h systm n lso onnting togthr th othr trminl of h systm. Th systms r thn si to onnt in prlll. Not tht n ltri urrnt trvling from g to h in th igrm will split in two t point, prt of th urrnt trvling th rout f n th othr prt trvling th prlll rout f. Knowing Aout TwoTrminl Systms Whih of ths systms ( n ) in Fig. D4 r twotrminl D1 systms? Suppos tht th tprorr in Fig. D4 is st on ply so tht no urrnt flows through th wirs to th mirophon. Is this tp rorr thn twotrminl systm? (Answr: 22) 25 26

14 MISN0423 D5 MISN0423 D6 () () S 1 S 1 S 2 g sris onntion S 2 prlll onntion Fig. D3: Spil onntions of two twotrminl systms S 1 n S 2. Knowing Aout Currnt n Work Pr Unit Chrg An ltron movs from ttry into wir t trminl A. D2 It thn movs with onstnt vrg vloity long th wir ( issiptiv twotrminl systm) until it rhs th point B. () As th ltron movs from A to B, os its oulom potntil nrgy inrs, rs, or rmin th sm? () Dos its vrg kinti nrgy inrs, rs or rmin th sm? () Dsri ny trnsfr of nrgy from th ltron to ny othr systm. (Answr: 19) Now: Go to tutoril stion D. Rlting Dsriptions of Ciruits (Cp. 4) Rognizing points with qul potntils: Whih of th ll D3 points in Fig. D5 must hv th sm potntil s th positiv trminl of th ttry? Whih must hv th sm potntil s th () urrnt in urrnt out Fig. D4. () mirophon or f urrnt in urrnt out urrnt in urrnt out h ttry Fig. D5. ngtiv trminl of th ttry? (Answr: 17) (Suggstion: [s3]) Rognizing quivlnt iruit igrms: Whih of th igrms D4 () n () in Fig. D6 is/r quivlnt to th igrm ()? (Answr: 14) ([2], [3]) Throughout this unit, w shll ssum tht trminls onnt y mtl wirs hv th sm potntil. [This ssumption is vry nrly orrt for short wirs. W shll isuss it furthr in Unit 424.] Rlting piturs n rwings of iruits: Figur D7 shows D5 thr ll light uls onnt to ttry. Whih iruit igrm (or igrms) in Fig. D6 sris this iruit? (Answr: 23) ([3], [4]) Mking iruit igrms: Figur D8 shows prts of two strns D6 of Christmstr lights. Drw iruit igrm for h string of lights. Us ll squr to init h light ul. (Answr: 26) (Suggstion: [s14]) Prlll n sris onntions: () Two ttris in flshlight D7 r onnt in sris. Us rtngl to rprsnt h ttry n rw igrm showing how thy r onnt. () A fluorsnt light ul n n innsnt light ul r onnt in prlll in ring lmp. Us rtngl to rprsnt h ul n rw igrm showing how thy r onnt. (Answr: 20) (Suggstion: [5]) Prlll n sris onntions: In th igrm shown in Fig. D D8 5, whih of th following pirs of systms r onnt in sris, whih r onnt in prlll, n whih r onnt nithr in prlll nor in sris: (1 n 3), (2 n 3), (ttry n 2)? (Answr: 30) (Suggstion: [5]) Mor prti for this Cpility: [p5], [p6], [p7] 27 28

15 MISN0423 D7 MISN0423 E1 SECT. E PRINCIPLES OF CIRCUIT ANALYSIS ttry 1 () Fig. D ttry () 1 2 ttry () f 1 2 Fig. D7. Lt us now summriz th gnrl prinipls whih llow us to rlt th urrnts through vrious twotrminl systms onnt in iruit, n lso to rlt th potntil iffrns twn th trminls of ths systms. By omining ths prinipls with knowlg of th rltion twn th urrnt n potntil iffrn for h twotrminl systm, w shoul thn l to nlyz ny iruit to fin unknown urrnts or potntils on th sis of limit informtion. Rltion twn urrnts Unlss spifilly stt othrwis, w shll onsir iruits in sty stt. If w onsir ny juntion whr two or mor trminls r onnt togthr, w thn know tht th hrg in this juntion rgion must rmin unhng. By Eq. (C2), th totl urrnt I in flowing into th juntion rgion must thn