rxiv.org > physis > rxiv:1405.7474 Does the eletromotive fore (lwys) represent work?. J. Pphristou 1, A. N. Mgouls 1 Deprtment of Physil Sienes, Nvl Ademy of Greee, Pireus, Greee E-mil: pphristou@snd.edu.gr Deprtment of Eletril Engineering, Nvl Ademy of Greee, Pireus, Greee E-mil: ris@snd.edu.gr Astrt In the literture of Eletromgnetism, the eletromotive fore of iruit is often defined s work done on unit hrge during omplete tour of the ltter round the iruit. We eplin why this sttement nnot e generlly regrded s true, lthough it is indeed true in ertin simple ses. Severl emples re used to illustrte these points. 1. Introdution In reent pper [1] the uthors suggested pedgogil roh to the eletromotive fore (emf) of iruit, fundmentl onept of Eletromgnetism. Rther thn defining the emf in n d ho mnner for eh prtiulr eletrodynmi system, this roh egins with the most generl definition of the emf nd then speilizes to ertin ses of physil interest, thus reovering the fmilir epressions for the emf. Among the vrious emples treted in [1], the se of simple ttery-resistor iruit ws of prtiulr interest sine, in this se, the emf ws shown to e equl to the work, per unit hrge, done y the soure (ttery) for omplete tour round the iruit. Now, in the literture of Eletrodynmis the emf is often defined s work per unit hrge. As we eplin in this pper, this is not generlly true eept for speil ses, suh s the forementioned one. In Setion, we give the generl definition of the emf, E, nd, seprtely, tht of the work per unit hrge, w, done y the genies responsile for the genertion nd preservtion of urrent flow in the iruit. We then stte the neessry onditions in order for the equlity E=w to hold. We stress tht, y their very definitions, E nd w re different onepts. Thus, the eqution E=w suggests the possile equlity of the vlues of two physil quntities, not the oneptul identifition of these quntities! Setion 3 reviews the se of iruit onsisting of ttery onneted to resistive wire, in whih se the equlity E=w is indeed vlid. In Se. 4, we study the prolem of wire moving through stti mgneti field. A prtiulr sitution where the equlity E=w is vlid is treted in Se. 5. Finlly, Se. 6 emines the se of sttionry wire inside time-vrying mgneti field. It is shown tht the equlity E=w is stisfied only in the speil se where the mgneti field vries linerly with time.. The generl definitions of emf nd work per unit hrge onsider region of spe in whih n eletromgneti (e/m) field eists. In the most generl sense, ny losed pth (or loop) within this region will e lled iruit (whether or not the whole or prts of onsist of mteril ojets suh s wires, resistors, pitors, tteries, et.). We ritrrily ssign positive diretion of trversing the loop, nd we onsider n element of oriented in the positive diretion (Fig. 1).
. J. Pphristou, A. N. Mgouls q F Figure 1 Imgine now test hrge q loted t the position of d l, nd let F e the fore on q t time t. This fore is eerted y the e/m field itself, s well s, possily, y dditionl energy soures (e.g., tteries or some eternl mehnil tion) tht my ontriute to the genertion nd preservtion of urrent flow round the loop. The fore per unit hrge t the position of t time t, is F f = q (1) Note tht f is independent of q, sine the eletromgneti fore on q is proportionl to the hrge. In prtiulr, reversing the sign of q will hve no effet on f (lthough it will hnge the diretion of F ). In generl, neither the shpe nor the size of is required to remin fied. Moreover, the loop my e in motion reltive to n eternl inertil oserver. Thus, for loop of (possily) vrile shpe, size or position in spe, we will use the nottion (t) to indite the stte of the urve t time t. We now define the eletromotive fore (emf) of the iruit t time t s the line integrl of f long, tken in the positive sense of : E (t) = f ( r, t) () ( t ) (where r is the position vetor of reltive to the origin of our oordinte system). Note tht the sign of the emf is dependent upon our hoie of the positive diretion of irultion of : y hnging this onvention, the sign of E is reversed. As mentioned ove, the fore (per unit hrge) defined in (1) n e ttriuted to two ftors: the intertion of q with the e/m field itself nd the tion on q due to ny dditionl energy soures. Eventully, this ltter intertion is eletromgneti in nture even when it origintes from some eternl mehnil tion. We write: f = f f em (3) where f em is the fore due to the e/m field nd f is the lied fore due to n dditionl energy soure. We note tht the fore (3) does not inlude ny resistive (dissiptive) fores tht oppose hrge flow long ; it only ontins fores tht my ontriute to the genertion nd preservtion of suh flow in the iruit. Now, suppose we llow single hrge q to mke full trip round the iruit under the tion of the fore (3). In doing so, the hrge desries urve in spe (not neessrily losed one!) reltive to n eternl inertil oserver. Let e n element of representing n infinitesiml displement of q in spe, in time. We define the work per unit hrge for this omplete tour round the iruit y the integrl: w= f (4) For sttionry iruit of fied shpe, oinides with the losed urve nd (4) redues to w= f ( fied ) (5) It should e noted refully tht the integrl () is evluted t fied time t, while in the integrls (4) nd (5) time is llowed to flow! In generl, the vlue of w depends on the time t 0 nd the point P 0 t whih q strts its round trip on. Thus, there is ertin miguity in the definition of work per unit hrge. On the other hnd, the miguity (so to spek) with respet to the emf is relted to the dependene of the ltter on time t.
