Does the electromotive force (always) represent work?

Transcription

1 Does the electromotive force (lwys) represent work?. J. Ppchristou *, A. N. Mgouls ** * Deprtment of Physicl Sciences, Nvl Acdemy of Greece, Pireus 18539, Greece E-mil: ppchristou@snd.edu.gr ** Deprtment of Electricl Engineering, Nvl Acdemy of Greece, Pireus 18539, Greece E-mil: ris@snd.edu.gr Astrct. In the literture of Electromgnetism, the electromotive force of circuit is often defined s work done on unit chrge during complete tour of the ltter round the circuit. We explin why this sttement cnnot e generlly regrded s true, lthough it is indeed true in certin simple cses. Severl exmples re used to illustrte these points. 1. Introduction In recent pper [1] the uthors suggested pedgogicl roch to the electromotive force (emf ) of circuit, fundmentl concept of Electromgnetism. Rther thn defining the emf in n d hoc mnner for ech prticulr electrodynmic system, this roch egins with the most generl definition of the emf nd then specilizes to certin cses of physicl interest, thus recovering the fmilir expressions for the emf. Among the vrious exmples treted in [1], the cse of simple ttery-resistor circuit ws of prticulr interest since, in this cse, the emf ws shown to e equl to the work, per unit chrge, done y the source (ttery) for complete tour round the circuit. Now, in the literture of Electrodynmics the emf is often defined s work per unit chrge. As we show in this pper, this is not generlly true except for specil cses, such s the forementioned one. In Section 2, we give the generl definition of the emf, E, nd, seprtely, tht of the work per unit chrge, w, done y the gencies responsile for the genertion nd preservtion of current flow in the circuit. We then stte the necessry conditions in order for the equlity E=w to hold. We stress tht, y their very definitions, E nd w re different concepts. Thus, the eqution E=w suggests the possile equlity of the vlues of two physicl quntities, not the conceptul identifiction of these quntities! Section 3 reviews the cse of circuit consisting of ttery connected to resistive wire, in which cse the equlity E=w is indeed vlid. In Sec. 4, we study the prolem of wire moving through sttic mgnetic field. A prticulr sitution where the equlity E=w is vlid is treted in Sec. 5. Finlly, Sec. 6 exmines the cse of sttionry wire inside time-vrying mgnetic field. It is shown tht the equlity E=w is stisfied only in the specil cse where the mgnetic field vries linerly with time.

2 2. J. Ppchristou, A. N. Mgouls 2. The generl definitions of emf nd work per unit chrge onsider region of spce in which n electromgnetic (e/m) field exists. In the most generl sense, ny closed pth (or loop) within this region will e clled circuit (whether or not the whole or prts of consist of mteril ojects such s wires, resistors, cpcitors, tteries, etc.). We ritrrily ssign positive direction of trversing the loop, nd we consider n element of oriented in the positive direction. Imgine now test chrge q locted t the position of, nd let F e the force on q t time t : q F + Figure 1 This force is exerted y the e/m field itself, s well s, possily, y dditionl energy sources (e.g., tteries or some externl mechnicl ction) tht my contriute to the genertion nd preservtion of current flow round the loop. The force per unit chrge t the position of t time t, is F f = q Note tht f is independent of q, since the electromgnetic force on q is proportionl to the chrge. In prticulr, reversing the sign of q will hve no effect on f (lthough it will chnge the direction of F ). In generl, neither the shpe nor the size of is required to remin fixed. Moreover, the loop my e in motion reltive to n externl inertil oserver. Thus, for loop of (possily) vrile shpe, size or position in spce, we will use the nottion (t) to indicte the stte of the curve t time t. We now define the electromotive force (emf ) of the circuit t time t s the line integrl of f long, tken in the positive sense of : E (t) ( t ) (1) = f ( r, t) (2) (where r is the position vector of reltive to the origin of our coordinte system). Note tht the sign of the emf is dependent upon our choice of the positive direction of circultion of : y chnging this convention, the sign of E is reversed. As mentioned ove, the force (per unit chrge) defined in (1) cn e ttriuted to two fctors: the interction of q with the e/m field itself nd the ction on q due to ny dditionl energy sources. Eventully, this ltter interction is electromgnetic in nture even when it origintes from some externl mechnicl ction. We write:

