Module 21: Faaday s Law of Induction 1 Table of Contents 10.1 Faaday s Law of Induction... 10-2 10.1.1 Magnetic Flux... 10-3 10.1.2 Lenz s Law... 10-5 1 10.2 Motional EMF... 10-7 10.3 Induced Electic Field... 10-10 10.4 Geneatos... 10-12 10.5 Eddy Cuents... 10-13 10.6 Summay... 10-15 10.7 Appendix: Induced Emf and Refeence Fames... 10-15 10.8 Poblem-Solving Tips: Faaday s Law and Lenz s Law... 10-16 10.9 Solved Poblems... 10-17 10.9.1 Rectangula Loop Nea a Wie... 10-17 10.9.2 Loop Changing Aea... 10-18 10.9.3 Sliding Rod... 10-19 10.9.4 Moving Ba... 10-21 10.9.5 Time-Vaying Magnetic Field... 10-22 10.9.6 Moving Loop... 10-23 10.10Conceptual Questions... 10-24 10.11Additional Poblems... 10-25 10.11.1 Sliding Ba... 10-25 10.11.2 Sliding Ba on Wedges... 10-26 10.11.3 RC Cicuit in a Magnetic Field... 10-26 10.11.4 Sliding Ba... 10-27 10.11.5 Rotating Ba... 10-27 10.11.6 Rectangula Loop Moving Though Magnetic Field... 10-28 10.11.7 Magnet Moving Though a Coil of Wie... 10-28 10.11.8 Altenating-Cuent Geneato... 10-29 10.11.9 EMF Due to a Time-Vaying Magnetic Field... 10-30 10.11.10 Squae Loop Moving Though Magnetic Field... 10-30 10.11.11 Falling Loop... 10-31 These notes ae excepted Intoduction to Electicity and Magnetism by Sen-Ben Liao, Pete Doumashkin, and John Belche, Copyight 2004, ISBN 0-536-81207-1. 10-1
Faaday s Law of Induction 10.1 Faaday s Law of Induction The electic fields and magnetic fields consideed up to now have been poduced by stationay chages and moving chages (cuents), espectively. Imposing an electic field on a conducto gives ise to a cuent which in tun geneates a magnetic field. One could then inquie whethe o not an electic field could be poduced by a magnetic field. In 1831, Michael Faaday discoveed that, by vaying magnetic field with time, an electic field could be geneated. The phenomenon is known as electomagnetic induction. Figue 10.1.1 illustates one of Faaday s expeiments. Figue 10.1.1 Electomagnetic induction Faaday showed that no cuent is egisteed in the galvanomete when ba magnet is stationay with espect to the loop. Howeve, a cuent is induced in the loop when a elative motion exists between the ba magnet and the loop. In paticula, the galvanomete deflects in one diection as the magnet appoaches the loop, and the opposite diection as it moves away. Faaday s expeiment demonstates that an electic cuent is induced in the loop by changing the magnetic field. The coil behaves as if it wee connected to an emf souce. Expeimentally it is found that the induced emf depends on the ate of change of magnetic flux though the coil. 10-2
10.1.1 Magnetic Flux Conside a unifom magnetic field passing though a suface S, as shown in Figue 10.1.2 below: Figue 10.1.2 Magnetic flux though a suface Let the aea vecto be A = Anˆ, whee A is the aea of the suface and nˆ its unit nomal. The magnetic flux though the suface is given by Φ = B A = BAcosθ (0.1.1) B whee θ is the angle between B and nˆ. If the field is non-unifom, Φ B then becomes Φ = d A (0.1.2) B B The SI unit of magnetic flux is the webe (Wb): S 1 Wb =1 T m 2 Faaday s law of induction may be stated as follows: The induced emf ε in a coil is popotional to the negative of the ate of change of magnetic flux: ε = dφ B (0.1.3) dt Fo a coil that consists of N loops, the total induced emf would be N times as lage: ε = N dφ B (0.1.4) dt Combining Eqs. (10.1.3) and (10.1.1), we obtain, fo a spatially unifom field B, 10-3
ε = d (BAcos θ ) = db Acos θ B da cos θ + BAsin θ dθ (0.1.5) dt dt dt dt Thus, we see that an emf may be induced in the following ways: (i) by vaying the magnitude of B with time (illustated in Figue 10.1.3.) Figue 10.1.3 Inducing emf by vaying the magnetic field stength (ii) by vaying the magnitude of (illustated in Figue 10.1.4.) A, i.e., the aea enclosed by the loop with time Figue 10.1.4 Inducing emf by changing the aea of the loop (iii) vaying the angle between B and the aea vecto A with time (illustated in Figue 10.1.5.) Figue 10.1.5 Inducing emf by vaying the angle between B and A. 10-4
