Chapter 34 Faraday s Law & Electromagnetic Induction

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1 Chapter 34 Faraday s Law & Electromagnetic Induction

2 Faraday s Discovery (~ 1831) Faraday found that a changing magnetic field creates a current in a wire. This is an informal statement of Faraday s law. Joseph Henry, a teacher at Albany Academy, discovered the principle a little earlier but didn t publish immediately. Henry is also the inventor of the solenoid and electromechanical relay.

3 What Faraday and Henry Observed (Show Film)

4 Motional EMF How motion in a magnetic field creates a current! F = q! v!! B q"v / l = qvb "V = vlb

5 Induced Current From Motional EMF The moving conducting bar is now connected via stationary rails to a circuit. There is only a voltage/emf induced in the moving bar I =!V / R I = vlb / R

6 Magnetic Counter-Force There is a SECOND magnetic force once the current starts to flow in the circuit:! F mag = I! l!! B This acts opposite to the motion of the bar.! F pull = "! F mag F pull = IlB = vlb R lb = vl 2 B 2 R F mag F pull

7 Conservation of Energy Rules... The energy expended by the pulling force to move the bar (against the magnetic counter-force) is equal to the energy dissipated by the circuit as heat. Input Power: P input = de input dt! P input = m dv " # dt P input = v2 l 2 B 2 R Dissipated Power: 1 2 mv2 = d dt $ % & v = F pull v P dissipate = I 2 R = v2 l 2 B 2 R

8 Eddy Currents: Moving a non-magnetic conductor through a magnetic field induces localized current loops. The B field then exerts a force on the currents opposite to the motion of the conductor. Mechanical motion --> Heat from current.

9 Faraday s Law An emf is induced in a conducting loop if the magnetic flux through the loop changes. The magnitude of the emf is!v = " = # d$ m dt = # d dt % loop! Bid! A!V = # d dt BAcos& (uniform B field) For N loops in series, the induced emf is:!v = " = #N d$ m dt ΔV is the voltage measured between the two ends of the loop

10 Magnetic Flux has the same definition as electric flux from Ch. 28 when we learned Gauss s Law. The magnetic flux measure the amount of magnetic field strength passing through a surface Units: 1 Weber= 1 Wb= 1 Tm 2 For a uniform magnetic field:! m =! Ai! B = AB " cos#! A = ˆnA For nonuniform magnetic fields!! m = Bid A! " area of loop Magnetic Flux

11 Faraday s Law: Two Separate Physical Effects! =! d" dt =! # Bi! d! A + d B! dt dt i A! & % ( $ '! Bi d! A= motional EMF. This can be explained in terms of the Lorentz force dt! F = q v! ) B! acting on charges in the moving wire d! B dt i! A * The conducting loop is static.

12 There is an induced current in a closed, conducting loop if and only if the magnetic flux through the loop is changing. The direction of the induced current is such that the induced magnetic field opposes the change in the flux. Lenz s Law In other words, the current creates a magnetic field in an attempt to resist any change in the flux through the loop.

13 Lenz s Law for Different Cases (For Students to Review)

14 Electric Fields Even Without the Conductor The induced electric field created by a changing magnetic field is independent of whether there is a conductor or not. The induced electric field is created in empty space. It s direction is determined by the right hand rule.! "! Eid s! = " d# m dt

15 Electrical Generators A mechanical force rotates the conducting loop at a constant rate B and A are constant but the angle varies harmonically! m =! Bi! A = BAcos"t # $ coil = %N d dt! m = NBA" sin"t Generator produces AC voltage

16 Transformers Τransformers are used to distribute electrical power by transforming the voltages from high to low or vice versa. An oscillating voltage in the first coil generates a magnetic flux: V 1 (t) =!N 1 d" m dt This time varying flux then induces a voltage in the second coil: V 2 (t) =!N 2 d" m dt V 2 (t) = V 1 (t) N 2 N 1 The power generated by the voltage V 1 equals the power induced in the second coil I 1 V 1 = I 2 V 2