3 Does the electromotive force (lwys) represent work? 3 f = f + f em (3) where f em is the force due to the e/m field nd f is the lied force due to n dditionl energy source. We note tht the force (3) does not include ny resistive (dissiptive) forces tht oppose chrge flow long ; it only contins forces tht my contriute to the genertion nd preservtion of such flow in the circuit. Now, suppose we llow single chrge q to mke full trip round the circuit under the ction of the force (3). In doing so, the chrge descries curve in spce (not necessrily closed one!) reltive to n externl inertil oserver. Let d l e n element of representing n infinitesiml displcement of q in spce, in time dt. We define the work per unit chrge for this complete tour round the circuit y the integrl: w= f For sttionry circuit of fixed shpe, coincides with the closed curve nd (4) reduces to w= f ( fixed ) (5) It should e noted crefully tht the integrl (2) is evluted t fixed 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 when q strts its round trip on. Thus, there is certin miguity in the definition of work per unit chrge. On the other hnd, the miguity (so to spek) with respect to the emf is relted to the dependence of the ltter on time t. The question now is: cn the emf e equl in vlue to the work per unit chrge, despite the fct 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 constnt, independent of time (de/dt=0) while w must not depend on the initil time t 0 of the round trip of q. These requirements re necessry conditions in order tht the equlity E=w e meningful. In the following sections we illustrte these ides y mens of severl exmples. As will e seen, the stisfction of the ove-mentioned conditions is n exception rther thn rule! (4) 3. A resistive wire connected to ttery onsider circuit consisting of n idel ttery (i.e., one with no internl resistnce) connected to metl wire of totl resistnce R. As shown in [1] (see lso [2]), the emf of the circuit in the direction of the current is equl to the voltge V of the ttery. Moreover, the emf in this cse represents the work, per unit chrge, done y the source (ttery). Let us review the proof of these sttements:

4 4. J. Ppchristou, A. N. Mgouls R I i E _ + f + I i Figure 2 A (conventionlly positive) moving chrge q is suject to two forces round the circuit : n electrosttic force Fe = qe t every point of nd force F inside the ttery, the ltter force crrying q from the negtive pole to the positive pole through the source. According to (3), the totl force per unit chrge is f = f + f = E+ f e The emf in the direction of the current (i.e., counterclockwise), t ny time t, is E = = + = f E f f d l (6) where we hve used the fcts tht E = 0 for n electrosttic field nd tht the ction of the source on q is limited to the region etween the poles of the ttery. Now, in stedy-stte sitution (Ι = constnt) the chrge q moves t constnt speed long the circuit. This mens tht the totl force on q in the direction of the pth is zero. In the interior of the wire, the electrosttic force Fe = qe is counterlnced y the resistive force on q due to the collisions of the chrge with the positive ions of the metl (s mentioned previously, this ltter force does not contriute to the emf ). In the interior of the (idel) ttery, however, where there is no resistnce, the electrosttic force must e counterlnced y the opposing force exerted y the source. Thus, in the section of the circuit etween nd, f = fe = E. By (6), then, we hve: E = E = V V = V (7) where V nd V re the electrosttic potentils t nd, respectively. We note tht the emf is constnt in time, s expected in stedy-stte sitution. Next, we wnt to find the work per unit chrge for complete tour round the circuit. To this end, we llow single chrge q to mke full trip round nd we use expression (5) (since the wire is sttionry nd of fixed shpe). In lying this reltion, time is ssumed to flow s q moves long. Given tht the sitution is sttic (time-independent), however, time is not relly n issue since it doesn t mtter t wht moment the chrge will pss y ny given point of. Thus, the integrtion in (5) will yield the sme result (7) s the integrtion in (6), despite the fct tht, in the ltter

5 Does the electromotive force (lwys) represent work? 5 cse, time ws ssumed fixed! We conclude tht the equlity w=e is vlid in this cse: the emf does represent work per unit chrge. 4. Moving wire inside sttic mgnetic field onsider wire moving in the xy-plne. The shpe nd/or size of the wire need not remin fixed during its motion. A sttic mgnetic field B( r ) is present in the region of spce where the wire is moving. For simplicity, we ssume tht this field is norml to the plne of the wire nd directed into the pge: y υ c + r υ ( r ) d B ( r ) ( t) z x Figure 3 In Fig. 3, the z-xis is norml to the plne of the wire nd directed towrds the reder. We cll d n infinitesiml norml vector representing n element of the plne surfce ounded y the wire (this vector is directed into the plne, consistently with the chosen clockwise direction of trversing the loop ). If u ˆz is the unit vector on the z- xis, then d= ( d) uˆ z nd B= B( r ) uˆ z, where B( r ) = B ( r ). onsider n element of the wire, locted t point with position vector r reltive to the origin of our inertil frme of reference. ll υ ( r ) the velocity of this element reltive to our frme. Let q e (conventionlly positive) chrge pssing y the considered point t time t. This chrge executes composite motion, hving velocity υc long the wire nd cquiring n extr velocity υ ( r ) due to the motion of the wire itself. The totl velocity of q reltive to us is υtot = υc+ υ. f m B ( r ) υ υc θ θ f r υ tot f f υ m tot f υ f υ r c c Figure 4