10.1.2 Lenz s Law The diection of the induced cuent is detemined by Lenz s law: The induced cuent poduces magnetic fields which tend to oppose the change in magnetic flux that induces such cuents. To illustate how Lenz s law woks, let s conside a conducting loop placed in a magnetic field. We follow the pocedue below: 1. Define a positive diection fo the aea vecto A. 2. Assuming that B is unifom, take the dot poduct of B and A. This allows fo the detemination of the sign of the magnetic flux Φ B. 3. Obtain the ate of flux change dφ B / dt by diffeentiation. Thee ae thee possibilities: > 0 induced emf ε < 0 dφ B : < 0 induced emf ε > 0 dt = 0 induced emf ε = 0 4. Detemine the diection of the induced cuent using the ight-hand ule. With you thumb pointing in the diection of A, cul the finges aound the closed loop. The induced cuent flows in the same diection as the way you finges cul if ε > 0, and the opposite diection if ε < 0, as shown in Figue 10.1.6. Figue 10.1.6 Detemination of the diection of induced cuent by the ight-hand ule In Figue 10.1.7 we illustate the fou possible scenaios of time-vaying magnetic flux and show how Lenz s law is used to detemine the diection of the induced cuent I. 10-5
(a) (b) (c) Figue 10.1.7 Diection of the induced cuent using Lenz s law (d) The above situations can be summaized with the following sign convention: Φ B dφ / B dt ε I + + + + + + + The positive and negative signs of I coespond to a counteclockwise and clockwise cuents, espectively. As an example to illustate how Lenz s law may be applied, conside the situation whee a ba magnet is moving towad a conducting loop with its noth pole down, as shown in Figue 10.1.8(a). With the magnetic field pointing downwad and the aea vecto A pointing upwad, the magnetic flux is negative, i.e., Φ B = BA < 0, whee A is the aea of the loop. As the magnet moves close to the loop, the magnetic field at a point on the loop inceases ( db / dt > 0), poducing moe flux though the plane of the loop. Theefoe, dφ B / dt = A ( db / dt ) < 0, implying a positive induced emf, ε > 0, and the induced cuent flows in the counteclockwise diection. The cuent then sets up an induced magnetic field and poduces a positive flux to counteact the change. The situation descibed hee coesponds to that illustated in Figue 10.1.7(c). Altenatively, the diection of the induced cuent can also be detemined fom the point of view of magnetic foce. Lenz s law states that the induced emf must be in the diection that opposes the change. Theefoe, as the ba magnet appoaches the loop, it expeiences 10-6
a epulsive foce due to the induced emf. Since like poles epel, the loop must behave as if it wee a ba magnet with its noth pole pointing up. Using the ight-hand ule, the diection of the induced cuent is counteclockwise, as view fom above. Figue 10.1.8(b) illustates how this altenative appoach is used. Figue 10.1.8 (a) A ba magnet moving towad a cuent loop. (b) Detemination of the diection of induced cuent by consideing the magnetic foce between the ba magnet and the loop 10.2 Motional EMF Conside a conducting ba of length l moving though a unifom magnetic field which points into the page, as shown in Figue 10.2.1. Paticles with chage q > 0 inside expeience a magnetic foce F = qv B B which tends to push them upwad, leaving negative chages on the lowe end. Figue 10.2.1 A conducting ba moving though a unifom magnetic field The sepaation of chage gives ise to an electic field E inside the ba, which in tun poduces a downwad electic foce F e = qe. At equilibium whee the two foces cancel, 10-7
we have qvb = qe, o E = vb. Between the two ends of the conducto, thee exists a potential diffeence given by V ab = V a V b = ε = El =Blv (0.2.1) Sinceε aises fom the motion of the conducto, this potential diffeence is called the motional emf. In geneal, motional emf aound a closed conducting loop can be witten as whee d s is a diffeential length element. ε = Ñ ( v B ) d s (0.2.2) Now suppose the conducting ba moves though a egion of unifom magnetic field B = Bkˆ (pointing into the page) by sliding along two fictionless conducting ails that ae at a distance l apat and connected togethe by a esisto with esistance R, as shown in Figue 10.2.2. Figue 10.2.2 A conducting ba sliding along two conducting ails Let an extenal foce F ext be applied so that the conducto moves to the ight with a constant velocity v = v î. The magnetic flux though the closed loop fomed by the ba and the ails is given by Thus, accoding to Faaday s law, the induced emf is Φ B = BA= Blx (0.2.3) ε = dφ B = d (Blx) = Bl dx = Blv (0.2.4) dt dt dt whee dx / dt = v is simply the speed of the ba. The coesponding induced cuent is ε I = = Blv (0.2.5) R R 10-8
