MAGNETIC INDUCTION. MISN MAGNETIC INDUCTION by J. S. Kovacs and P. Signell Michigan State University

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1 MAGNETIC INDUCTION V V in MISN MAGNETIC INDUCTION by J. S. Kovacs and P. Signe Michigan State University 1. Introduction a. Overview b. Magnetic Fu c. The Fu Integrand d. The Direction of Positive Fu e. Induced Votage f. Induced Votage and Fu Change Rate g. The Power Transferred Eampe of Lorentz-Force Induction a. A Loop Enters A Magnetic Fied b. The Loop Traves Through the Magnetic Fied c. The Loop Leaves the Magnetic Fied d. The Time Rate of Change of the Fu Induction by Time-Changing B a. The Faraday-Henry Law b. Eampe: Lineary Increasing B Lenz s Law Sef-Inductance, Inductors Acknowedgments Gossary Project PHYSNET Physics Bdg. Michigan State University East Lansing, MI 1

2 ID Sheet: MISN Tite: Magnetic Induction Author: J. Kovacs and P. Signe, Michigan State University Version: 2/1/2000 Evauation: Stage 0 Length: 1 hr; 24 pages Input Skis: 1. Vocabuary: energy-power-time reation (MISN-0-20), eectric fied (MISN-0-115), battery(misn-0-117), power (eectrica) (MISN-0-118), votage, resistor (MISN-0-119), magnetic fied, Lorentz force (MISN-0-122), surface integra (MISN-0-132), ine integra, Ampere s aw (MISN-0-138). Output Skis (Knowedge): K1. Vocabuary: Faraday-Henry aw, fu, induced current, induced potentia difference, induced votage, Lenz s aw. K2. Write down the Faraday-Henry Law, reating the ine-integra of the induced eectric fied to the time rate of change of the surface integra of the magnetic fied across a surface bounded by the path of the ine-integra, taking care to epain the sign convention. Epain in words what the physica phenomenon is that is described by this equation. Output Skis (Probem Soving): S1. Determine the votage induced around given cosed paths when a changing magnetic fied in the region is specified. S2. Determine the votage induced around given cosed paths when the veocity of the path through a uniform static magnetic fied is specified. S3. Use Lenz s aw to determine the directions of induced votages, currents, and magnetic fieds. THIS IS A DEVELOPMENTAL-STAGE PUBLICATION OF PROJECT PHYSNET The goa of our project is to assist a network of educators and scientists in transferring physics from one person to another. We support manuscript processing and distribution, aong with communication and information systems. We aso work with empoyers to identify basic scientific skis as we as physics topics that are needed in science and technoogy. A number of our pubications are aimed at assisting users in acquiring such skis. Our pubications are designed: (i) to be updated quicky in response to fied tests and new scientific deveopments; (ii) to be used in both cassroom and professiona settings; (iii) to show the prerequisite dependencies eisting among the various chunks of physics knowedge and ski, as a guide both to menta organization and to use of the materias; and (iv) to be adapted quicky to specific user needs ranging from singe-ski instruction to compete custom tetbooks. New authors, reviewers and fied testers are wecome. PROJECT STAFF Andrew Schnepp Eugene Kaes Peter Signe Webmaster Graphics Project Director ADVISORY COMMITTEE D. Aan Bromey Yae University E. Leonard Jossem The Ohio State University A. A. Strassenburg S. U. N. Y., Stony Brook Views epressed in a modue are those of the modue author(s) and are not necessariy those of other project participants. c 2001, Peter Signe for Project PHYSNET, Physics-Astronomy Bdg., Mich. State Univ., E. Lansing, MI 48824; (517) For our ibera use poicies see: 3 4