6 6. J. Ppchristou, A. N. Mgouls The lnce of forces cting on q is shown in the digrm of Fig. 4. The mgnetic force on q is norml to the chrge s totl velocity nd equl to Fm = q ( υtot B). Hence, the mgnetic force per unit chrge is fm = υtot B. Its component long the wire (i.e., in the direction of ) is counterlnced y the resistive force f r, which opposes the motion of q long (this force, s mentioned previously, does not contriute to the emf ). However, the component of the mgnetic force norml to the wire will tend to mke the wire move ckwrds (in direction opposing the desired motion of the wire) unless it is counterlnced y some externl mechnicl ction (e.g., our hnd, which pulls the wire forwrd). Now, the chrge q tkes shre of this ction y mens of some force trnsferred to it y the structure of the wire. This force (which will e clled n lied force) must e norml to the wire (in order to counterlnce the norml component of the mgnetic force). We denote the lied force per unit chrge y f. Although this force origintes from n externl mechnicl ction, it is delivered to q through n electromgnetic interction with the crystl lttice of the wire (not to e confused with the resistive force, whose role is different!). According to (3), the totl force contriuting to the emf of the circuit is f = f + f. By (2), the emf t time t is m E (t) = f + f m ( t) ( t) The second integrl vnishes since the lied force is norml to the wire element t every point of. The integrl of the mgnetic force is equl to ( υ B) = ( υ B) + ( υ B) tot c The first integrl on the right vnishes, s cn e seen y inspecting Fig. 4. Thus, we finlly hve: E (t) As shown nlyticlly in [1], the emf of is equl to [ ( ) ( r B r )] = υ (8) ( t) d E (t) = Φ m( t) (9) dt where we hve introduced the mgnetic flux through, m ( t) B ( r ) d Φ = = B( r ) d S ( t) S( t) (10) [By S(t) we denote ny open surfce ounded y t time t ; e.g., the plne surfce enclosed y the wire.] Now, let e the pth of q in spce reltive to the externl oserver, for full trip of q round the wire (if every prt of the wire is moving, will e n open curve). According to (4), the work done per unit chrge for this trip is

7 Does the electromotive force (lwys) represent work? 7 w f = + f m The first integrl vnishes (cf. Fig. 4), while for the second one we notice tht f = f + f = f (since the lied force is norml to the wire element everywhere; see Fig. 4). Thus we finlly hve: w= f f = f = f υ dt with where =υ dt is the infinitesiml displcement of the wire element in time dt. (11) 5. An exmple: Motion inside uniform mgnetic field onsider metl r () of length h, sliding prllel to itself with constnt speed υ on two prllel rils tht form prt of U-shped wire, s shown in Fig. 5: y h O z c d d B I x x + υ = const. Figure 5 A uniform mgnetic field B, pointing into the pge, fills the entire region. A circuit (t) of vrile size is formed y the rectngulr loop (cd). The field nd the surfce element re written, respectively, s B= Buˆz (where B= B = const. ) nd d= ( d) uˆ z (note tht the direction of trversing the loop is now counterclockwise). The generl digrm of Fig. 4, representing the lnce of forces, reduces to the one shown in Fig. 6, elow. Note tht this ltter digrm concerns only the moving prt () of the circuit, since it is in this prt only tht the velocity υ nd the lied force f re nonzero. The emf of the circuit t time t is, ccording to (8), E (t) ( ) = υ B = υb = υb = υb h ( t )

8 8. J. Ppchristou, A. N. Mgouls f m υ υc υ tot θ θ f υ υ c B f r x Figure 6 Alterntively, the mgnetic flux through is m ( ) t B ( r ) Φ = d= B d = B d = Bhx S ( t) S ( t) S ( t) (where x is the momentry position of the r t time t), so tht d d x E (t) = Φ m( t) = B h = Bhυ dt dt We note tht the emf is constnt (time-independent). Next, we wnt to use (11) to evlute the work per unit chrge for complete tour of chrge round. Since the lied force is nonzero only on the section () of, the pth of integrtion, (which is stright line, given tht the chrge moves t constnt velocity in spce) will correspond to the motion of the chrge long the metl r only, i.e., from to. (Since the r is eing displced in spce while the chrge is trveling long it, the line will not e prllel to the r!) According to (11), w= f f = f = f = f υ dt with (cf. Fig. 6). Now, the role of the lied force is to counterlnce the x-component of the mgnetic force in order tht the r my move t constnt speed in the x direction. Thus, f = fm cosθ = υtot B cosθ = Bυc nd fυ dt= Bυυc dt= Bυ (since υ c dt represents n elementry displcement of the chrge long the metl r in time dt). We finlly hve: w = B υ = B υ = B υ h We note tht, in this specific exmple, the vlue of the work per unit chrge is equl to tht of the emf, oth these quntities eing constnt nd unmiguously defined. This would not hve een the cse, however, if the mgnetic field were nonuniform!