and its diection is counteclockwise, accoding to Lenz s law. The equivalent cicuit diagam is shown in Figue 10.2.3: Figue 10.2.3 Equivalent cicuit diagam fo the moving ba The magnetic foce expeienced by the ba as it moves to the ight is 2 2 ˆ ˆ ˆ B lv F ˆ B = Il ( j ) ( B k) = IlB i = i (0.2.6) R which is in the opposite diection of v. Fo the ba to move at a constant velocity, the net foce acting on it must be zeo. This means that the extenal agent must supply a foce F ext = F 2 2 =+ B B lv î (0.2.7) R The powe deliveed by F ext is equal to the powe dissipated in the esisto: as equied by enegy consevation. 2 2 2 2 Blv (Blv) ε 2 P = F ext v = Fext v = v = = = I R (0.2.8) R R R Fom the analysis above, in ode fo the ba to move at a constant speed, an extenal agent must constantly supply a foce F ext. What happens if at t = 0, the speed of the od is v 0, and the extenal agent stops pushing? In this case, the ba will slow down because of the magnetic foce diected to the left. Fom Newton s second law, we have o 2 2 B lv dv F B = =ma = m (0.2.9) R dt 2 2 dv B l dt = dt = (0.2.10) v mr τ 10-9
2 2 whee τ = mr / B l. Upon integation, we obtain vt () = ve 0 t /τ (0.2.11) Thus, we see that the speed deceases exponentially in the absence of an extenal agent doing wok. In pinciple, the ba neve stops moving. Howeve, one may veify that the total distance taveled is finite. 10.3 Induced Electic Field In Chapte 3, we have seen that the electic potential diffeence between two points A and B in an electic field E can be witten as ΔV B = V B V A = E d s A (0.3.1) When the electic field is consevative, as is the case of electostatics, the line integal of E d s is path-independent, which implies Ñ E d s = 0. Faaday s law shows that as magnetic flux changes with time, an induced cuent begins to flow. What causes the chages to move? It is the induced emf which is the wok done pe unit chage. Howeve, since magnetic field can do not wok, as we have shown in Chapte 8, the wok done on the mobile chages must be electic, and the electic field in this situation cannot be consevative because the line integal of a consevative field must vanish. Theefoe, we conclude that thee is a non-consevative electic field E nc associated with an induced emf: ε = Ñ E Combining with Faaday s law then yields Ñ E nc d s (0.3.2) dφ B nc ds = (0.3.3) dt The above expession implies that a changing magnetic flux will induce a nonconsevative electic field which can vay with time. It is impotant to distinguish between the induced, non-consevative electic field and the consevative electic field which aises fom electic chages. As an example, let s conside a unifom magnetic field which points into the page and is confined to a cicula egion with adius R, as shown in Figue 10.3.1. Suppose the magnitude of B inceases with time, i.e., db / dt > 0. Let s find the induced electic field eveywhee due to the changing magnetic field. 10-10
Since the magnetic field is confined to a cicula egion, fom symmety aguments we choose the integation path to be a cicle of adius. The magnitude of the induced field E nc at all points on a cicle is the same. Accoding to Lenz s law, the diection of E nc must be such that it would dive the induced cuent to poduce a magnetic field opposing the change in magnetic flux. With the aea vecto A pointing out of the page, the magnetic flux is negative o inwad. With db / dt > 0, the inwad magnetic flux is inceasing. Theefoe, to counteact this change the induced cuent must flow counteclockwise to poduce moe outwad flux. The diection of E nc is shown in Figue 10.3.1. Figue 10.3.1 Induced electic field due to changing magnetic flux Let s poceed to find the magnitude of E nc. In the egion < R, the ate of change of magnetic flux is Using Eq. (10.3.3), we have which implies dφ d d db B A dt dt dt dt Ñ 2 B = ( ) = ( BA) = π (0.3.4) dφ B db 2 E nc ds = E nc (2π ) = = π (0.3.5) dt dt db E nc = (0.3.6) 2 dt Similaly, fo > R, the induced electic field may be obtained as o E nc (2π ) = dφ B db = π R 2 dt dt (0.3.7) 10-11
A plot of E nc as a function of is shown in Figue 10.3.2. R 2 db E nc = (0.3.8) 2 dt Figue 10.3.2 Induced electic field as a function of 10.4 Geneatos One of the most impotant applications of Faaday s law of induction is to geneatos and motos. A geneato convets mechanical enegy into electic enegy, while a moto convets electical enegy into mechanical enegy. Figue 10.4.1 (a) A simple geneato. (b) The otating loop as seen fom above. Figue 10.4.1(a) is a simple illustation of a geneato. It consists of an N-tun loop otating in a magnetic field which is assumed to be unifom. The magnetic flux vaies with time, theeby inducing an emf. Fom Figue 10.4.1(b), we see that the magnetic flux though the loop may be witten as The ate of change of magnetic flux is Φ = B A= BAcos θ = BA cos ωt (0.4.1) B dφ B = BAω sinωt (0.4.2) dt 10-12
