# Chapter 31. Faraday s Law

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1 Chapter 31 Faraday s Law 1

2 Ampere s law Magnetic field is produced by time variation of electric field dφ B ( I I ) E d s = µ o + d = µ o I+ µ oεo ds E B 2

3 Induction A loop of wire is connected to a sensitive ammeter When a magnet is moved toward the loop, the ammeter deflects 3

4 Induction An induced current is produced by a changing magnetic field There is an induced emf associated with the induced current A current can be produced without a battery present in the circuit Faraday s law of induction describes the induced emf 4

5 Induction When the magnet is held stationary, there is no deflection of the ammeter Therefore, there is no induced current Even though the magnet is in the loop 5

6 Induction The magnet is moved away from the loop The ammeter deflects in the opposite direction 6

7 Induction The ammeter deflects when the magnet is moving toward or away from the loop The ammeter also deflects when the loop is moved toward or away from the magnet Therefore, the loop detects that the magnet is moving relative to it We relate this detection to a change in the magnetic field This is the induced current that is produced by an induced emf 7

8 Faraday s law Faraday s law of induction states that the emf induced in a circuit is directly proportional to the time rate of change of the magnetic flux through the circuit Mathematically, ε = dφ B 8

9 Faraday s law Assume a loop enclosing an area A lies in a uniform magnetic field B The magnetic flux through the loop is Φ B = BA cos θ The induced emf is d( BAcos θ ) ε = Ways of inducing emf: The magnitude of B can change with time The area A enclosed by the loop can change with time The angle θ can change with time Any combination of the above can occur 9

10 Motional emf A motional emf is one induced in a conductor moving through a constant magnetic field The electrons in the conductor experience a force, F B = qv x B that is directed along l 10

11 Motional emf F B = qv x B Under the influence of the force, the electrons move to the lower end of the conductor and accumulate there As a result, an electric field E is produced inside the conductor The charges accumulate at both ends of the conductor until they are in equilibrium with regard to the electric and magnetic forces qe = qvb or E = vb 11

12 Motional emf E = vb A potential difference is maintained between the ends of the conductor as long as the conductor continues to move through the uniform magnetic field If the direction of the motion is reversed, the polarity of the potential difference is also reversed 12

13 Example: Sliding Conducting Bar E = vb ε = El = Blv 13

14 Example: Sliding Conducting Bar The induced emf is dφ dx = = = B ε B B v I = ε Bv R = R 14

15 Lenz s law dφb ε = Faraday s law indicates that the induced emf and the change in flux have opposite algebraic signs This has a physical interpretation that has come to be known as Lenz s law Lenz s law: the induced current in a loop is in the direction that creates a magnetic field that opposes the change in magnetic flux through the area enclosed by the loop The induced current tends to keep the original magnetic flux through the circuit from changing 15

16 Lenz s law dφb ε = Lenz s law: the induced current in a loop is in the direction that creates a magnetic field that opposes the change in magnetic flux through the area enclosed by the loop The induced current tends to keep the original magnetic flux through the circuit from changing B is increasing with time B B is decreasing with time B I B I I B I 16

17 Electric and Magnetic Fields Ampere-Maxwell law Faraday s law B E t ( ) E B t ( ) 17

18 Example 1.A long solenoid has n turns per meter and carries a current I = I max 1 Inside the solenoid and coaxial with it is a coil that has a radius R and consists of a total of N turns of fine wire. What emf is induced in the coil by the changing current? ( ) I ( ) B t µ n t = o 2 2 ( t ) πr NBt ( ) µπr NnI ( t) Φ = = o α t ( e ) ( ) ( ) dφ t di t ε = = µ oπr Nn = µπr NnαI e 2 2 αt o max 18

19 Example 2 A single-turn, circular loop of radius R is coaxial with a long solenoid of radius r and length l and having N turns. The variable resistor is changed so that the solenoid current decreases linearly from I 1 to I 2 in an interval t. Find the induced emf in the loop. N B t µ o I t l () = () N l 2 2 () t πr B() t µ πr I () t Φ = = o ( ) N ( ) dφ t di t N I I ε = = µ oπr = µ πr l l t o 19

20 Example 3 A square coil (20.0 cm 20.0 cm) that consists of 100 turns of wire rotates about a vertical axis at rev/min. The horizontal component of the Earth s magnetic field at the location of the coil is T. Calculate the maximum emf induced in the coil by this field. Φ = BAcosθ ε = d( BAcos θ ) θ = ωt d(cos ωt) ε = BA = BAωsinωt ε max = BAω = 12.6mV 20

21 Chapter 32 Induction 21

22 Self-Inductance When the switch is closed, the current does not immediately reach its maximum value Faraday s law can be used to describe the effect As the current increases with time, the magnetic flux through the circuit loop due to this current also increases with time This corresponding flux due to this current also increases This increasing flux creates an induced emf in the circuit 22