9 Does the electromotive force (lwys) represent work? 9 6. Sttionry wire inside time-vrying mgnetic field Our finl exmple concerns sttionry wire inside time-vrying mgnetic 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 υ c + z r d B ( r, t) x Figure 7 As is well known [1-6], the presence of time-vrying mgnetic field implies the presence of n electric field E s well, such tht B E = t As discussed in [1], the emf of the circuit t time t is given y (12) d E (t) = Ε ( r, t) = Φ m( t) (13) dt where m ( t) B ( r, t) d B( r Φ = =, t) d S S (14) is the mgnetic flux through t this time. On the other hnd, the work per unit chrge for full trip round is given y (5): w= f, where f = fem= E+ ( υc B), so tht w E = + ( υ B ) As is esy to see (cf. Fig. 7), the second integrl vnishes, thus we re left with c w= E (15) The similrity of the integrls in (13) nd (15) is deceptive! The integrl in (13) is evluted t fixed time t, while in (15) time is llowed to flow s the chrge moves long. Is it, nevertheless, possile tht the vlues of these integrls coincide? As

10 10. J. Ppchristou, A. N. Mgouls mentioned t the end of Sec. 2, necessry condition for this to e the cse is tht the two integrtions yield time-independent results. In order tht E e time-independent (ut nonzero), the mgnetic flux (14) thus the mgnetic field itself must increse linerly with time. On the other hnd, the integrtion (15) for w will e timeindependent if so is the electric field. By (12), then, the mgnetic field must e linerly dependent on time, which rings us ck to the previous condition. As n exmple, ssume tht the mgnetic field is of the form B= B t uˆ ( B = const.) 0 z 0 A possile solution of (12) for E is, in cylindricl coordintes, E= B ρ 2 0 uˆ ϕ We ssume tht these solutions re vlid in limited region of spce (e.g., in the interior of solenoid whose xis coincides with the z-xis) so tht ρ is finite in the region of interest. Now, consider circulr wire of rdius R, centered t the origin of the xy-plne. Then, given tht = ( ) uˆ ϕ, E B R = = = E B 0π R Alterntively, m Bd B 2 S 0 π R t Φ = =, so tht E= dφ / dt = B π R. We expect tht, due to the time constncy of the electric field, the sme result will e found for the work w y using (15). m oncluding remrks No single, universlly ccepted definition of the emf seems to exist in the literture of Electromgnetism. The definition given in this rticle (s well s in [1]) comes close to those of [2] nd [3]. In prticulr, y using n exmple similr to tht of Sec. 5 in this pper, Griffiths [2] mkes cler distinction etween the concepts of emf nd work per unit chrge. In [4] (s well s in numerous other textooks) the emf is identified with work per unit chrge, in generl, while in [5] nd [6] it is defined s closed line integrl of the non-conservtive prt of the electric field, ccompnying timevrying mgnetic flux. The lnce of forces nd the origin of work in conducting circuit moving through mgnetic field re nicely discussed in [2, 7, 8]. Of course, the list of references cited ove is y no mens exhustive. It only serves to illustrte the diversity of ides concerning the concept of the emf. The sutleties inherent in this concept mke it n interesting suject for continuing reserch, for the dvnced student of clssicl Electrodynmics!

11 Does the electromotive force (lwys) represent work? 11 References 1.. J. Ppchristou, A. N. Mgouls, Electromotive force: A guide for the perplexed, rxiv: ( 2. D. J. Griffiths, Introduction to Electrodynmics, 3 rd Edition (Prentice-Hll, 1999). 3. W. N. ottinghm, D. A. Greenwood, Electricity nd Mgnetism (mridge, 1991). 4. R. K. Wngsness, Electromgnetic Fields, 2 nd Edition (Wiley, 1986). 5. J. D. Jckson, lssicl Electrodynmics, 2 nd Edition (Wiley, 1975). 6. W. K. H. Pnofsky, M. Phillips, lssicl Electricity nd Mgnetism, 2 nd Edition (Addison-Wesley, 1962). 7. E. P. Mosc, Mgnetic forces doing work?, Am. J. Phys. 42 (1974) J. A. Redinz, Forces nd work on wire in mgnetic field, Am. J. Phys. 79 (2011) 774.