Since thee ae N tuns in the loop, the total induced emf acoss the two ends of the loop is ε = N dφ B = NBA ω sinωt (0.4.3) dt If we connect the geneato to a cicuit which has a esistance R, then the cuent geneated in the cicuit is given by I = ε = NBAω sinωt (0.4.4) R R The cuent is an altenating cuent which oscillates in sign and has an amplitude I 0 = NBAω / R. The powe deliveed to this cicuit is On the othe hand, the toque exeted on the loop is Thus, the mechanical powe supplied to otate the loop is Since the dipole moment fo the N-tun cuent loop is = (NBA ω)2 2 P= I ε sin ωt (0.4.5) R τ = μb sinθ = μb sin ωt (0.4.6) P =τω = μb ω sinωt (0.4.7) m the above expession becomes 2 2 N A B ω μ = NIA = sinωt (0.4.8) R 2 2 2 P m = N A B ω sinωt Bω sin ωt = (NAB ω) sin 2 ωt (0.4.9) R R As expected, the mechanical powe put in is equal to the electical powe output. 10.5 Eddy Cuents We have seen that when a conducting loop moves though a magnetic field, cuent is induced as the esult of changing magnetic flux. If a solid conducto wee used instead of a loop, as shown in Figue 10.5.1, cuent can also be induced. The induced cuent appeas to be ciculating and is called an eddy cuent. 10-13
Figue 10.5.1 Appeaance of an eddy cuent when a solid conducto moves though a magnetic field. The induced eddy cuents also geneate a magnetic foce that opposes the motion, making it moe difficult to move the conducto acoss the magnetic field (Figue 10.5.2). Figue 10.5.2 Magnetic foce aising fom the eddy cuent that opposes the motion of the conducting slab. Since the conducto has non-vanishing esistance R, Joule heating causes a loss of powe by an amount P = ε 2 / R. Theefoe, by inceasing the value of R, powe loss can be educed. One way to incease R is to laminate the conducting slab, o constuct the slab by using gluing togethe thin stips that ae insulated fom one anothe (see Figue 10.5.3a). Anothe way is to make cuts in the slab, theeby disupting the conducting path (Figue 10.5.3b). Figue 10.5.3 Eddy cuents can be educed by (a) laminating the slab, o (b) making cuts on the slab. Thee ae impotant applications of eddy cuents. Fo example, the cuents can be used to suppess unwanted mechanical oscillations. Anothe application is the magnetic baking systems in high-speed tansit cas. 10-14
10.6 Summay The magnetic flux though a suface S is given by Φ = d A B B S Faaday s law of induction states that the induced emf ε in a coil is popotional to the negative of the ate of change of magnetic flux: dφb ε = N dt The diection of the induced cuent is detemined by Lenz s law which states that the induced cuent poduces magnetic fields which tend to oppose the changes in magnetic flux that induces such cuents. A motional emf ε is induced if a conducto moves in a magnetic field. The geneal expession foε is ε = Ñ (v B ) d s In the case of a conducting ba of length l moving with constant velocity v though a magnetic field which points in the diection pependicula to the ba and v, the induced emf is ε = Bvl. An induced emf in a stationay conducto is associated with a non-consevative electic field E : nc ε = E d s = nc dφ B dt 10.7 Appendix: Induced Emf and Refeence Fames In Section 10.2, we have stated that the geneal equation of motional emf is given by ε = Ñ ( v B ) d s whee v is the velocity of the length element d s of the moving conducto. In addition, we have also shown in Section 10.4 that induced emf associated with a stationay conducto may be witten as the line integal of the non-consevative electic field: 10-15
ε = Ñ E Howeve, whethe an object is moving o stationay actually depends on the efeence fame. As an example, let s examine the situation whee a ba magnet is appoaching a conducting loop. An obseve O in the est fame of the loop sees the ba magnet moving towad the loop. An electic field E nc is induced to dive the cuent aound the loop, and a chage on the loop expeiences an electic foce F e = qe nc. Since the chage is at est accoding to obseve O, no magnetic foce is pesent. On the othe hand, an obseve O ' in the est fame of the ba magnet sees the loop moving towad the magnet. Since the conducting loop is moving with a velocity v, a motional emf is induced. In this fame, O ' sees the chage q moving with a velocity v, and concludes that the chage expeiences a magnetic foce F = qv B B. nc d s Figue 10.7.1 Induction obseved in diffeent efeence fames. In (a) the ba magnet is moving, while in (b) the conducting loop is moving. Since the event seen by the two obseve is the same except the choice of efeence fames, the foce acting on the chage must be the same, F e = F B, which implies E nc = v B (0.7.1) In geneal, as a consequence of elativity, an electic phenomenon obseved in a efeence fame O may appea to be a magnetic phenomenon in a fame O ' that moves at a speed v elative too. 10.8 Poblem-Solving Tips: Faaday s Law and Lenz s Law In this chapte we have seen that a changing magnetic flux induces an emf: ε = N dφ B dt 10-16