3 MISN MAGNETIC INDUCTION by J. S. Kovacs and P. Signe Michigan State University 1. Introduction 1a. Overview. When magnetic fieds and/or eectric fieds change with time, effects occur which are profoundy different from what woud be epected from just the static properties of these fieds. When eectric fieds change with time, magnetic fieds are produced just as woud be produced by currents. When magnetic fieds change with time, eectric fieds are produced just as woud be produced by charges. The fieds soproduced are caed induced fieds and they simpy add to the fieds produced by charges and currents. In this modue we study: (1) moving circuits in stationary magnetic fieds, so the Lorentz Force provides the induction; and (2) stationary circuits in time-changing magnetic fieds so the induction is governed by the so-caed Faraday-Henry aw. 1b. Magnetic Fu. Throughout the study of magnetic induction we make use of the concept of the magnetic fu through a surface S bounded by a directed ine L, written Φ SL and defined by: Φ SL B ds. (1) SL The reationships between these quantities are shown in Fig. 1. The word fu, stricty speaking, merey refers to vaue of the integra on the right side of Eq. (1). However, the word fu originated with peope who iked to think of it as a sort of measure of the number of (ficticious) magnetic fied ines that go through the surface of integration. ` B ds` Figure 1. The ange θ is defined as the ange between the magnetic fied B and the vector d S whose direction is norma to the surface eement of area ds. MISN ` B ds` ` B ds` Figure 2. The case where ds is determined by L: (a) for one choice of the direction L; and (b) for the other choice. 1c. The Fu Integrand. To construct the integrand in Eq. (1), one needs to take the scaar product of the two vectors B and d S and for that their directions and magnitudes must be known. The vector B is normay known in any particuar appication but we must determine the direction of the infinitesima vector surface eement d S by eamining that eement and its reationship to the directed ine L. After a, there are two sides to the surface and each side has a unit vector norma to the surface at that point. Note that the unit vector ˆn, norma to the surface, enters Eq. (1) in the form: d S = ˆn ds. The two norma unit vectors on opposite sides of the surface point in eacty opposite directions. The scaar product in Eq.(1) wi be positive with one choice for positive d S, negative with the other choice. However, the sign of the integrand must be determined correcty because the sign has important physica consequences. 1d. The Direction of Positive Fu. The positive fu direction for a surface S is tied to the positive direction around the ine L that borders the surface. Once you choose one of those two directions, the other is thereby fied. You may choose one of them in any manner that appeas to you, but then you must define the other in conformity with the choice you just made. The directions of L and S are reated the same way that the magnetic fied surrounding a wire is reated to the direction of the current in the wire, or in the same way that the direction of the current in a oop of wire is reated to the magnetic fied at the center of the oop, as iustrated in Fig. 2. There is a oop of area 4.0 m 2 in the -z pane: take the surface the oop encoses as a fat disc in the pane. The magnetic fied is, at some time, say, B = 12 T ŷ at a space points encosed by the oop. Show that the magnetic fu through the oop is just the area times the fied: Φ SL = 48 T m 2. If the fied was tited so it was at an ange θ to the norma to the surface, then Eq. (1) shows that the fu through the surface woud be: Φ SL = 5 6

4 MISN MISN V c d (a) R (b) v Figure 3. The votage drop around a compete path: (a) zero for a noninduced votage; (b) non-zero for an induced votage. v 48 cos θ T m 2. Fu is sometimes given the unit weber, abbreviated W: W T m 2. 1e. Induced Votage. When magnetic effects induce an eectric fied in a circuit, that eectric fied can be integrated aong the circuit to obtain the induced potentia difference: V ind = E ind d. (2) This induced potentia difference is caed the induced votage for short, and it wi produce a current in the circuit if the circuit resistance is ess than infinite. The induced votage represents the work per unit charge that the eectric fied is wiing to do. It is non-zero around a compete cosed path, which is quite a difference from ordinary votages (such as those suppied by batteries) that are zero around a cosed path. We iustrate this in Fig. 3 with a specific battery-driven circuit and a specific induction-driven circuit where a magnetic fied is changing with time. In Fig. 3a the votage drop cockwise from c to d is 2.0 V and the votage drop cockwise from d to c is 2.0 V (a votage rise). Thus the votage drop around the tota circuit, cockwise from c to d to c, is zero. In Fig. 3b the votage drop from any one point cockwise around the circuit and back to the origina point is not zero and is, in fact, the induced votage. 1f. Induced Votage and Fu Change Rate. When the magnetic fu, through the surface that is surrounded by a oop, changes with time, the induced votage equas the rate of change of the magnetic fu through any surface bounded by the oop: V ind = dφ dt. (3) r ine =0 Figure 4. A square-oop circuit enters a region of constant magnetic fied that is directed into the page. 1g. The Power Transferred. If the magnetic fied is constant but the circuit is moving (the Lorentz-Force case) the power dissipated in the circuit resistance comes from the mechanica energy needed to move the circuit in the fied. If the circuit is stationary but the magnetic fied is changing with time (the Faraday-Henry case) the power dissipated in the circuit resistance comes from the energy that must be epended to change that fied. 2. Eampe of Lorentz-Force Induction 2a. A Loop Enters A Magnetic Fied. In Fig. 4 we show an eampe of Lorentz force induction. There a oop circuit traveing at speed v enters a region of constant magnetic fied B. Note that is the distance from the center ine of the oop to the center ine of the region of magnetic fied; and that the ength of each side of the oop is and the width of the magnetic fied is 2. Unti the eading edge of the oop reaches the fied there is no Lorentz force on the mobie charges in the wire that constitutes the oop. When = 3/2 the wire enters the fied and there is a Lorentz force on the mobie charges in part of the oop that has entered the fied. That force is: F = q v B so the induced eectric fied, the force per unit charge, is E ind = v B. Integrating this aong the eading edge of the oop, 7 8