qul to th totl urrnt I out flowing out of this rgion. Thus t ny juntion, I in = I out (E1) Rltion twn potntils Th potntil t vry point hs som uniqu vlu. Hn th potntil rop V from ny point to ny othr point hs lso som uniqu vlu. But this potntil rop is qul to sum of th potntil rops long sussiv sgmnts of ny pth from to. * To summriz: () f () f Fig. D8. From ny point to ny othr point, th potntil rop V is th sum of th potntil rops long ny pth from to. (E2) * This sttmnt is quivlnt to sying tht th work on on hrg prtil moving long ny pth from to is just th sum of th works on long sussiv sgmnts of this pth. In prtiulr, this onlusion implis tht th sum of th potntil rops lult long ny on of svrl ltrnt pths from to must th sm, qul to th potntil rop V. * 29 30

16 MISN0423 E2 MISN0423 E3 * This sttmnt is onsqun of th ft tht th work on y oulom fors on hrg prtil moving twn two points is inpnnt of th pth. I B I B I = 3.5 A B Fus I B I = 3.2 A L I R I R ANALOGY TO ELEVATION Bttry 6.2 V Lmp 6.1 V Rio Th pring onlusions r nlogous to fts fmilir in vryy lif. Evry point in trrin hs som uniqu lvtion ov s lvl. In going on hik from point to nothr point, th rop in lvtion is thn simply th sum of th rops in lvtion long sussiv portions of ny hiking pth from to. Corrsponingly, th sum of th rops in lvtion long ltrnt hiking pths from to must th sm, qul to th rop in lvtion from to. APPLICABILITY OF ANALOGOUS PRINCIPLES TO FLU IDS Prinipls nlogous to Eq. (E1) n Rul (E2) pply in isussing th sty flow of fluis (suh s tht of wtr through th pips of pluming systm, or loo through loo vssls). Th prinipl Eq. (E 1) thn rlts th flui (or mss) urrnts into n out of ny rgion. Th othr rlvnt quntity is th prssur (inst of th ltri potntil). Sin th prssur hs uniqu vlu t vry point, th prinipl orrsponing to Rul (E2) is tht th prssur rop from ny point to ny othr point is th sum of th prssur rops long ny pth from to. Exmpl E1: Fining unknown quntitis in iruit Figur E1 shows iruit igrm rprsnting th storg ttry of r onnt through fus to rio n lmp. Th ttry mintins twn its trminls n potntil rop V = 6.2 volt. Th urrnt I B flowing through th ttry is 3.5 mpr in th init sns. Th msur potntil rop V from trminl to trminl of th lmp is V = 6.1 volt n th urrnt I L through th lmp is 3.2 mpr in th init sns. Wht thn is th urrnt through th fus n th potntil rop twn th trminls of th fus? Also, wht is th urrnt through th rio n th potntil rop twn th trminls of th rio? In th sty stt, th urrnt I B flowing out of th ttry into th wir nr must qul to th urrnt flowing out of th wir I B I B I L Fig. E1: Ciruit onsisting of storg ttry onnt to lmp n rio. into th fus. Thus th urrnt through th fus is lso I B = 3.5 mpr in th sns init in th igrm. This urrnt I B thn flows out of th fus into th juntion point. Th urrnt thn flows out of this juntion point long two possil pths, on prt I L of this urrnt flowing through th lmp n nothr prt I R of th urrnt flowing through th rio. In sty stt, th urrnts into n out of th juntion r rlt so tht I B = I L I R. Hn th urrnt through th rio is I R = I B I L = 3.5 mpr 3.2 mpr = 0.3 mpr. (Not tht th urrnts I L n I R romin t th nxt juntion point so tht th urrnt flowing long th pth from to is I L I R = I B, whih is proprly just th urrnt flowing k through th ttry.) Lt us now look t th potntil rops. Consiring th points n, th potntil rop V long th pth through