accoding to Faaday s law of induction. Fo a conducto which foms a closed loop, the emf sets up an induced cuent I = ε / R, whee R is the esistance of the loop. To compute the induced cuent and its diection, we follow the pocedue below: 1. Fo the closed loop of aea A on a plane, define an aea vecto A and let it point in the diection of you thumb, fo the convenience of applying the ight-hand ule late. Compute the magnetic flux though the loop using Detemine the sign of Φ B. B A (B is unifom) Φ B = B d A (B is non-unifom) 2. Evaluate the ate of change of magnetic flux dφ B / dt. Keep in mind that the change could be caused by (i) changing the magnetic field db / dt 0, (ii) changing the loop aea if the conducto is moving ( da / dt 0 ), o (iii) changing the oientation of the loop with espect to the magnetic field ( dθ / dt Detemine the sign of dφ B / dt. 0 ). 3. The sign of the induced emf is the opposite of that of dφ B / dt. The diection of the induced cuent can be found by using Lenz s law discussed in Section 10.1.2. 10.9 Solved Poblems 10.9.1 Rectangula Loop Nea a Wie An infinite staight wie caies a cuent I is placed to the left of a ectangula loop of wie with width w and length l, as shown in the Figue 10.9.1. Figue 10.9.1 Rectangula loop nea a wie 10-17
(a) Detemine the magnetic flux though the ectangula loop due to the cuent I. (b) Suppose that the cuent is a function of time with I ( t ) = a + bt, whee a and b ae positive constants. What is the induced emf in the loop and the diection of the induced cuent? Solutions: (a) Using Ampee s law: B d s = μ I (0.9.1) Ñ 0 enc the magnetic field due to a cuent-caying wie at a distance away is B = μ 0 I (0.9.2) 2π The total magnetic flux Φ B though the loop can be obtained by summing ove contibutions fom all diffeential aea elements da =l d: + B d B d = μ Il 0 s w d μ = 0 Il s + w ln Φ = Φ = B A 2π s 2π s (0.9.3) Note that we have chosen the aea vecto to point into the page, so that Φ >0. B (b) Accoding to Faaday s law, the induced emf is ε = dφ B = d μ Il 0 ln s + w = μ l 0 ln s + w di dt dt 2π s 2π s dt = μ bl 0 ln s + w (0.9.4) 2π s whee we have used di / dt = b. The staight wie caying a cuent I poduces a magnetic flux into the page though the ectangula loop. By Lenz s law, the induced cuent in the loop must be flowing counteclockwise in ode to poduce a magnetic field out of the page to counteact the incease in inwad flux. 10.9.2 Loop Changing Aea A squae loop with length l on each side is placed in a unifom magnetic field pointing into the page. Duing a time inteval Δt, the loop is pulled fom its two edges and tuned 10-18
into a hombus, as shown in the Figue 10.9.2. Assuming that the total esistance of the loop is R, find the aveage induced cuent in the loop and its diection. Solution: Using Faaday s law, we have Figue 10.9.2 Conducting loop changing aea ε = ΔΦ B = B ΔA (0.9.5) Δt Δt Since the initial and the final aeas of the loop ae A l 2 and A = l 2 i = f sin θ, espectively (ecall that the aea of a paallelogam defined by two vectos l 1 and l 2 is A= l 1 l 2 = ll sin θ ), the aveage ate of change of aea is 1 2 which gives Thus, the aveage induced cuent is ΔA A = f A i l 2 (1 sin θ ) = < 0 (0.9.6) Δt Δt Δt ε = Bl 2 (1 sin θ ) > 0 (0.9.7) Δt I = ε = Bl 2 (1 sin θ ) R ΔtR (0.9.8) Since (ΔA/ Δt) < 0, the magnetic flux into the page deceases. Hence, the cuent flows in the clockwise diection to compensate the loss of flux. 10.9.3 Sliding Rod A conducting od of length l is fee to slide on two paallel conducting bas as in Figue 10.9.3. 10-19
