# Chapter 20: Electromagnetic Induction. PHY2054: Chapter 20 1

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1 Chapter 20: Electromagnetic Induction PHY2054: Chapter 20 1

2 Electromagnetic Induction Magnetic flux Induced emf Faraday s Law Lenz s Law Motional emf Magnetic energy Inductance RL circuits Generators and transformers Topics PHY2054: Chapter 20 2

3 Reading Quiz 1 Magnetic flux through a wire loop depends on: 1) thickness of the wire 2) resistivity of the wire 3) geometrical layout of the wire 4) material that the wire is made of 5) none of the above Φ B = BAcosθ Flux depends only on geometrical properties PHY2054: Chapter 20 3

4 Reading Quiz 2 An induced emf produced in a motionless circuit is due to 1) a static (steady) magnetic field 2) a changing magnetic field Faraday s law 3) a strong magnetic field 4) the Earth s magnetic field 5) a zero magnetic field PHY2054: Chapter 20 4

5 Reading Quiz 3 Motional emf relates to an induced emf in a conductor which is: 1) long 2) sad 3) stationary 4) insulated 5) moving Potential difference proportional to velocity PHY2054: Chapter 20 5

6 Reading Quiz 4 Faraday s law says that 1. an emf is induced in a loop when it moves through an electric field 2. the induced emf produces a current whose magnetic field opposes the original change 3. the induced emf is proportional to the rate of change of magnetic flux Faraday s law PHY2054: Chapter 20 6

7 A generator is a device that: Reading Quiz 5 a) transforms mechanical into electrical energy b) transforms electrical into mechanical energy c) transforms low voltage to high voltage PHY2054: Chapter 20 7

8 Magnetic Flux Define magnetic flux Φ B Φ B = B A= BAcosθ θ is angle between B and the normal to the plane Flux units are T-m 2 = webers When B field is not constant or area is not flat Integrate over area Φ B = B d A A Won t cover this case here PHY2054: Chapter 20 8

9 Φ B = B A= BAcosθ 1 Φ = 0 Φ = BA Φ B = BA B B 2 PHY2054: Chapter 20 9

10 Experimental Observation of Induction This effect can be quantified by Faraday s Law PHY2054: Chapter 20 10

11 Faraday s Law of Induction induced emf ε = N ΔΦ Δt B rate of change of flux with time number of loops The faster the change, the larger the induced emf Flux change caused by changes in either B, area, orientation Minus sign explained next slide PHY2054: Chapter 20 11

12 Faraday s Law of Induction induced emf ε = N ΔΦ Δt B rate of change of flux with time number of loops Minus sign from Lenz s Law: Induced current produces a magnetic field which opposes the original change in flux PHY2054: Chapter 20 12

13 Comment on Lenz s Law Why does the induced current oppose the change in flux? Consider the alternative If the induced current reinforced the change then the change would get bigger which would then induce a larger current and then the change would get even bigger and so on... A clear violation of conservation of energy!! PHY2054: Chapter 20 13

14 Direction of Induced Current Bar magnet moves through coil Current induced in coil A S N v Reverse pole Induced current changes sign B N S v v Coil moves past fixed bar magnet Current induced in coil as in (A) C S N Bar magnet stationary inside coil D S N No current induced in coil PHY2054: Chapter 20 14

15 ConcepTest: Lenz s Law If a North pole moves towards the loop from above the page, in what direction is the induced current? (a) clockwise (b) counter-clockwise (c) no induced current Must counter flux change in downward direction with upward B field PHY2054: Chapter 20 15

16 ConcepTest: Induced Currents A wire loop is being pulled through a uniform magnetic field. What is the direction of the induced current? (a) clockwise (b) counter-clockwise (c) no induced current No change in flux, no induced current x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x PHY2054: Chapter 20 16

17 ConcepTest: Induced Currents x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 1 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Counterclockwise x x x x x x x x x x x x x x x x x None x x x x x x 2 In the cases above, what is the direction of the induced current? PHY2054: Chapter 20 17

18 ConcepTest: Lenz s Law If a coil is shrinking in a B field pointing into the page, in what direction is the induced current? (a) clockwise (b) counter-clockwise (c) no induced current Downward flux is decreasing, so need to create downward B field PHY2054: Chapter 20 18

19 Induced currents A circular loop in the plane of the paper lies in a 3.0 T magnetic field pointing into the paper. The loop s diameter changes from 100 cm to 60 cm in 0.5 s What is the magnitude of the average induced emf? What is the direction of the induced current? If the coil resistance is 0.05Ω, what is the average induced current? V ( ) ΔΦ π B = = 3.0 = Volts Δt 0.5 Direction = clockwise (Lenz s law) Current = / 0.05 = 60.3 A PHY2054: Chapter 20 19

