1. Faraday’s Law and Induction
Chapter 23
Faraday’s Law
and
Induction
Re-arrangement By Thang Hai Book Publishing Co.,

2. In a commercial electric power plant, large generators transform energy that is transferred out of the plant by electrical transmission. These generators use magnetic induction to generate a potential difference when coils of wire in the generator are rotated in a magnetic field. The source of energy to rotate the coils might be falling water, burning fossil fuels, or a nuclear reaction.
Fig. 23-CO, p. 765

3. 23.1 Michael Faraday 1791 – 1867 Great experimental physicist
Contributions to early electricity include
Invention of motor, generator, and transformer
Electromagnetic induction
Laws of electrolysis

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

5. Figure 23.1: A straight electrical conductor of length ℓ moving with a velocity v→ through a uniform magnetic field B→ directed perpendicular to v→. A current is induced in the conductor due to the magnetic force on charged particles in the conductor.
Fig. 23-1, p. 766

6. EMF Produced by a Changing Magnetic Field, 1
Fig 23.2
A loop of wire is connected to a sensitive ammeter
When a magnet is moved toward the loop, the ammeter deflects
The deflection indicates a current induced in the wire

7. EMF Produced by a Changing Magnetic Field, 2
Fig 23.2
When the magnet is held stationary, there is no deflection of the ammeter
Therefore, there is no induced current
Even though the magnet is inside the loop

8. EMF Produced by a Changing Magnetic Field, 3
Fig 23.2
The magnet is moved away from the loop
The ammeter deflects in the opposite direction

9. Active Figure
23.2
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10. EMF Produced by a Changing Magnetic Field, Summary
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
An electric current is set up in the coil as long as relative motion occurs between the magnet and the coil
This is the induced current that is produced by an induced emf

11. Faraday’s Experiment – Set Up
A primary coil is connected to a switch and a battery
The wire is wrapped around an iron ring
A secondary coil is also wrapped around the iron ring
There is no battery present in the secondary coil
The secondary coil is not electrically connected to the primary coil
Fig 23.3

12. Active Figure
23.3
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13. Faraday’s Experiment – Findings
At the instant the switch is closed, the galvanometer (ammeter) needle deflects in one direction and then returns to zero
When the switch is opened, the galvanometer needle deflects in the opposite direction and then returns to zero
The galvanometer reads zero when there is a steady current or when there is no current in the primary circuit

14. Faraday’s Experiment – Conclusions
An electric current can be produced by a time-varying magnetic field
This would be the current in the secondary circuit of this experimental set-up
The induced current exists only for a short time while the magnetic field is changing
This is generally expressed as: an induced emf is produced in the secondary circuit by the changing magnetic field
The actual existence of the magnetic field is not sufficient to produce the induced emf, the field must be changing

15. Magnetic Flux
To express Faraday’s finding mathematically, the magnetic flux is used
The flux depends on the magnetic field and the area:
Fig 23.4

16. Flux and Induced emf
An emf is induced in a circuit when the magnetic flux through the surface bounded by the circuit changes with time
This summarizes Faraday’s experimental results

17. Faraday’s Law – Statements
Faraday’s Law of Induction states that the emf induced in a circuit is equal to the time rate of change of the magnetic flux through the circuit
Mathematically,

18. Faraday’s Law – Statements, cont
If the circuit consists of N identical and concentric loops, and if the field lines pass through all loops, the induced emf becomes
The loops are in series, so the emfs in the individual loops add to give the total emf

19. Faraday’s Law – Example
Assume a loop enclosing an area A lies in a uniform magnetic field
The magnetic flux through the loop is FB = B A cos q
The induced emf is
Fig 23.5

20. Ways of Inducing an emf
The magnitude of the field can change with time
The area enclosed by the loop can change with time
The angle q between the field and the normal to the loop can change with time
Any combination of the above can occur

21. Applications of Faraday’s Law – Pickup Coil
The pickup coil of an electric guitar uses Faraday’s Law
The coil is placed near the vibrating string and causes a portion of the string to become magnetized
When the string vibrates at the some frequency, the magnetized segment produces a changing flux through the coil
The induced emf is fed to an amplifier
Fig 23.6

22. Figure 23.6b: The pickups (the circles beneath the metallic strings) of this electric guitar detect the vibrations of the strings and send this information through an amplifier and into speakers. (A switch on the guitar allows the musician to select which set of six pickups is used.)
Fig. 23-6b, p. 769