Figue 10.9.3 Sliding od In addition, two esistos R 1 and R 2 ae connected acoss the ends of the bas. Thee is a unifom magnetic field pointing into the page. Suppose an extenal agent pulls the ba to the left at a constant speed v. Evaluate the following quantities: (a) The cuents though both esistos; (b) The total powe deliveed to the esistos; (c) The applied foce needed fo the od to maintain a constant velocity. Solutions: (a) The emf induced between the ends of the moving od is The cuents though the esistos ae ε = dφ B = Blv (0.9.9) dt ε I 1 = ε, I 2 = (0.9.10) R 1 R 2 Since the flux into the page fo the left loop is deceasing, I 1 flows clockwise to poduce a magnetic field pointing into the page. On the othe hand, the flux into the page fo the ight loop is inceasing. To compensate the change, accoding to Lenz s law, I 2 must flow counteclockwise to poduce a magnetic field pointing out of the page. (b) The total powe dissipated in the two esistos is P R = I ε +I (I + I ) = ε 2 1 1 2 2 2 1 1 1 2 ε = 1 2 ε R + = B l v 1 R 2 R + 1 R 2 (0.9.11) (c) The total cuent flowing though the od is I = I 1 +I 2. Thus, the magnetic foce acting on the od is 1 1 2 2 1 1 F B = IlB = ε lb + = B l v + (0.9.12) R 1 R 2 R 1 R 2 10-20
and the diection is to the ight. Thus, an extenal agent must apply an equal but opposite foce F ext = F B to the left in ode to maintain a constant speed. Altenatively, we note that since the powe dissipated in the esistos must be equal to P ext, the mechanical powe supplied by the extenal agent. The same esult is obtained since P ext = F v ext = Fext v (0.9.13) 10.9.4 Moving Ba A conducting od of length l moves with a constant velocity v pependicula to an infinitely long, staight wie caying a cuent I, as shown in the Figue 10.9.4. What is the emf geneated between the ends of the od? Solution: Figue 10.9.4 A ba moving away fom a cuent-caying wie Fom Faaday s law, the motional emf is ε = Blv (0.9.14) whee v is the speed of the od. Howeve, the magnetic field due to the staight cuentcaying wie at a distance away is, using Ampee s law: Thus, the emf between the ends of the od is given by B = μ 0 I (0.9.15) 2π μ ε = I 0 2 π lv (0.9.16) 10-21
10.9.5 Time-Vaying Magnetic Field A cicula loop of wie of adius a is placed in a unifom magnetic field, with the plane of the loop pependicula to the diection of the field, as shown in Figue 10.9.5. Figue 10.9.5 Cicula loop in a time-vaying magnetic field The magnetic field vaies with time accoding to B ( ) positive constants. (a) Calculate the magnetic flux though the loop at t = 0. (b) Calculate the induced emf in the loop. t = B + bt, whee B 0 and b ae (c) What is the induced cuent and its diection of flow if the oveall esistance of the loop is R? (d) Find the powe dissipated due to the esistance of the loop. 0 Solution: (a) The magnetic flux at time t is given by Φ = BA = B + bt π a 2 = π B 0 + bt a 2 B ( 0 )( ) ( ) (0.9.17) whee we have chosen the aea vecto to point into the page, so that Φ >0. At t = 0, B we have (b) Using Faaday s Law, the induced emf is Φ =π B a 2 (0.9.18) B 0 d B + bt dφ B db 2 ( 0 ) 2 ε = = A = (π a ) = πba (0.9.19) dt dt dt 10-22
(c) The induced cuent is ε I = = πba2 (0.9.20) R R and its diection is counteclockwise by Lenz s law. (d) The powe dissipated due to the esistance R is 10.9.6 Moving Loop P= I 2 R = 2 πba2 R = (πba2 ) 2 (0.9.21) R R A ectangula loop of dimensions l and w moves with a constant velocity v away fom an infinitely long staight wie caying a cuent I in the plane of the loop, as shown in Figue 10.9.6. Let the total esistance of the loop be R. What is the cuent in the loop at the instant the nea side is a distance fom the wie? Solution: Figue 10.9.6 A ectangula loop moving away fom a cuent-caying wie The magnetic field at a distance s fom the staight wie is, using Ampee s law: μ I B = 0 (0.9.22) 2π s The magnetic flux though a diffeential aea element da = lds of the loop is μ0 I dφ B = B d A = l ds (0.9.23) 2π s whee we have chosen the aea vecto to point into the page, so that Φ ove the entie aea of the loop, the total flux is B >0. Integating 10-23
Φ = μ 0 Il + wds 2π s = μ 0 Il + w B ln 2π (0.9.24) Diffeentiating with espect to t, we obtain the induced emf as ε = dφ B = μ Il d 0 ln + w = μ Il 0 1 1 d = μ Il wv 0 (0.9.25) dt 2 π dt 2 π + w dt 2 π ( + w) whee v= d / dt. Notice that the induced emf can also be obtained by using Eq. (10.2.2): The induced cuent is ε = ( ) d μ s = [ ( ) ( + w ] = vl I 0 μ 0 I v B vl B B ) 2 π 2π ( + w) (0.9.26) μ 0 Il vw = 2π ( + w) ε μ 0 Il vw I = = R 2π R ( + w) (0.9.27) 10.10 Conceptual Questions 1. A ba magnet falls though a cicula loop, as shown in Figue 10.10.1 Figue 10.10.1 (a) Descibe qualitatively the change in magnetic flux though the loop when the ba magnet is above and below the loop. (b) Make a qualitative sketch of the gaph of the induced cuent in the loop as a function of time, choosing I to be positive when its diection is counteclockwise as viewed fom above. 2. Two cicula loops A and B have thei planes paallel to each othe, as shown in Figue 10.10.2. 10-24