23 Self-Inductance Lenz Law: The direction of the induced emf is such that it would cause an induced current in the loop which would establish a magnetic field opposing the change in the original magnetic field The direction of the induced emf is opposite the direction of the emf of the battery This results in a gradual increase in the current to its final equilibrium value This effect is called self-inductance The emf ε L is called a self-induced emf 23

24 Self-Inductance: Coil Example A current in the coil produces a magnetic field directed toward the left If the current increases, the increasing flux creates an induced emf of the polarity shown in (b) The polarity of the induced emf reverses if the current decreases 24

25 Solenoid Assume a uniformly wound solenoid having N turns and length l The interior magnetic field is N B = µ on I = µ o I The magnetic flux through each turn is Φ = BA = µ B o NA I The magnetic flux through all N turns Φ = N Φ = µ t B o If I depends on time then self-induced emf can found from the Faraday s law dφt ε si = = µ o 2 NA I 2 NAdI 25

26 Solenoid The magnetic flux through all N turns 2 NA Φ t = µ o I = L I Self-induced emf: 2 dφt NAdI di ε si = = µ o = L 26

27 Inductance ε L d I = L Φ = L I L is a constant of proportionality called the inductance of the coil and it depends on the geometry of the coil and other physical characteristics The SI unit of inductance is the henry (H) Named for Joseph Henry V s 1H = 1 A 27

28 Inductor ε L d I = L Φ = L I A circuit element that has a large self-inductance is called an inductor The circuit symbol is We assume the self-inductance of the rest of the circuit is negligible compared to the inductor However, even without a coil, a circuit will have some self-inductance Φ 1 = L1 I Flux through solenoid Φ 2 = L2 I Flux through the loop I L >> L 1 2 I 28

29 The effect of Inductor ε L d I = L Φ=LI The inductance results in a back emf Therefore, the inductor in a circuit opposes changes in current in that circuit 29

30 RL circuit ε L d I = L Φ=LI An RL circuit contains an inductor and a resistor When the switch is closed (at time t = 0), the current begins to increase At the same time, a back emf is induced in the inductor that opposes the original increasing current 30

31 RL circuit ε L d I = L Kirchhoff s loop rule: d I ε I R L = 0 Solution of this equation: I ε R ε R where τ = L/ R - time constant ( Rt L = 1 e ) ( t τ I = 1 e ) 31

32 RL circuit I ε = R ( t τ 1 e ) d I = ε e L t τ 32

33 Chapter 32 Energy Density of Magnetic Field 33

34 Energy of Magnetic Field ε L d I = L I ε = I R+ L d Iε = I R+ LI d 2 I Let U denote the energy stored in the inductor at any time The rate at which the energy is stored is du = d I To find the total energy, integrate and 2 I I U = L I d I = L L I

35 Energy of a Magnetic Field Given U = ½ L I 2 For Solenoid: L = µ n 2 A I = o B µn o B B U = µ on A = A 2 µn o 2µ o Since Al is the volume of the solenoid, the magnetic energy density, u B is u B U B = = A 2µ This applies to any region in which a magnetic field exists (not just the solenoid) 2 o 35

36 Energy of Magnetic and Electric Fields U C Q = C 2 2 U L = L 2 I 2 36

37 Chapter 32 LC Circuit 37

38 LC Circuit A capacitor is connected to an inductor in an LC circuit Assume the capacitor is initially charged and then the switch is closed Assume no resistance and no energy losses to radiation 38

39 LC Circuit With zero resistance, no energy is transformed into internal energy The capacitor is fully charged The energy U in the circuit is stored in the electric field of the capacitor The energy is equal to Q 2 max / 2C The current in the circuit is zero No energy is stored in the inductor The switch is closed 39

40 LC Circuit I = dq The current is equal to the rate at which the charge changes on the capacitor As the capacitor discharges, the energy stored in the electric field decreases Since there is now a current, some energy is stored in the magnetic field of the inductor Energy is transferred from the electric field to the magnetic field 40

41 LC circuit I max Q = 0 I = dq The capacitor becomes fully discharged It stores no energy All of the energy is stored in the magnetic field of the inductor The current reaches its maximum value The current now decreases in magnitude, recharging the capacitor with its plates having opposite their initial polarity 41

42 LC circuit I = dq Eventually the capacitor becomes fully charged and the cycle repeats The energy continues to oscillate between the inductor and the capacitor The total energy stored in the LC circuit remains constant in time and equals 2 Q 1 2 U = U 2 2 I C + UL = + L C 42

43 LC circuit Q C = L di I = dq Q C Solution: = L 2 d Q 2 Q = Q cos max ( ωt + φ ) Q max C cos 2 ( ωt + φ) = LQ ω cos ( ωt + φ) max ω = LC It is the natural frequency of oscillation of the circuit