23. Figure 23. 8: (Thinking Physics 23
Figure 23.8: (Thinking Physics 23.1) Essential components of a ground fault interrupter.
Fig. 23-8, p. 769

27. Figure 23.9: (Example 23.2) Exponential decrease in the magnitude of the magnetic field with time. The induced emf and induced current vary with time in the same way.
Fig. 23-9, p. 770

29. 23.2 Motional emf
A motional emf is one induced in a conductor moving through a magnetic field
The electrons in the conductor experience a force that is directed along l
Fig 23.10

30. Motional emf, cont
Under the influence of the force, the electrons move to the lower end of the conductor and accumulate there
As a result of the charge separation, an electric field 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

31. Motional emf, final For equilibrium, q E = q v B or E = v B
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

32. Sliding Conducting Bar
Fig 23.11
A bar moving through a uniform field and the equivalent circuit diagram
Assume the bar has zero resistance
The work done by the applied force appears as internal energy in the resistor R

33. Sliding Conducting Bar, cont
The induced emf is
Since the resistance in the circuit is R, the current is

34. Active Figure
23.11
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35. Sliding Conducting Bar, Energy Considerations
The applied force does work on the conducting bar
This moves the charges through a magnetic field
The change in energy of the system during some time interval must be equal to the transfer of energy into the system by work
The power input is equal to the rate at which energy is delivered to the resistor

46. AC Generators
Electric generators take in energy by work and transfer it out by electrical transmission
The AC generator consists of a loop of wire rotated by some external means in a magnetic field
Fig 23.14

47. Induced emf In an AC Generator
The induced emf in the loops is
This is sinusoidal, with emax = N A B w
Fig 23.14

48. Active Figure
23.14
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49. 23.3 Lenz’ Law
Faraday’s Law indicates 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’ Law
It was developed by a German physicist, Heinrich Lenz

50. Lenz’ Law, cont
Lenz’ Law states the polarity of the induced emf in a loop is such that it produces a current whose magnetic field opposes the change in magnetic flux through the loop
The induced current tends to keep the original magnetic flux through the circuit from changing

51. Figure 23. 15: (Thinking Physics 23
Figure 23.15: (Thinking Physics 23.2) An ideal transformer consists of two coils of wire wound on the same iron core. An alternating voltage ΔV1 is applied to the primary coil, and the output voltage ΔV2 appears across the resistance R.
Fig , p. 775

52. Figure 23.16: (a) As the conducting bar slides on the two fixed conducting rails, the flux due to the magnetic field directed inward through the area enclosed by the loop increases in time. By Lenz’s law, the induced current must be counterclockwise so as to produce a counteracting magnetic field directed outward from the page. (b) When the bar moves to the left, the induced current must be clockwise. Why?
Fig , p. 776

53. Lenz’ Law – Example 1
Fig 23.17
When the magnet is moved toward the stationary loop, a current is induced as shown in a
This induced current produces its own magnetic field that is directed as shown in b to counteract the increasing external flux

54. Lenz’ Law – Example 2
Fig 23.18
When the magnet is moved away the stationary loop, a current is induced as shown in c
This induced current produces its own magnetic field that is directed as shown in d to counteract the decreasing external flux

58. 23.5 Induced emf and Electric Fields
An electric field is created in the conductor as a result of the changing magnetic flux
Even in the absence of a conducting loop, a changing magnetic field will generate an electric field in empty space
This induced electric field has different properties than a field produced by stationary charges

59. Figure 23.20: A conducting loop of radius r in a uniform magnetic field perpendicular to the plane of the loop. If B→ changes in time, an electric field is induced in a direction tangent to the loop.
Fig , p. 778

60. Induced emf and Electric Fields, cont
The emf for any closed path can be expressed as the line integral of over the path
Faraday’s Law can be written in a general form

61. Induced emf and Electric Fields, final
The induced electric field is a nonconservative field that is generated by a changing magnetic field
The field cannot be an electrostatic field because if the field were electrostatic, and hence conservative, the line integral would be zero and it isn’t

69. 23.5 Self-Induction
When the switch is closed, the current does not immediately reach its maximum value
Faraday’s Law can be used to describe the effect

70. Self-Induction, 2
As the current increases with time, the magnetic flux through the circuit loop due to this current also increases with time
The corresponding flux due to this current also increases
This increasing flux creates an induced emf in the circuit

71. Self-Inductance, 3
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
Sometimes called a back emf
This results in a gradual increase in the current to its final equilibrium value