Figue 10.10.2 Loop A has a cuent moving in the counteclockwise diection, viewed fom above. (a) If the cuent in loop A deceases with time, what is the diection of the induced cuent in loop B? Will the two loops attact o epel each othe? (b) If the cuent in loop A inceases with time, what is the diection of the induced cuent in loop B? Will the two loops attact o epel each othe? 3. A spheical conducting shell is placed in a time-vaying magnetic field. Is thee an induced cuent along the equato? 4. A ectangula loop moves acoss a unifom magnetic field but the induced cuent is zeo. How is this possible? 10.11 Additional Poblems 10.11.1 Sliding Ba A conducting ba of mass m and esistance R slides on two fictionless paallel ails that ae sepaated by a distance l and connected by a battey which maintains a constant emf ε, as shown in Figue 10.11.1. Figue 10.11.1 Sliding ba A unifom magnetic field B is diected out of the page. The ba is initially at est. Show that at a late time t, the speed of the ba is 2 2 whee τ = mr / B l. v = ε (1 e t /τ ) Bl 10-25
10.11.2 Sliding Ba on Wedges A conducting ba of mass m and esistance R slides down two fictionless conducting ails which make an angle θ with the hoizontal, and ae sepaated by a distance l, as shown in Figue 10.11.2. In addition, a unifom magnetic field B is applied vetically downwad. The ba is eleased fom est and slides down. Figue 10.11.2 Sliding ba on wedges (a) Find the induced cuent in the ba. Which way does the cuent flow, fom a to b o b to a? (b) Find the teminal speed v t of the ba. Afte the teminal speed has been eached, (c) What is the induced cuent in the ba? (d) What is the ate at which electical enegy has been dissipated though the esisto? (e) What is the ate of wok done by gavity on the ba? 10.11.3 RC Cicuit in a Magnetic Field Conside a cicula loop of wie of adius lying in the xy plane, as shown in Figue 10.11.3. The loop contains a esisto R and a capacito C, and is placed in a unifom magnetic field which points into the page and deceases at a ate db / dt = α, with α > 0. Figue 10.11.3 RC cicuit in a magnetic field 10-26
(a) Find the maximum amount of chage on the capacito. (b) Which plate, a o b, has a highe potential? What causes chages to sepaate? 10.11.4 Sliding Ba A conducting ba of mass m and esistance R is pulled in the hoizontal diection acoss two fictionless paallel ails a distance l apat by a massless sting which passes ove a fictionless pulley and is connected to a block of mass M, as shown in Figue 10.11.4. A unifom magnetic field is applied vetically upwad. The ba is eleased fom est. Figue 10.11.4 Sliding ba (a) Let the speed of the ba at some instant be v. Find an expession fo the induced cuent. Which diection does it flow, fom a to b o b to a? You may ignoe the fiction between the ba and the ails. (b) Solve the diffeential equation and find the speed of the ba as a function of time. 10.11.5 Rotating Ba A conducting ba of length l with one end fixed otates at a constant angula speed ω, in a plane pependicula to a unifom magnetic field, as shown in Figue 10.11.5. Figue 10.11.5 Rotating ba (a) A small element caying chage q is located at a distance away fom the pivot point O. Show that the magnetic foce on the element is F B = qbω. 10-27
1 (b) Show that the potential diffeence between the two ends of the ba is Δ V = B ω l 2. 2 10.11.6 Rectangula Loop Moving Though Magnetic Field A small ectangula loop of length l = 10 cm and width w = 8.0 cm with esistance R = 2.0 Ω is pulled at a constant speed v = 2.0 cm/s though a egion of unifom magnetic field B = 2.0 T, pointing into the page, as shown in Figue 10.11.6. Figue 10.11.6 At t = 0, the font of the ectangula loop entes the egion of the magnetic field. (a) Find the magnetic flux and plot it as a function of time (fom t = 0 till the loop leaves the egion of magnetic field.) (b) Find the emf and plot it as a function of time. (c) Which way does the induced cuent flow? 10.11.7 Magnet Moving Though a Coil of Wie Suppose a ba magnet is pulled though a stationay conducting loop of wie at constant speed, as shown in Figue 10.11.7. Figue 10.11.7 Assume that the noth pole of the magnet entes the loop of wie fist, and that the cente of the magnet is at the cente of the loop at time t = 0. (a) Sketch qualitatively a gaph of the magnetic flux Φ B though the loop as a function of time. 10-28