th ttry is known to 6.2 volt. But this potntil rop must th sm s tht long th ltrnt pth through th fus, th lmp, n th onnting lin (long whih th potntil rop is zro). Th potntil rop long this pth is thn V = V V 0. Hn 6.2 volt = V 6.1 volt so tht th potntil rop V from trminl to trminl of th fus must qul to V = 0.1 volt. Not lso tht th known potntil rop V = 6.1 volt long th pth through th lmp must th sm s long th pth,, f, through th rio. But th potntil rop long this ltrnt pth is just th potntil rop V f through th rio (sin th potntil rops long th onnting pths from to, n from f to, r zro). Thus V = V f so tht th potntil rop twn th trminls n f of th rio is qul to th potntil rop V = 6.1 volt twn th trminls of th lmp. I R f I R 31 32

17 MISN0423 E4 MISN0423 E5 I I 1 S 1 S 2 I = I = I 1 2 I 2 V = V V 1 2 I V = V 1 V = V 2 V = V Fig. E2: Twotrminl systms S 1 n S 2 onnt in sris. SERIES AND PARALLEL CONNECTIONS Th prinipls of iruit nlysis hv immit implitions for systms onnt in sris or prlll (s fin t th n of S. D.) Sris onntion Consir two systms onnt in sris, s shown in Fig. E2. In sty stt, th urrnt flowing out of on systm must qul to th urrnt flowing into th nxt systm. Thus th urrnt through h systm is th sm, qul to th urrnt I whih flows through th omposit twotrminl systm. On th othr hn, th totl potntil rop V ross th omposit systm (i.., th potntil rop from trminl to trminl in Fig. E2) is qul to th sum of th potntil rops ross h of th iniviul systms. Prlll onntion Consir now two systms onnt in prlll, s shown in Fig. E 3. Sin th orrsponing trminls of ths systms r onnt togthr, thy must hv th sm potntil. Hn th potntil rop V ross th omposit systm (i.., from trminl to trminl f in Fig. E 3) must th sm s th potntil rop ross h iniviul systm. On th othr hn, th urrnt I whih flows into th omposit systm flows prtilly through h of th onnt systms. In sty stt, th urrnt I flowing into th systm must thn qul to th sum of th urrnts flowing through th iniviul systms. (Thus, in Fig. E3, th urrnt I flowing into th juntion t th point must qul to th sum of th urrnts I 1 n I 2 flowing out of this juntion through th iniviul systms.) I I 1 I 2 I 1 I 1 S 1 S 2 I 2 I 2 I = I I 1 2 f I 1 I 2 I V = V = V 1 2 V = V 1 V = V 2 V = V f Fig. E3: Twotrminl systms S 1 n S 2 onnt in prlll. Rlting Currnts n Potntils In Ciruit (Cp. 3) Now: Go to tutoril stion E. A urrnt of 2.0 mpr flows into twotrminl systm onsisting of two lightuls onnt in prlll. Th potntil rop E1 ross on light ul is 100 volt, n th urrnt through it is 1.4 mpr. () Drw iruit igrm showing th two light uls n th snss of th urrnts flowing through thm. () Apply I in = I out to fin th mgnitu of th urrnt flowing through th son light ul. () Apply V = sum of potntil rops long ny pth to fin th potntil rop ross this son light ul. (Answr: 27) ([1], [2]) A urrnt of 2.0 mpr flows into on trminl of twotrminl E2 systm onsisting of two light uls onnt in sris. Th potntil rop V ross th omin systm is 100 volt. Th potntil rop V ross th first light ul is 70 volt. Wht is th potntil rop V ross th son light ul, n wht is th mgnitu of th urrnt through this ul? (Answr: 21) (Suggstion: [s13]) Figur E4 shows th iruit igrm of n mrgny ltril E3 systm for summr in oftn lft without outsi ltril powr. Th rrows init th sns of th urrnt in h prt of th iruit. () If h ttry is n utomoil ttry whih prous onstnt potntil iffrn of 6.0 volt twn its trminls, wht is th potntil rop from to? () Suppos tht th urrnt through th rio is 0.5 mpr, th urrnt through th ttry is 1.4 mpr, n th wtr pump is turn off so tht no urrnt flows through it. Wht thn is th urrnt through h of th two light uls? (Answr: 24) 33 34