72. Self-Induction, 4 This effect is called self-inductance
Because the changing flux through the circuit and the resultant induced emf arise from the circuit itself
The emf eL is called a self-induced emf

73. Self-Inductance, Equations
An induced emf is always proportional to the time rate of change of the current
L is a constant of proportionality called the inductance of the coil
It depends on the geometry of the coil and other physical characteristics

74. Inductance of a Coil
A closely spaced coil of N turns carrying current I has an inductance of
The inductance is a measure of the opposition to a change in current
Compared to resistance which was opposition to the current

75. Inductance Units The SI unit of inductance is a Henry (H)
Named for Joseph Henry

76. Joseph Henry 1797 – 1878 Improved the design of the electromagnet
Constructed one of the first motors
Discovered the phenomena of self-inductance

77. Inductance of a Solenoid
Assume a uniformly wound solenoid having N turns and length l
Assume l is much greater than the radius of the solenoid
The interior magnetic field is

78. Inductance of a Solenoid, cont
The magnetic flux through each turn is
Therefore, the inductance is
This shows that L depends on the geometry of the object

83. 23.6 RL Circuit, Introduction
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

84. RL Circuit, Analysis 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
Fig 23.23

85. Active Figure
23.23
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86. RL Circuit, Analysis, cont
Applying Kirchhoff’s Loop Rule to the previous circuit gives
Looking at the current, we find

87. RL Circuit, Analysis, Final
The inductor affects the current exponentially
The current does not instantly increase to its final equilibrium value
If there is no inductor, the exponential term goes to zero and the current would instantaneously reach its maximum value as expected

88. RL Circuit, Time Constant
The expression for the current can also be expressed in terms of the time constant, t, of the circuit
where t = L / R
Physically, t is the time required for the current to reach 63.2% of its maximum value

89. RL Circuit, Current-Time Graph, 1
The equilibrium value of the current is e/R and is reached as t approaches infinity
The current initially increases very rapidly
The current then gradually approaches the equilibrium value
Fig 23.24

90. RL Circuit, Current-Time Graph, 2
The time rate of change of the current is a maximum at t = 0
It falls off exponentially as t approaches infinity
In general,
Fig 23.25

91. Active Figure
23.26
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95. 23.7 Energy in a Magnetic Field
In a circuit with an inductor, the battery must supply more energy than in a circuit without an inductor
Part of the energy supplied by the battery appears as internal energy in the resistor
The remaining energy is stored in the magnetic field of the inductor

96. Energy in a Magnetic Field, cont
Looking at this energy (in terms of rate)
Ie is the rate at which energy is being supplied by the battery
I2R is the rate at which the energy is being delivered to the resistor
Therefore, LI dI/dt must be the rate at which the energy is being delivered to the inductor

97. Energy in a Magnetic Field, final
Let U denote the energy stored in the inductor at any time
The rate at which the energy is stored is
To find the total energy, integrate and UB = ½ L I2

98. Energy Density of a Magnetic Field
Given U = ½ L I2,
Since Al is the volume of the solenoid, the magnetic energy density, uB is
This applies to any region in which a magnetic field exists
not just the solenoid

103. Inductance Example – Coaxial Cable
Calculate L and energy for the cable
The total flux is
Therefore, L is
The total energy is
Fig 23.32

107. 23.8 Magnetic Levitation – Repulsive Model
A second major model for magnetic levitation is the EDS (electrodynamic system) model
The system uses superconducting magnets
This results in improved energy effieciency

108. Magnetic Levitation – Repulsive Model, 2
The vehicle carries a magnet
As the magnet passes over a metal plate that runs along the center of the track, currents are induced in the plate
The result is a repulsive force
This force tends to lift the vehicle
There is a large amount of metal required
Makes it very expensive

109. Japan’s Maglev Vehicle
The current is induced by magnets passing by coils located on the side of the railway chamber
Fig 23.33

110. EDS Advantages Includes a natural stabilizing feature
If the vehicle drops, the repulsion becomes stronger, pushing the vehicle back up
If the vehicle rises, the force decreases and it drops back down
Larger separation than EMS
About 10 cm compared to 10 mm

111. EDS Disadvantages Levitation only exists while the train is in motion
Depends on a change in the magnetic flux
Must include landing wheels for stopping and starting
The induced currents produce a drag force as well as a lift force
High speeds minimize the drag
Significant drag at low speeds must be overcome every time the vehicle starts up