Does the eletromotive fore (lwys) represent work? The question now is: n the emf e equl in vlue to the work per unit hrge, despite the ft tht these quntities re defined differently? For the equlity E=w to hold, oth E nd w must e defined unmiguously. Thus, E must e onstnt, independent of time (de/=0) while w must not depend on the initil time t 0 or the initil point P 0 of the round trip of q on. These requirements re neessry onditions in order for the equlity E=w to e meningful. In the following setions we illustrte these ides y mens of severl emples. As will e seen, the stisftion of the ove-mentioned onditions is the eeption rther thn the rule! 3. A resistive wire onneted to ttery onsider iruit onsisting of n idel ttery (i.e., one with no internl resistne) onneted to metl wire of totl resistne R (Fig. ). As shown in [1] (see lso []), the emf of the iruit in the diretion of the urrent is equl to the voltge V of the ttery. Moreover, the emf in this se represents the work, per unit hrge, done y the soure (ttery). Let us review the proof of these sttements. I i E _ R f i I Figure A (onventionlly positive) moving hrge q is sujet to two fores round the iruit : n eletrostti fore F = qe t every point of e nd fore F inside the ttery, the ltter fore rrying q from the negtive pole to the positive pole through the soure. Aording to (3), the totl fore per unit hrge is f = f f = E f e The emf in the diretion of the urrent (i.e., ounterlokwise), t ny time t, is E = f = E f = f (6) where we hve used the fts tht E = 0 for n eletrostti field nd tht the tion of the soure on q is limited to the region etween the poles of the ttery. Now, in stedy-stte sitution (Ι = onstnt) the hrge q moves t onstnt speed long the iruit. This mens tht the totl fore on q in the diretion of the pth is zero. In the interior of the wire, the eletrostti fore Fe = qe is ounterlned y the resistive fore on q due to the ollisions of the hrge with the positive ions of the metl (s mentioned previously, this ltter fore does not ontriute to the emf ). In the interior of the (idel) ttery, however, where there is no resistne, the eletrostti fore must e ounterlned y the opposing fore eerted y the soure. Thus, in the setion of the iruit etween nd f = f = E. By (6), then, we hve:, e E = E = V V = V (7) where V nd V re the eletrostti potentils t nd, respetively. We note tht the emf is onstnt in time, s epeted in stedy-stte sitution. Net, we wnt to find the work per unit hrge for omplete tour round the iruit. To this end, we llow single hrge q to mke full trip round nd we use epression (5) (sine the wire is sttionry nd of fied shpe). In lying this reltion, time is ssumed to flow s q moves long. Given tht the sitution is stti (time-independent), however, time is not relly n issue sine it doesn t mtter t wht moment the hrge will pss y ny given point of. Thus, the integrtion in (5) will yield the sme result (7) s the integrtion in (6), de- 3
. J. Pphristou, A. N. Mgouls spite the ft tht, in the ltter se, time ws ssumed fied. We onlude tht the equlity w=e is vlid in this se: the emf does represent work per unit hrge. 4. Moving wire inside stti mgneti field onsider wire moving in the y-plne. The shpe nd/or size of the wire need not remin fied during its motion. A stti mgneti field B( r ) is present in the region of spe where the wire is moving. For simpliity, we ssume tht this field is norml to the plne of the wire nd direted into the pge. y z r υ υ ( r ) Figure 3 ( t) d B ( r ) In Fig. 3, the z-is is norml to the plne of the wire nd direted towrds the reder. We ll d n infinitesiml norml vetor representing n element of the plne surfe ounded y the wire (this vetor is direted into the plne, onsistently with the hosen lokwise diretion of trversing the loop ). If u ˆz is the unit vetor on the z-is, then d= ( d) uˆ z nd B= B( r ) uˆ z, where B( r ) = B ( r ). onsider n element of the wire, loted t point with position vetor r reltive to the origin of our inertil frme of referene. ll υ ( r ) the veloity of this element reltive to our frme. Let q e (onventionlly positive) hrge pssing y the onsidered point t time t. This hrge eeutes omposite motion, hving veloity υ long the wire nd quiring n etr veloity υ ( r ) due to the motion of the wire itself. The totl veloity of q reltive to us υ = υ υ. is tot f m B ( r ) υ υ θ θ f r υ tot Figure 