18 MISN0423 E6 ttris (Prti: [p8]) wtr pump lights 1 2 rio Fig. E4. Knowing Aout Sris n Prlll Conntions Two twotrminl systms (A n B) n onnt ithr in E4 sris or in prlll. Whih of th following sttmnts sris systm ompos of A n B onnt in sris, n whih sris systm ompos of A n B onnt in prlll? () Currnt through A quls urrnt through B. () Potntil rop ross A quls potntil rop ross B. () Currnt into omin systm quls sum of urrnts through A n through B. () Potntil rop ross ntir systm quls sum of potntil rop ross A n ross B. (Answr: 33) MISN0423 F1 SECT. F CHARACTERISTICS OF A TWOTERMINAL SYS TEM Currnt n nrgy issiption As isuss in S. D, hrg prtils moving through issiptiv twotrminl systm intrt with th toms in th systm. As rsult, ths hrg prtils los thir nrgy so s to inrs th rnom intrnl nrgy of ll th prtils in th systm. Hn sty ltri urrnt through th systm n only mintin if th nrgy thus trnsform into rnom intrnl nrgy is qul to th nonrnom nrgy suppli to (or work on on) th moving hrg prtils. * * Chngs in th kinti nrgy of th moving hrg prtils r so smll s to ontriut ngligily to th rnom intrnl nrgy of th systm. Rlting urrnt n work Consir twotrminl systm with trminls n. (S Fig. F1.) To mintin sty urrnt I through this systm from to, som positiv nonrnom work pr unit hrg w must on on th hrg prtils (moving through th systm from to ) in orr to rsupply th nrgy lost y ths prtils to th rnom intrnl nrgy of th systm. On th othr hn, th sty urrnt through th systm is zro if no work is on to rsupply nrgy lost to rnom intrnl nrgy. Thus I 0 only if w 0, n I = 0 if w = 0. If th urrnt I is suffiintly smll, this urrnt must thn simply proportionl to th work w. * * Th rson is tht grph showing th rltion twn I n w must nrly stright lin in suffiintly smll rgion nr th point whr I = 0 n w = 0. Aoringly, w n writ I = Gw = 1 w, or RI = w R (F1) whr th quntity G (or R = 1/G) is som onstnt, inpnnt of I or w, whih hrtrizs th prtiulr twotrminl systm. Th quntity R is ll th rsistn of th systm n its riprol G = 1/R is ll th onutn of th systm

19 MISN0423 F2 MISN0423 F3 I I Fig. F1: Sty stt urrnt I flowing through twotrminl systm. This hrtristi rltion twn th urrnt I through th systm n th potntil rop V ross its trminls is vli s long s I is not too lrg. W shll us this rltion throughout th nxt two units to isuss vrious kins of prtilly usful twotrminl systms. Kins of work on Wht is th nonrnom work pr unit hrg w, on on hrg prtil moving through th systm from to? Som of this work is th oulom ltri work pr unit hrg. This work is simply qul to th potntil rop V = V = V V from to (i.., to th rop in oulom potntil nrgy pr unit hrg from to ). But in omplx systms nonoulom work my lso on. (For xmpl, in ttry hmil rtions n supply hmil nrgy, n thus o hmil work, whn hrg prtils mov from on trminl to th othr.) Suh nonoulom work is ll mf in orn with this finition:* Df. * Th trm mf is n rvition (pronoun s th sprt lttrs,m,f) for th phrs ltromotiv for. Th full wors r toy lmost nvr us n r misling, rronously using for inst of work pr unit hrg. Emf: Th mf E = E from point to point long som spifi pth, is