General Review. LECTURE 16 Faraday s Law of Induction

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1 Electrostatics General Review Motion of q in eternal E-field E-field generated b Sq i Magnetostatics Motion of q and I in eternal B-field B-field generated b I Electrodnamics Time dependent B-field generates E-field AC circuits, inductors, transformers, etc. Time dependent E-field generates B-field Electromagnetic radiation, light LECTURE 16 Farada s Law of Induction 1

2 Current flows onl if there is relative motion between the loop and the magnet Current disappears when the relative motion ceases DEMO Induction Faster motion produces a greater current ammeter Caution this picture is not an eample of right hand rule! Induction Effects Bar magnet moves through coil, or Coil moves past fied bar magnet: Current induced in coil S v Change pole that enters: Induced current changes sign S v Bar magnet stationar inside coil: o current induced in coil S Change direction of motion in an of above eamples v S Induced current changes sign 2

3 DEMO Induction Effects From Currents Coil A Coil B electromagnetic_induction Open or close the switch Current induced in coil B Stead state current in coil A o current induced in coil B Conclusion: A change in magnetic field in a loop induces a current in the loop Man was to do this How can we quantif this? Induction Eample A wire loop falling into a B field (increasing) time Force acting on moving charges Magnetic Field B Downward Velocit v!" "!" F = qv B 3

4 Magnetic Flu Perpendicular component of vector field through a surface S Φ B = B! nda ^ = B! d A! S If S is flat and B is uniform S Φ B = BAcosθ da Unit: 1 Weber = 1 Wb = 1 Tm 2 If S is bounded b a conducting path (e.g., a wire) An EMF is induced along the path Magnitude of the emf e induced in a conducting loop equals the rate of change of the magnetic flu F B through the loop Farada s Law ε = dφ B egative sign indicates emf opposes change (Lenz s Law) 4

5 Farada s Law Restated EMF induced in loop EMF = work/charge e = W/q Magnetic forces do no work Moving charges => E is present ot conservative in this case ot zero around closed path Static E is conservative! " E d L! = 0 E nc onl does work if conducting loop present Still present with or without conductor! ε = " E nc d L! = dφ B C EMF for a Coil of Loops Flu through one loop Φ B = BAcosθ Flu through loops Φ B = BAcosθ EMF through loops ε = dφ B = d = πr 2 cosθ db ( BAcosθ ) 5

6 DEMO How to Change Magnetic Flu in a Coil 1.! B changes: ΔΦ B Δt = ΔB Δt Acosθ 3. θ changes: ΔΦ B Δt = BA Δ cosθ Δt 2. A changes: ΔΦ B Δt = B ΔA Δt cosθ 4. changes: ΔΦ B = Δ Δt Δt BAcosθ Unlikel Lenz s Law An induced current has a direction such that its associated magnetic field opposes the change in the magnetic flu that induces the current B induced alwas opposes the change in the flu of B, but does not alwas point opposite it!!! Move towards loop, F B increases B induced opposes this increase I must be counterclockwise 6

7 Lenz s Law An induced current has a direction such that its associated magnetic field opposes the change in the magnetic flu that induces the current B induced alwas opposes the change in the flu of B, but does not alwas point opposite it!!! Move loop awa from, F B decreases B induced opposes this decrease I must be clockwise Direction of I must be to oppose change Otherwise violates conservation of energ Small change would be magnified continuousl Eample 2 A conducting rectangular loop moves with constant velocit v in the - direction and a constant current I flows in a wire in the + direction as shown. What is the direction of the induced current in the loop? v I (a) CCW (b) CW (c) o induced current 7

8 Eample 2 A conducting rectangular loop moves with constant velocit v in the - direction and a constant current I flows in a wire in the + direction as shown. What is the direction of the induced current in the loop? B B B = µ 0 I 2πr v B B I (a) CCW (b) CW (c) o induced current db 0 Loop moving from higher magnetic field to lower field B points into picture Lenz Law: an emf will be induced to oppose the reduction in flu Clockwise current is induced to restore the flu. i.e., induced current causes etra B into plane of figure DEMO E&M Cannon v Connect solenoid to a source of alternating voltage The flu through the area ^ to ais of solenoid therefore changes with time A conducting ring placed on top of the solenoid will have a current induced in it opposing this change There will then be a force on the ring Contains a current circulating in the presence of a magnetic field It s the off-ais component of B (the fringe field ) that flings the ring side view F F B B B B top view ~ 8