5 MISN MISN À - -½ Bv V in ½ -Bv À 2 2d. The Time Rate of Change of the Fu. The induced potentia difference is simpy reated to the time rate of change of the magnetic fu encosed by the oop: V ind,dcba = Φ. (8) This comes about because the magnetic fied is constant so the fu integra, Eq. (1), is just Φ = B da/dt where A is the area of the oop containing non-zero B. That area is A = r so da/dt = dr/dt = v. Then Φ = B v and Eq. (8) is proved. We wi find that Eq. (8) is not restricted to Lorentz force cases but is aso true where the oop is stationary (not moving) in the fied but the magnetic fied itsef is increasing or decreasing with time. Figure 5. The induced potentia difference around the oop of Fig. 4 as a function of the position of the oop. we get for the induced potentia of point b with respect to point a: V ind,ba = v B. (4) Thus any induced current wi fow from a to b (from higher potentia to ower, in the direction of the Lorentz Force). 2b. The Loop Traves Through the Magnetic Fied. As the oop traves across the magnetic fied the induced potentia difference aong the top and bottom arms of the oop wi be zero as is readiy seen from the vector form of the Lorentz force. The potentia difference between the ends of the traiing edge of the oop wi be the same as for the eading edge: V ind,cd = v B. (5) As we make a compete circuit around the oop the induced potentia differences aong the two ends oppose each other and the net potentia difference is zero: V ind,oop = 0. (6) 2c. The Loop Leaves the Magnetic Fied. When the eading edge of the coi reaches the far edge of the fied and emerges into a region of zero magnetic fied, its induced potentia drops to zero and the oop potentia difference is: V ind,dc = + v B, (7) which woud cause current to fow around the oop in a cockwise direction. Fig. 5 shows a pot of the oop potentia difference, V dcba, vs.. 3. Induction by Time-Changing B 3a. The Faraday-Henry Law. Time-changing magnetic fieds produce induced eectric fieds, hence potentia differences, in a manner described by the so-caed Faraday-Henry aw. This aw reates the ine integra of the induced eectric fied to the time-derivative of the surfaceintegrated magnetic fied: E ind d = d B ds dt. (9) LS Now we can substitute ( V ind ) for the eft side and ( Φ) for the right side to get the simper-appearing form: SL V ind,ls = Φ SL. (10) Note that this equation is eacty the same as the equation for Lorentzforce induction, Eq. (8). 3b. Eampe: Lineary Increasing B. Suppose a coi having N turns and area A is paced perpendicuar to a magnetic fied that is increasing ineary with time: B = B 0 + B 1 t, where B 0 and B 1 are constants. The induced votage is, by Eq. (10): V ind = Φ = N A B