th nonoulom work pr unit hrg on on hrg prtil moving long this pth. (F2) Th nonoulom work pr unit hrg on on hrg prtil moving through th systm from to is thus ll th mf E = E of th systm from to. (This mf is orinrily inpnnt of th urrnt I or potntil rop V through th systm.) Hn th totl nonrnom work pr unit hrg w, on on hrg prtil moving through th systm from to is simply th sum of th oulom n nonoulom fors pr unit hrg, i.., w = V E (F3) Rltion twn I n V Aoringly, th rltion (F1) twn urrnt n nonrnom work n writtn mor xpliitly s RI = V E. (F4) DISCUSSION Th rltion (F4) shows how th ltri urrnt through twotrminl systm is u oth to th oulom work on on th moving hrg prtils us of th potntil rop twn th trminls, n lso to th nonoulom work on on ths prtils. [Anlogously, th flui urrnt of liqui through horizontl pip is u oth to th work on on th moving liqui us of th prssur rop twn th ns of th pip, n lso to othr kins of work on y vis suh s pumps.] Utility of th rltion Th rltion (F4) is vli for ny issiptiv twotrminl systm s long s th urrnt through it is not too lrg. (In prti, this rltion is usully vli for ll urrnts of intrst.) Not tht Eq. (F4) sris xpliitly how th sty urrnt I through ny suh twotrminl systm is rlt to th potntil rop V twn th trminls of th systm. This rltion involvs two onstnts hrtrizing prtiulr systm, nmly th ltri rsistn R of this systm n th mf E of this systm. Mning of R By Eq. (F1) or Eq. (F4), th rsistn R of systm is numril quntity sriing th rltion twn th urrnt I through th systm n th work w = V E on pr unit hrg. Thus th rsistn is fin s follows (for urrnts whih r suffiintly smll): Df. Rsistn: R = w I = V E (F5) I If givn mount of work pr unit hrg w prous lrg urrnt, th rsistn R of th systm is thus smll (n th onutn 1/R of th systm is orrsponingly lrg). On th othr hn, if th sm givn mount of work pr unit hrg w prous only smll urrnt, th rsistn R of th systm is thus lrg (n th onutn 1/R of th systm is orrsponingly smll). Ths ommnts suggst why th quntity R sris th rsistn to urrnt flow

20 MISN0423 F4 Unit of R Th work pr unit hrg w hs th sm unit s th potntil rop, i.., volt. Aoring to Df. (F5) th unit of R is thn; unit of R = volt mpr = ohm. (F6) Hr th nw unit ohm [rvit y th Grk lttr Ω (omg)] is just onvnint rvition for th omintion volt/mpr. This unit is nm in honor of Gorg S. Ohm ( ), th Grmn physiist who stui sty ltri urrnts in wirs. Sign of R To mintin sty positiv urrnt flowing through twotrminl systm from trminl to trminl, th work pr unit hrg w, on on prtil moving from to, must positiv (or zro in th spil s whr th issiption of nrgy into rnom intrnl nrgy is ngligily smll). For only thn will th work on on ny positiv mount of hrg, pssing through th systm from to, proprly positiv so s to inrs th rnom intrnl nrgy of th systm. Hn w onlu tht th rsistn R = w/i of systm must lwys positiv or zro. Illustrtion: Work Don on Prtils In Bttry Figur F2 shows iruit, inluing ttry through whih F1 flows urrnt I. Consir positivly hrg prtil moving through th ttry from its ngtivly hrg trminl to its positivly hrg trminl. () Is th oulom for on this prtil irt roughly from towrs, or from towrs? () Wht is th sign of th oulom ltri work on on this prtil s it movs from to? (Answr: 28) (Suggstion: [s1]) MISN0423 F5 ttry Fig. F2. Proprtis: Answr ths qustions for potntil rop, mf, n F3 rsistn: () Wht lgri symol most oftn rprsnts this