20 ConcepTest: Induced Currents A wire loop is pulled away from a current-carrying wire. What is the direction of the induced current in the loop? (a) clockwise (b) counter-clockwise (c) no induced current Downward flux through loop decreases, so need to create downward field I PHY2054: Chapter 20 20

21 ConcepTest: Induced Currents A wire loop is moved in the direction of the current. What is the direction of the induced current in the loop? (a) clockwise (b) counter-clockwise (c) no induced current Flux does not change when moved along wire I PHY2054: Chapter 20 21

22 ConcepTest: Lenz s Law If the B field pointing out of the page suddenly drops to zero, in what direction is the induced current? (a) clockwise (b) counter-clockwise (c) no induced current Upward flux through loop decreases, so need to create upward field If a coil is rotated as shown, in a B field pointing to the left, in what direction is the induced current? (a) clockwise (b) counter-clockwise (c) no induced current Flux into loop is increasing, so need to create field out of loop PHY2054: Chapter 20 22

23 ConcepTest: Induced Currents Wire #1 (length L) forms a one-turn loop, and a bar magnet is dropped through. Wire #2 (length 2L) forms a two-turn loop, and the same magnet is dropped through. Compare the magnitude of the induced currents in these two cases. (a) I 1 = 2 I 2 (b) I 2 = 2 I 1 (c) I 1 = I 2 0 (d) I 1 = I 2 = 0 Voltage doubles, but R also doubles, leaving current the same (e) Depends on the strength of the magnetic field PHY2054: Chapter 20 23

24 Motional EMF Consider a conducting rod moving on metal rails in a uniform magnetic field: ε dφ B d( BA) d( BLx) dx = = = = BL dt dt dt dt Current will flow counter-clockwise in this circuit. Why? x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x v L ε = BLv PHY2054: Chapter 20 24

25 Force and Motional EMF Pull conducting rod out of B field Current is clockwise. Why? ε i = = BLv R R Current within B field causes force F 2 2 = ilb = B Lv R Force opposes pull (RHR) Also follows from Lenz s law We must pull with this force to maintain constant velocity PHY2054: Chapter 20 25

26 Power and Motional EMF Force required to pull loop: Power required to pull loop: Energy dissipation through resistance 2 2 = ilb = B L v R Same as pulling power! So power is dissipated as heat Kinetic energy is constant, so energy has to go somewhere Rod heats up as you pull it F P 2 BLv P = i R = R = R = Fv= B L v R B L v R PHY2054: Chapter 20 26

27 Example Pull a 30cm x 30cm conducting loop of aluminum through a 2T B field at 30cm/sec. Assume it is 1cm thick. EMF Circumference = 120cm = 1.2m, cross sectional area = 10-4 m 2 R = ρl/a = 2.75 x 10-8 * 1.2 / 10-4 = 3.3 x 10-4 Ω Current Force ε = BLv = = 0.18V i F 4 = ε / R = 0.18/ = 545A = ilb = = 327 N 74 lbs! Power P 2 = i R =98W About C per sec (from specific heat, density) PHY2054: Chapter 20 27

28 Electric Generators Rotate a loop of wire in a uniform magnetic field: changing θ changing flux induced emf Φ B = B A cos θ = B A cos(ωt) Rotation: θ = ωt PHY2054: Chapter 20 28

29 Electric Generators Flux is changing in a sinusoidal manner ε Leads to an alternating emf (AC generator) dφ dcos( ωt) B = N = NBA = NBAω sin( ωt) dt dt This is how electricity is generated Water or steam (mechanical power) turns the blades of a turbine which rotates a loop Mechanical power converted to electrical power PHY2054: Chapter 20 29

30 ConcepTest: Generators A generator has a coil of wire rotating in a magnetic field. If the B field stays constant and the area of the coil remains constant, but the rotation rate increases, how is the maximum output voltage of the generator affected? (a) Increases (b) Decreases (c) Stays the same (d) Varies sinusoidally ε = NBAω sin( ωt) PHY2054: Chapter 20 30

31 Induction in Stationary Circuit Switch closed (or opened) Current induced in coil B (direction shown is for closing) Steady state current in coil A No current induced in coil B B A PHY2054: Chapter 20 31

32 Self - Inductance Consider a single isolated coil: Current (red) starts to flow clockwise due to the battery But the buildup of current leads to changing flux in loop Induced emf (green) opposes the change This is a self-induced emf (also called back emf) ε dφ di = N = L dt dt L is the self-inductance units = Henry (H) 12V induced emf PHY2054: Chapter 20 32