6 MISN r v ine =0 Figure 6. As Fig. 4 but with circuit resistance shown epicity. 4. Lenz s Law Lenz s aw states that magnetic induction is aways in such a direction as to tend to oppose the change that produced the induction. For eampe, as the oop in Fig. 4 enters the magnetic fied the induced potentia difference is such as to produce a counter-cockwise current in the oop. That countercockwise current wi itsef produce a magnetic fied that comes out of the page and so to some etent opposes the into-thepage continuing increase in the origina oop fu. Simiary, the cockwise current induced as the oop eaves the fied itsef produces a magnetic fied that is into the page: this tends to oppose the continuing decrease that is taking pace in the oop fu. Lentz s aw aso states that if a time-changing potentia difference causes a time-changing magnetic fied, then the time-changing magnetic fied produces an eectric fied and a resuting current fow that wi oppose the change in the origina potentia difference. Simiary, a changing current induces a current that tends to oppose the continuing change in the origina current. MISN Sef-Inductance, Inductors The term inductor refers to a circuit eement that usuay has the shape of a soenoid, a torus, or simpy many oops of wire gued together. A current fowing through an inductor of course sets up a magnetic fied so changes in the current resut in changes in the magnetic fied. Those in turn produce an induced votage drop in the eement, a votage drop that opposes the change in the current. The magnitude of the induced votage is proportiona to the time-rate-of-change of the current, as we have seen, so we write: V ind = L di dt, (11) where L is determined soey from the geometry of the inductor and is caed the inductor s inductance. The S.I. unit of inductance is the henry, abbreviated H: H V s A 1 as can be seen from Eq. (11). Acknowedgments Preparation of this modue was supported in part by the Nationa Science Foundation, Division of Science Education Deveopment and Research, through Grant #SED to Michigan State University. Gossary Faraday-Henry aw: the integra form of a Mawe equation, stating that the ine integra of the eectric fied around a oop is equa to the negative of the time derivative of the integra of the norma component of the magnetic fied (integrated over the surface bounded by the oop): E d = (d/dt) B ds. This aw is often quoted in its short LS LS form: the votage induced in the oop equas the time derivative of the magnetic fu through the surface bounded by the oop: V ind = Φ. fu: denoted Φ; the integra, over a surface with a particuar boundary, of the component of the fied norma to the surface. For magnetic fu: Φ SL = B ds. SL induced current: the current produced in a circuit as a resut of an induced potentia difference (which see). The usua reations between votage, resistance, and current appy

7 MISN MISN PS-1 induced potentia difference: the tota distributed potentia difference encountered in traversing a circuit due to motion of the circuit in a magnetic fied or due to the circuit being in a time-changing magnetic fied. just another name for induced potentia differ- induced votage: ence (which see). sef-inductance: the negative of the induced votage around a oop divided by the time-rate-of-change of magnetic fu through any surface bounded by that oop: L = V induced / Φ. The minus sign shows that the induced votage opposes the change in the fu. The SI unit of inductance is the henry, abbreviated H: H V s A 1. inductor: a circuit eement whose purpose is to provide sefinductance, the eectrica circuit anaog of mechanica inertia (mass). An inductor is usuay in the shape of a soenoid or a toroid. The sefinductance of an inductor depends on the geometry of the inductor, and the magnetic susceptibiity of the materias of which the inductor is constructed. Inductors in eectronic circuits typicay are in the mh range. Lenz s aw: a aw stating that induced phenomena oppose the inducing phenomena. For eampe, an induced current produces a magnetic fied which opposes the changes in the magnetic fied that caused the current to be induced. PROBLEM SUPPLEMENT Note: Probems 1-3 aso appear in this modue s Mode Eam. 1. A circuar oop, of radius R = 0.20 meters, in the pane of the page is in a region where a magnetic fied B is directed into the page (see sketch). The magnitude of B varies with time according to: B = B 0 At 2, where B 0 = 2.0 T (tesas) and A = 0.30 T/s 2 are constants. ` B directed into page a. Find the magnitude and direction of the induced potentia difference, V ind, around this oop at the instant t = 0.20 sec. [F] Is V ind constant with time? [A] Check that the dimensions of V ind foow correcty from the dimensions of what goes into your cacuations. [H] b. Find the magnitude and direction of the tangentia component of the eectric fied vector at any point on this oop. [J] c. What can the Faraday-Henry Law te you about the norma (radia) component of E at any point on the oop? [E] What does symmetry te you? [I] 13 14