4 f f υ m tot f υ f υ The lne of fores ting on q is shown in the digrm of Fig. 4. The mgneti fore on q is norml to the hrge s totl veloity nd equl to Fm = q ( υtot B). Hene, the mgneti fore per unit hrge is fm = υtot B. Its omponent long the wire (i.e., in the diretion of ) is ounterlned y the resistive fore f r, whih opposes the motion of q long (this fore, s mentioned previously, does not ontriute to the emf ). However, the omponent of the mgneti fore norml to the wire will tend to mke the wire move kwrds (in diretion opposing the desired motion of the wire) unless it is ounterlned y some eternl mehnil tion (e.g., our hnd, whih pulls the wire forwrd). Now, the hrge q tkes shre of this tion y mens of some fore trnsferred to it y the struture of the wire. This fore (whih will e lled n lied fore) must e norml to the wire (in order to ounterlne the norml omponent of the mgneti fore). We denote the lied fore per unit hrge y f. Although this fore origintes from n eternl mehnil tion, it is delivered to q through n eletromgneti intertion with the rystl lttie of the wire (not to e onfused with the resistive fore, whose role is different!). Aording to (3), the totl fore ontriuting to the emf of the iruit is f = fm f. By (), the emf t time t is r 4
Does the eletromotive fore (lwys) represent work? E (t) = f f m ( t) ( t) The seond integrl vnishes sine the lied fore is norml to the wire element t every point of. The integrl of the mgneti fore is equl to ( υ B) = ( υ B) ( υ B) tot The first integrl on the right vnishes, s n e seen y inspeting Fig. 4. Thus, we finlly hve: E (t) [ ( ) ( r B r )] = υ (8) ( t) As shown nlytilly in [1, ], the emf of is equl to d E (t) = Φ m( t) (9) where we hve introdued the mgneti flu through, m ( t) B ( r ) d Φ = = B( r ) d S ( t) S( t) (10) [By S(t) we denote ny open surfe ounded y t time t ; e.g., the plne surfe enlosed y the wire.] Now, let e the pth of q in spe reltive to the eternl oserver, for full trip of q round the wire (in generl, will e n open urve). Aording to (4), the work done per unit hrge for this trip is w f = f m The first integrl vnishes (f. Fig. 4), while for the seond one we notie tht f = f f = f w= f with f = f = f υ (11) where =υ is the infinitesiml displement of the wire element in time. 5. An emple: Motion inside uniform mgneti field onsider metl r () of length h, sliding prllel to itself with onstnt speed υ on two prllel rils tht form prt of U-shped wire, s shown in Fig. 5. A uniform mgneti field B, pointing into the pge, fills the entire region. y h O z d d B I Figure 5 υ = onst. A iruit (t) of vrile size is formed y the retngulr loop (d). The field nd the surfe element re written, respetively, s B= Buˆz (where B= B = onst. ) nd d= ( d) uˆ z (note tht the diretion of trversing the loop is now ounterlokwise). The generl digrm of Fig. 4, representing the lne of fores, redues to the one shown in Fig. 6. Note tht this ltter digrm onerns only the moving prt () of the iruit, sine it is in this prt only tht the veloity υ nd the lied fore f re nonzero. (sine the lied fore is norml to the wire element everywhere; see Fig. 4). Thus we finlly hve: 5
. J. Pphristou, A. N. Mgouls f m B υ υ υ tot θ θ f f r Figure 6 υ υ (f. Fig. 6). Now, the role of the lied fore is to ounterlne the -omponent of the mgneti fore in order tht the r my move t onstnt speed in the diretion. Thus, nd f = f osθ = υ B osθ = Bυ m tot f υ = Bυυ = Bυ The emf of the iruit t time t is, ording to (8), E (t) ( ) = υ B ( t ) = υb = υb = υb h Alterntively, the mgneti flu through is m ( ) t B ( r ) Φ = d= B d = B d S ( t) S ( t) S ( t) = Bh (where is the momentry position of the r t time t), so tht d d E (t) = Φ m( t) = B h = Bhυ We note tht the emf is onstnt (timeindependent). Net, we wnt to use (11) to evlute the work per unit hrge for omplete tour of hrge round. Sine the lied fore is nonzero only on the setion () of, the pth of integrtion, (whih is stright line, given tht the hrge moves t onstnt veloity in spe) will orrespond to the motion of the hrge long the metl r only, i.e., from to. (Sine the r is eing displed in spe while the hrge is trveling long it, the line will not e prllel to the r.) Aording to (11), w= f with f = f = f = f υ (sine υ represents n elementry displement of the hrge long the metl r in time ). We finlly hve: w = B υ = B υ = B υ h We note tht, in this speifi emple, the vlue of the work per unit hrge is equl to tht of the emf, oth these quntities eing onstnt nd unmiguously defined. This would not hve een the se, however, if the mgneti field were nonuniform! 6. Sttionry wire inside time-vrying mgneti field Our finl emple onerns sttionry wire inside time-vrying mgneti field of the form B ( r, t) = B ( r, t) uˆ z (where B ( r, t) = B ( r, t) ), s shown in Fig. 7. y z r υ d Figure 7 B ( r, t) As is well known [1-7], the presene of time-vrying mgneti field implies the presene of n eletri field E s well, suh tht 6