33 Total flux (length l) B= μ 0 in Inductance of Solenoid NΦ = nl BA = n Ali ε B μ 0 ( )( ) 2 dφb 2 di di = N = μ0n Al = L dt dt dt L 0 2 = μ n Al To make large inductance: Lots of windings Big area Long PHY2054: Chapter 20 33

34 LR Circuits Inductance and resistor in series with battery of EMF V Start with no initial current in circuit Close switch at t = 0 Current is initially 0 (initial increase causes voltage drop across inductor) Find i(t) Resistor: ΔV R = Ri Inductor: ΔV L = L di/dt Use back emf of inductor () it V ΔVL V Ldi / dt = = R R V R L PHY2054: Chapter 20 34

35 Analysis of LR Circuit General solution ( )( / / 1 ) i= V R e tr L Rise from 0 with time constant τ = L / R Final current (maximum) PHY2054: Chapter 20 35

36 L-R Circuits (2) Switch off battery: Find i(t) if current starts at i 0 / i i0e tr L = Exponential fall to 0 with time constant τ = L / R Initial current (maximum) PHY2054: Chapter 20 36

37 Exponential Behavior τ = L/R is the characteristic time of any RL circuit Only t / τ is meaningful, just like for RC circuit t = τ Current falls to 1/e = 37% of maximum value Current rises to 63% of maximum value t =2τ Current falls to 1/e 2 = 13.5% of maximum value Current rises to 86.5% of maximum value t =3τ Current falls to 1/e 3 = 5% of maximum value Current rises to 95% of maximum value t =5τ Current falls to 1/e 5 = 0.7% of maximum value Current rises to 99.3% of maximum value PHY2054: Chapter 20 37

38 ConcepTest: Generators and Motors A current begins to flow in a wire loop placed in a magnetic field as shown. What does the loop do? (a) moves to the right (b) moves up (c) rotates around horizontal axis (d) rotates around vertical axis (e) moves out of the page B This is how a motor works!! PHY2054: Chapter 20 38

39 Electric Motors Current is supplied from an external source of emf (battery or power supply) Forces act to rotate the wire loop A motor is essentially a generator operated in reverse! PHY2054: Chapter 20 39

40 Motor Forces act to rotate the loop towards the vertical. When loop is vertical, current switches sign and the forces reverse, in order to keep the loop in rotation. This is why alternating current is necessary for a motor to operate. PHY2054: Chapter 20 40

41 Motors Generators Electrical mechanical energy Mechanical electrical energy PHY2054: Chapter 20 41

42 Energy of an Inductor Energy is stored in inductor when current flows through it U L = 1 2 Li 2 Energy is stored in the B field Similar to energy in charged capacitor U C 2 q = = 2C 1 2 CV 2 Energy is stored in the E field (between plates) PHY2054: Chapter 20 42

43 Energy of Solenoid Recall B field of solenoid (current i, n turns per meter) B = μ ni 0 Inductance of solenoid L Φ B = BA = μ nia ( )( μ ) N Φ B nl nia = = = i i 0 2 μ0 n Al Energy of solenoid can be written in terms of B field and volume (V = la) ( 2 ) B B UL = Li = μ 2 2 0n la = Al μ0n 2μ0 0 Energy density of B field Volume PHY2054: Chapter 20 43

44 Typical MRI magnet B = T R = 0.25 m, l = 2.5 m V = Al = π R l Energy of Solenoid (2) m Assume B = 1 T typically U B 2 2 B 1 = V = ( 0.5) 200kJ 2μ 7 2 4π 10 0 This is a lot of energy Needs to be held in by strong materials PHY2054: Chapter 20 44

45 Gigajoule Magnet at CERN CMS experiment at CERN p-p collisions at world s highest energy in 2007 Hope to discover new particles, find the origin of mass and new fundamental forces Compact Muon Solenoid PHY2054: Chapter 20 45

46 Human PHY2054: Chapter 20 46

47 CMS Experiment Magnet Large central solenoid magnet to study particle production B = 4T, R = 3.15 m, l = 12.5 m U B = 2.6 x 10 9 J = 2.6 gigajoules!! 2 2 B 4 UB = Al = π 2μ 7 2 4π 10 0 ( )( 12.5) PHY2054: Chapter 20 47

48 Home page CMS Articles and Pictures Pictures of detector orinfo.html Interesting article on solenoid, with pictures hure/magnetbrochure.pdf Other documents & pictures about CMS nts.html PHY2054: Chapter 20 48

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