9 Eample 3 Suppose two aluminum rings are used in the demo; Ring 2 is identical to Ring 1 ecept that it has a small slit as shown. Let F 1 be the force on Ring 1; F 2 be the force on Ring 2. Ring 1 Ring 2 (a) F 2 < F 1 (b) F 2 = F 1 (c) F 2 > F 1 Eample 3 Suppose two aluminum rings are used in the demo; Ring 2 is identical to Ring 1 ecept that it has a small slit as shown. Let F 1 be the force on Ring 1; F 2 be the force on Ring 2. Ring 1 Ring 2 (a) F 2 < F 1 (b) F 2 = F 1 (c) F 2 > F 1 Force is due to a current flowing in the ring Ring located in a magnetic field with a component perpendicular to the current EMF is induced in Ring 2 equal to that of Ring 1, but O CURRET is induced in Ring 2 because of the slit! Therefore, O force on Ring 2! 9

10 Eample 4 Suppose two identicall shaped rings are used in the demo. Ring 1 is made of copper (resistivit = Wm); Ring 2 is made of aluminum (resistivit = Wm). Let F 1 be the force on Ring 1; F 2 be the force on Ring 2. What is the relationship between the two rings? Ring 1 Ring 2 (a) F 2 < F 1 (b) F 2 = F 1 (c) F 2 > F 1 Eample 4 Suppose two identicall shaped rings are used in the demo. Ring 1 is made of copper (resistivit = Wm); Ring 2 is made of aluminum (resistivit = Wm). Let F 1 be the force on Ring 1; F 2 be the force on Ring 2. What is the relationship between the two rings? Ring 1 Ring 2 (a) F 2 < F 1 (b) F 2 = F 1 (c) F 2 > F 1 The emf s induced in each case are equal. The currents induced in the ring are OT equal because of the different resistivities of the materials. The copper ring will have a larger current induced (smaller resistance) and therefore will eperience a larger force (F proportional to current). 10

11 Eample 5 The magnetic field in a region of space of radius 2R is aligned with the z direction and changes in time as shown in the plot. What is sign [direction] of the induced emf in a ring of radius R at time t=t 1? (a) e < 0 (E cw) (b) e = 0 (c) e > 0 (E ccw) B into the screen X X X X X X X X B z t 1 t Eample 5 The magnetic field in a region of space of radius 2R is aligned with the z direction and changes in time as shown in the plot. What is sign [direction] of the induced emf in a ring of radius R at time t=t 1? B z X X X X X X X X (a) e < 0 (E cw) (b) e = 0 (c) e > 0 (E ccw) t 1 t B field becoming more positive at t = t 1 => induced EMF Doesn t matter that B = 0 at t = t 1 Induces EMF that opposes change in flu E in CW sense to generate B induced in -z direction (into screen) 11

12 Eample 6 The magnetic field in a region of space of radius 2R is aligned with the z direction (into the page) and changes in time as shown in the plot What is the relation between the magnitudes of the induced electric fields E R at radius R and E 2R at radius 2R? X X X X R 2R X B z X X X t 1 B into the screen t (a) E 2R = E R (b) E 2R = 2E R (c) E 2R = 4E R Eample 6 The magnetic field in a region of space of radius 2R is aligned with the z direction (into the page) and changes in time as shown in the plot What is the relation between the magnitudes of the induced electric fields E R at radius R and E 2R at radius 2R? X X X X R 2R X B z X X X t 1 B into the screen t (a) E 2R = E R (b) E 2R = 2E R (c) E 2R = 4E R The rate of change of the flu is proportional to the area: dφ B = πr 2 db E R 12

13 Eample 6 The magnetic field in a region of space of radius 2R is aligned with the z direction (into the page) and changes in time as shown in the plot What is the relation between the magnitudes of the induced electric fields E R at radius R and E 2R at radius 2R? B z X X X X R 2R t 1 B into the screen X X X X t (a) E 2R = E R (b) E 2R = 2E R (c) E 2R = 4E R The path integral of the induced E field is proportional to R ε =! E d! l! = E(2πR) = dφ B = πr 2 db E = R 2 db Summar Farada s Law (Lenz Law) A changing magnetic flu through a loop induces a current in that loop ε = dφ M egative sign indicates that the induced EMF opposes the change in flu Farada's Law in terms of Electric Fields! E d! l " = dφ M 13