Does the eletromotive fore (lwys) represent work? B E= t (1) As disussed in [1], the emf of the iruit t time t is given y where d E (t) = Ε ( r, t) = Φ m( t) (13) m ( t) B ( r, t) d B( r Φ = =, t) d S S (14) is the mgneti flu through t this time. On the other hnd, the work per unit hrge for full trip round is given y (5): w= f, where f = f ( ) em= E υ B, so tht w E = ( υ B ) As is esy to see (f. Fig. 7), the seond integrl vnishes, thus we re left with w= E (15) The similrity of the integrls in (13) nd (15) is deeptive! The integrl in (13) is evluted t fied time t, while in (15) time is llowed to flow s the hrge moves long. Is it, nevertheless, possile tht the vlues of these integrls oinide? As mentioned t the end of Se., neessry ondition for this to e the se is tht the two integrtions yield timeindependent results. In order tht E e timeindependent (ut nonzero), the mgneti flu (14) thus the mgneti field itself must inrese linerly with time. On the other hnd, the integrtion (15) for w will e time-independent if so is the eletri field. By (1), then, the mgneti field must e linerly dependent on time, whih rings us k to the previous ondition. As n emple, ssume tht the mgneti field is of the form B= B t uˆ ( B = onst.) 0 z 0 A possile solution of (1) for E is, in ylindril oordintes, E= B ρ 0 uˆ ϕ [We ssume tht these solutions re vlid in limited region of spe (e.g., in the interior of solenoid whose is oinides with the z-is) so tht ρ is finite in the region of interest.] Now, onsider irulr wire of rdius R, entered t the origin of the y-plne. Then, given tht = ( ) uˆ ϕ, E B R = = = Alterntively, 0 E B 0π R m Bd B S 0 π R t 0π Φ = =, so tht E= dφ / = B R. We ntiipte tht, m due to the time onstny of the eletri field, the sme result will e found for the work w y using (15). 7. onluding remrks No single, universlly epted definition of the emf seems to eist in the literture of Eletromgnetism. The definition given in this rtile (s well s in [1]) omes lose to those of [] nd [3]. In prtiulr, y using n emple similr to tht of Se. 5 in this pper, Griffiths [] mkes ler distintion etween the onepts of emf nd work per unit hrge. In [4] nd [5] (s well s in numerous other tetooks) the emf is identified with work per unit hrge, in generl, while in [6] nd [7] it is defined s losed line integrl of the non-onservtive prt of the eletri field tht ompnies timevrying mgneti flu. The lne of fores nd the origin of work in onduting iruit moving through mgneti field re niely disussed in [, 8, 9]. An interesting roh to the reltion etween work nd emf, utilizing the onept of virtul work, is desried in [10]. 7
. J. Pphristou, A. N. Mgouls Of ourse, the list of referenes ited ove is y no mens ehustive. It only serves to illustrte the diversity of ides onerning the onept of the emf. The sutleties inherent in this onept mke it n interesting sujet of study for oth the reserher nd the dvned student of lssil Eletrodynmis. Referenes [1]. J. Pphristou, A. N. Mgouls, Eletromotive fore: A guide for the perpleed (http://riv.org/s/111.6463). [] D. J. Griffiths, Introdution to Eletrodynmis, 3 rd Edition (Prentie-Hll, 1999). [3] W. N. ottinghm, D. A. Greenwood, Eletriity nd Mgnetism (mridge, 1991). [4] D. M. ook, The Theory of the Eletromgneti Field (Dover, 003). [5] R. K. Wngsness, Eletromgneti Fields, nd Edition (Wiley, 1986). [6] J. D. Jkson, lssil Eletrodynmis, nd Edition (Wiley, 1975). [7] W. K. H. Pnofsky, M. Phillips, lssil Eletriity nd Mgnetism, nd Edition (Addison-Wesley, 196). [8] E. P. Mos, Mgneti fores doing work?, Am. J. Phys. 4 (1974) 95. [9] J. A. Redinz, Fores nd work on wire in mgneti field, Am. J. Phys. 79 (011) 774. [10] R. A. Diz, W. J. Herrer, S. Gomez, The role of the virtul work in Frdy s Lw (http://riv.org/s/1104.1718). 8