44 LC circuit Q = Q cos max ( ωt + φ ) ω = 2 1 LC The current can be expressed as a function of time I = dq = ωqmax sin( ωt + φ ) The total energy can be expressed as a function of time Q 1 Q U U U ωt L ωt 2c 2 2c 2 2 = max C + L = cos + Imax sin = max 2 Qmax 1 = L 2c 2 2 I max 44

45 LC circuit Q = Q cos max ( ωt + φ ) I = ωq sin( ωt + φ ) max The charge on the capacitor oscillates between Q max and -Q max The current in the inductor oscillates between I max and -I max Q and I are 90 o out of phase with each other So when Q is a maximum, I is zero, etc. 45

46 LC circuit The energy continually oscillates between the energy stored in the electric and magnetic fields When the total energy is stored in one field, the energy stored in the other field is zero 46

47 LC circuit In actual circuits, there is always some resistance Therefore, there is some energy transformed to internal energy Radiation is also inevitable in this type of circuit The total energy in the circuit continuously decreases as a result of these processes 47

48 Problem 2 A capacitor in a series LC circuit has an initial charge Q max and is being discharged. Find, in terms of L and C, the flux through each of the N turns in the coil, when the charge on the capacitor is Q max /2. The total energy is conserved: 2 2 Qmax Q 1 = + L I 2C 2C 2 2 Q Qmax = Qmax Q Qmax Qmax 3Qmax 1 1 L I = = = 2 2C 2C 2C 4 2C 8C I = 2 3 CL Q max Φ= = 3 L Q LI C 2 max Φ 3L Q Φ 1 = = N C 2N max 48

49 Chapter 31 Maxwell s Equations 49

50 Maxwell s Equations S S E da= B da= E ds q ε Gauss's law ( electric) 0 Gauss's law in magnetism B d s= µ I + ε µ o dφ = B o o o Faraday's law dφ E Ampere-Maxwell law 50

51 Chapter 34 Electromagnetic Waves 51

52 Maxwell Equations Electromagnetic Waves q E da= B da= ε o dφb E ds= B d s= µ I + µ ε 0 o o o dφ E Electromagnetic waves solutions of Maxwell equations Empty space: q = 0, I = 0 E da= 0 B da= 0 dφb E ds= B d s= µ oε o dφ E Solution Electromagnetic Wave 52

53 Plane Electromagnetic Waves Assume EM wave that travel in x-direction Then Electric and Magnetic Fields are orthogonal to x This follows from the first two Maxwell equations E da= 0 B da= 0 53

54 Plane Electromagnetic Waves If Electric Field and Magnetic Field depend only on x and t then the third and the forth Maxwell equations can be rewritten as dφb E ds= B d s= µ oε o dφ E E B B = µ 2 oε o and = µ 2 2 oε o 2 x t x t E 54

55 Plane Electromagnetic Waves E E B B = µ 2 oε o and = µ 2 2 oε o 2 x t x t Solution: E = E cos( kx ωt) max 2 E x = 2 E 2 maxk cos( kx ωt) E t 2 2 = Emaxω cos( kx ωt) 2 E k cos( kx ωt )= µ ε E ω cos( kx ωt) max max k= ω µ ε

56 Plane Electromagnetic Waves E = E cos( kx ωt) max k= ω µ ε 0 0 The angular wave number is k = 2π/λ - λ is the wavelength The angular frequency is ω = 2πƒ - ƒ is the wave frequency c 2 π =2πf µ ε λ = = µ ε c λ = = f µ ε f m/ s speed of light 56

57 Plane Electromagnetic Waves E = E cos( kx ωt) max H = H cos( kx ωt) max c = 1 µ ε 0 0 ω = ck λ = c f Emax ω E = = = c B k B max E and B vary sinusoidally with x 57

58 Time Sequence of Electromagnetic Wave 58

59 Poynting Vector Electromagnetic waves carry energy As they propagate through space, they can transfer that energy to objects in their path The rate of flow of energy in an em wave is described by a vector, S, called the Poynting vector The Poynting vector is defined as 1 S E B µ o 59

60 Poynting Vector The direction of Poynting vector is the direction of propagation Its magnitude varies in time Its magnitude reaches a maximum at the same instant as E and B 1 S E B µ o 60

61 Poynting Vector The magnitude S represents the rate at which energy flows through a unit surface area perpendicular to the direction of the wave propagation This is the power per unit area The SI units of the Poynting vector are J/s. m 2 = W/m 2 1 S E B µ o 61

62 The EM spectrum Note the overlap between different types of waves Visible light is a small portion of the spectrum Types are distinguished by frequency or wavelength 62

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