8 MISN PS-2 MISN PS-3 2. A S P Q This same oop is paced in a region of uniform fied B, B = 2.0 T, directed into the page. The oop is rotated about the ais AA so that point P comes out of the page, point Q goes into the page. In 0.02 sec the oop is fipped over so that P is where Q was and vice versa. a. What is the average V ind induced around this oop? [B] b. What is the direction of this V ind? (Woud it drive the induced current in the direction SP T or T P S? [D] 3. As shown in the figure, ines of magnetic fied are directed perpendicuar to a circuar oop of wire of radius r. a. If the magnitude of the fied is given by B = B 0 e t/τ, where B 0 and τ are constants, what is the V ind induced in the oop as a function of time? [G] (In the figure, d s denotes the positive direction around the oop.) B T A' 5. R ½ S A square oop of wire is stationary within a time-dependent magnetic fied that is confined to a square region of space. The ength of one side of the oop is and the oop etends /2 into the fied (see sketch). The magnetic fied varies with time as B = B 0 sin(ωt). Use these vaues to answer the questions beow: B 0 = 1.0 T, ω = 2.0π/ s, R = 1.0 Ω and = 1.0 m. a. At t = 3.5 s what is V ind? [M] b. At that same time what is the power dissipation through the resistor? [O] Hep: [S-3] c. What is the work done by the fied from t = 0.0 s to t = 3.5 s? [N] Hep: [S-2] r b. Answer the same question if the oop is tited through 60.0, [L] or [K] 4. Using the definitions of the tesa (p. 359) and the vot, in terms of fundamenta dimensions, show that both sides of the Faraday-Henry Law have the same units. [C] ds 15 16

9 MISN PS-4 MISN AS-1 Brief Answers: A. No. SPECIAL ASSISTANCE SUPPLEMENT B. 25 vots. Hep: [S-1] C. See tet. D. Again, use Lenz s Law to find that the direction of decreasing V ind, the downhi direction for the current, the direction the current wi move, is SP T. E. Nothing, in the case of this circuar path. F vots, cockwise. G. πr 2 B. τ Lenz s Law verifies, direction same as d s. Dimensions: m 2 T s 1 = kg m 2 s 2 C 1 = N C 1 = vot S-1 (from PS, Probem 2a) The average vaue of a sine function over a haf cyce is: sin = π 0 S-2 (from PS, Probem 5b) sin d π 0 d = cos π 0 π =... The power-votage-resistance reation is referenced in this modue s ID Sheet. S-3 (from PS, Probem 5c) The energy-power-time reation is referenced in this modue s ID Sheet. H. See definition of the dimension of tesa, p. 359 of AF. I. Symmetry tes you that whatever E n (the component of E norma to page) is, it is the same at every point on the circe. Because the change in B is entirey perpendicuar to the page, E n in fact is zero. (Refer to Lenz s Law in this modue s tet). J. E t = vot/m = N/C. K. Zero. L. πr 2 B. 2τ M. V ind = 2 ωb 0 cos(ωt) = π V 2 N. W = 1 ( ) 2 2 [ t R 2 ωb ] 4ω sin(2ωt) = 1.75π 2 J Hep: [S-3] O. P = V 2 ind /R = π2 V Ω = π2 A V = π 2 W Hep: [S-2] 17 18

10 MISN ME-1 MISN ME-2 MODEL EXAM 1. See Output Skis K1-K2 in this modue s ID Sheet. The actua eam may have one or more of these skis, or none. 2. A circuar oop, of radius R = 0.20 meters, in the pane of the page is in a region where a magnetic fied B is directed into the page (see sketch). The magnitude of B varies with time according to: B = B 0 At 2, where B 0 = 2.0 T (tesas) and A = 0.30 T/s 2 are constants. ` B directed into page a. Find the magnitude and direction of the induced potentia difference, V ind, around this oop at the instant t = 0.20 sec. Is V ind constant with time? Check that the dimensions of V ind foow correcty from the dimensions of what goes into your cacuations. b. Find the magnitude and direction of the tangentia component of the eectric fied vector at any point on this oop. c. What can the Faraday-Henry Law te you about the norma (radia) component of E at any point on the oop? What does symmetry te you? 3. A S P Q This same oop is paced in a region of uniform fied B, B = 2.0 T, directed into the page. The oop is rotated about the ais AA so that point P comes out of the page, point Q goes into the page. In 0.02 sec the oop is fipped over so that P is where Q was and vice versa. T a. What is the average V ind induced around this oop? b. What is the direction of this V ind? (Woud it drive the induced current in the direction SP T or T P S? 4. As shown in the figure, ines of magnetic fied are directed perpendicuar to a circuar oop of wire of radius r. a. If the magnitude of the fied is given by B = B 0 e t/τ, where B 0 and τ are constants, what is the V ind induced in the oop as a function of time? (In the figure, d s denotes the positive direction around the oop.) B b. Answer the same question if the oop is tited through 60.0, or ds A' r Brief Answers: 1. See this modues s tet

11 MISN ME-3 2. See this modue s Probem Suppement, probem See this modue s Probem Suppement, probem See this modue s Probem Suppement, probem

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