(b) Sketch qualitatively a gaph of the cuent I in the loop as a function of time. Take the diection of positive cuent to be clockwise in the loop as viewed fom the left. (c) What is the diection of the foce on the pemanent magnet due to the cuent in the coil of wie just befoe the magnet entes the loop? (d) What is the diection of the foce on the magnet just afte it has exited the loop? (e) Do you answes in (c) and (d) agee with Lenz's law? (f) Whee does the enegy come fom that is dissipated in ohmic heating in the wie? 10.11.8 Altenating-Cuent Geneato An N-tun ectangula loop of length a and width b is otated at a fequency f in a unifom magnetic field B which points into the page, as shown in Figue 10.11.8 At time t = 0, the loop is vetical as shown in the sketch, and it otates counteclockwise when viewed along the axis of otation fom the left. Figue 10.11.8 (a) Make a sketch depicting this geneato as viewed fom the left along the axis of otation at a time Δt shotly afte t = 0, when it has otated an angle θ fom the vetical. Show clealy the vecto B, the plane of the loop, and the diection of the induced cuent. (b) Wite an expession fo the magnetic flux Φ B passing though the loop as a function of time fo the given paametes. (c) Show that an induced emf ε appeas in the loop, given by ε = 2π fnbab sin(2 π ft ) = ε 0 sin(2 π ft) (d) Design a loop that will poduce an emf with ε 0 = 120 V when otated at 60 evolutions/sec in a magnetic field of 0.40 T. 10-29
10.11.9 EMF Due to a Time-Vaying Magnetic Field A unifom magnetic field B is pependicula to a one-tun cicula loop of wie of negligible esistance, as shown in Figue 10.11.9. The field changes with time as shown (the z diection is out of the page). The loop is of adius = 50 cm and is connected in seies with a esisto of esistance R = 20 Ω. The "+" diection aound the cicuit is indicated in the figue. Figue 10.11.9 (a) What is the expession fo EMF in this cicuit in tems of B z ( t ) fo this aangement? (b) Plot the EMF in the cicuit as a function of time. Label the axes quantitatively (numbes and units). Watch the signs. Note that we have labeled the positive diection of the emf in the left sketch consistent with the assumption that positive B is out of the pape. [Patial Ans: values of EMF ae 1.96 V, 0.0 V, 0.98 V]. (c) Plot the cuent I though the esisto R. Label the axes quantitatively (numbes and units). Indicate with aows on the sketch the diection of the cuent though R duing each time inteval. [Patial Ans: values of cuent ae 98 ma, 0.0 ma, 49 ma] (d) Plot the ate of themal enegy poduction in the esisto. [Patial Ans: values ae 192 mw, 0.0 mw, 48 mw]. 10.11.10 Squae Loop Moving Though Magnetic Field An extenal foce is applied to move a squae loop of dimension l l and esistance R at a constant speed acoss a egion of unifom magnetic field. The sides of the squae loop make an angle θ = 45 with the bounday of the field egion, as shown in Figue 10.11.10. At t = 0, the loop is completely inside the field egion, with its ight edge at the bounday. Calculate the powe deliveed by the extenal foce as a function of time. 10-30
Figue 10.11.10 10.11.11 Falling Loop A ectangula loop of wie with mass m, width w, vetical length l, and esistance R falls out of a magnetic field unde the influence of gavity, as shown in Figue 10.11.11. The magnetic field is unifom and out of the pape ( B = B î ) within the aea shown and zeo outside of that aea. At the time shown in the sketch, the loop is exiting the magnetic field at speed v = vkˆ. Figue 10.11.11 (a) What is the diection of the cuent flowing in the cicuit at the time shown, clockwise o counteclockwise? Why did you pick this diection? (b) Using Faaday's law, find an expession fo the magnitude of the emf in this cicuit in tems of the quantities given. What is the magnitude of the cuent flowing in the cicuit at the time shown? (c) Besides gavity, what othe foce acts on the loop in the ±kˆ diection? Give its magnitude and diection in tems of the quantities given. (d) Assume that the loop has eached a teminal velocity and is no longe acceleating. What is the magnitude of that teminal velocity in tems of given quantities? (e) Show that at teminal velocity, the ate at which gavity is doing wok on the loop is equal to the ate at which enegy is being dissipated in the loop though Joule heating. 10-31
MIT OpenCouseWae http://ocw.mit.edu 8.02SC Physics II: Electicity and Magnetism Fall 2010 Fo infomation about citing these mateials o ou Tems of Use, visit: http://ocw.mit.edu/tems.