17 Transformers for Electrical Power Distribution The power delivered by the first (primary) coil: P 1 = I 1 V 1 The power delivered by the second (secondary) coil: P 2 = I 2 V 2 1)Step Up Transformer: Convert from high I and low V to low I and high V 2)Electrical Transmission Lines: The power dissipated by the power lines is P dissipated = I 2 R 3)Step down transformer: Convert from low I and high V to high I and low V Step UpTransformer: Low V to High V Step Down Transformer: High V to Low V

18 Wireless Power Transmission

19 Inductors Inductors are passive circuit elements that store energy in magnetic fields. They resist changes in the voltage/current in a circuit. They are typically a coil of wire such as a solenoid. Inductors are to magnetic fields what capacitors are to electric fields.

20 The coil of wire sees a flux generated by itself. The flux induces a voltage ( a backreaction) in circuit according to Lenz s law Self Inductance: L = N! m / I Self-Inductance Induced "Backreaction" Voltage: "V L = #N d! m dt = #L di dt

21 What Does an Inductor Do??? Inductors only do something when the current in a circuit is changing. An inductor generates a voltage difference that opposes changes in the current.

22 The Voltage Across an Inductor:! m = N! per loop = LI "V = #N d dt! per loop di = #L dt The unit of inductance is the henry:1 henry= 1 H=1 Tm 2 /A Inductance of a solenoid having N turns, length l and cross-section area A is! B solenoid = µ 0 # " N l L solenoid = µ 0 N 2 A l $! &I ' ( 1coil = µ 0 # % " N l $ &IA %

23 Energy Stored in an Inductor The power lost from a current in a circuit to an inductor is P = I!V = "LI di dt This energy gets stored in the potential energy of the inductor du L = "P = +LI di dt dt Integrating this expression gives the energy of the inductor: U L = 1 2 LI 2 For a solenoid, we can rewrite this in terms of the B field inside the solenoid: ( ) U L = Al # 1 & B 2 $ % 2µ 0 ' ( U L = (volume) ) (energy density of magnetic field)

24 Energy Stored in a Capacitor and Inductor

25 An LR circuit consists of an inductor and resistor in series. It is similar to the RC circuit from Ch. 32 in its behaviour. Like the RC circuit, the LR circuit can be used to make analogue low pass filters (removes the high frequency components of an AC signal) LR Circuits

26 Solving for the LR Circuit (1) The figure is the simplest type of LR circuit. What happens when the switch is moved to b?

27 Solving for the LR Circuit (2) Initially when the switch is in position a the current is: I 0 =!V bat / R Constant current=no effect from the inductor! Changing the switch to b, the current is initially I 0. Use Kirchoff's law for the voltages when switch is in b:!v R +!V L = 0 "IR " L di dt = 0 di dt = " 1 # I # = L / R This is the same equation we saw for the charge in an RC circuit: dq = " 1 Q # =RC (For RC Circuit!!) dt # We know the solution then! "t /# I(t)=I 0 e The current in an LR circuit behaves the same as that in an RC circuit!

28 Decay of Current in LR Circuit: The inductance prevents the current from turning off instantly when the switch is moved from a to b. Every real circuit has a finite amount of inductance.!t /" I(t) = I 0 e " = L / R

29 What happens when switch is flipped back to position a? Kirchoff's Law:!V bat " IR " L di dt = 0 I =!V bat R! = L / R ( 1" ) e"t/!

30 The Inductance of a Circuit Generates Sparks When You Unplug It!

31 Delay in Current for Large L (1)

32 Delay in Current for Large L (2)

33 Delay in Current for Large L (3) t<0

34 Delay in Current for Large L (4) t 0 =0

35 Delay in Current for Large L (5) t 1 >>t 0 + L/(R+R bulb )

36 Delay in Current for Large L (6) t 2 >t 1

37 Delay in Current for Large L (7) t 3 >>t 2 +L/(R+R bulb )

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