quntity? () Is this quntity numr or vtor? If it is numr, wht signs n it hv (,, 0)? () Wht is th SI unit of this quntity? (Answr: 31) Dpnn of work pr unit hrg on rsistn: A thin wir A F4 hs rsistn qul to four tims th rsistn of thik wir B. () Suppos th sm urrnt I flows through oth wirs. To ompr th works w A n w B pr unit hrg rquir to prou ths urrnts; stt whthr th work pr unit hrg w A for wir A is qul to 4, or 1/4 tims th work pr unit hrg w B for wir w B. () For wir with lrg rsistn, is th work pr unit hrg w lrgr or smllr thn th work pr unit hrg rquir to prou th sm urrnt through wir with smllr rsistn? (Answr: 34) Knowing Aout Enrgy Trnsfr Consir n ltron in wir whih is onnt to th trminls F5 of ttry. As th ltron movs long th wir, ltri work is ontinully on on it, n so its ltri potntil nrgy rss. Yt in sty stt, th vrg sp v of th ltron rmins onstnt, n so its vrg kinti nrgy, (1/2)mv 2, os not inrs. () Wht kin of nrgy os inrs s th ltron s potntil nrgy rss? () Us your nswr to xplin why lit light ul is hot. (Answr: 29) Unrstning RI = V E (Cp. 1) Exmpl: As urrnt flows from to through th ttry shown F2 in Fig. F2, th oulom work pr unit hrg is V = 1.4 volt, whil th nonoulom work (suppli y th hmil rtions in th ttry) is E = 1.5 volt. Wht is th totl work pr unit hrg on on prtils moving from to through this ttry? If th rsistn of th ttry is 0.1 ohm, wht is th urrnt I? (Answr: 25) 39 40

21 MISN0423 G1 SECT. G SUMMARY DEFINITIONS ltri urrnt; Df. (A1) mpr; Eq. (A2) sty stt; Df. (C1) twotrminl systm; Df. (D1) sris onntion; S. D, Fig. D3 prlll onntion; S. D, Fig. D3 mf; Df. (F2) ltri rsistn; Df. (F5) ohm; Eq. (F6) IMPORTANT RESULTS Dfinition of ltri urrnt: Df. (A1) I = Q/t Rltion twn urrnt n hrg prtil motion: Eq. (B2) I = (N/l)qv Rltion twn urrnts in sty stt: Eq. (C2), Eq. (E1) For ny rgion, I in = I out Rltion twn potntil rops: Rul (E2) V is th sum of th potntil rops long ny pth from to. Rltion twn I n V for twotrminl systm: Eq. (F4) RI = V E (if I is not too lrg) USEFUL KNOWLEDGE Efftiv urrnts n potntil rops (S. C) iruit igrms (S. D) Enrgy trnsfr in iruits (S.F) MISN0423 G2 NEW CAPABILITIES (1) Unrstn ths rltions: () Th finition I = Q/t of urrnt (S. A); () I in = I out for urrnts in sty stt (Sts. C n E, [p3]); () V = sum of potntil rops long ny pth from to (S. E); () RI = V E = w, th totl work pr unit hrg (S. F). (2) Rlt urrnt s mgnitu n sns (or sign rltiv to hosn sns): () To sription of th vrg motion of hrg prtils (ross ounry or long wir.) (Sts. A n C, [p1]). () To th numr of moving hrg prtils pr unit lngth of wir, th vrg vloity of ths prtils, n th sign of thir hrgs. (S. B, [p2]). (3) Us I in = I out n V = sum of potntil rops long ny pth to rlt th urrnts n potntil rops in iruit. (Sts. C, D, n E, [p4], [p8]). (4) Us intrhngly th following sriptions of iruit: () Pitur or rwing of th iruit; () Vrious quivlnt iruit igrms; () Vrl sription of th iruit in trms of prlll n sris onntions. (S. D, [p5], [p6], [p7]). Tutoril is: S. C: Visulizing urrnts S. D: Rlting iruits sri in vrious wys S. E: Rlting urrnts n potntils in iruit S. H: Aitionl prolms Rlting Dsriptions of Currnt (Cp. 1,2,3) Th lt rrying hrg to lrg VnGrff gnrtor hs G1 with of 50 m n hrg nsity of C/m 2. Thus th hrg Q in 1 mtr lngth of lt is C. If th lt movs t sp of 30 m/s, wht is th mgnitu of th urrnt u to th hrg rri y th lt? Wht is th hrg livr y th lt to 41 42