1. Lecture 3-5 Faraday’ s Law (pg. 24 – 35)
ECT1026 Field Theory
Lecture 3-5
Faraday’ s Law
(pg. 24 – 35)
In 1831, Michael Faraday discovers that a changing magnetic flux can induce an electromotive force.

2. ? Magnetic Field  Electric Current In the previous lectures
ECT1026 Field Theory
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In the previous lectures
Electric Current  Magnetic Field
How to determine the magnetic field?
Biot-Savart Law
Ampere’s Law
Long Straight Wire
Pie-shaped Wire Loop
Circular Loop Wire
Long Straight Wire
Long Solenoid
Toroid
Magnetic Field  Electric Current ?

3. This mechanism is called “Electromagnetic Induction”
ECT1026 Field Theory
2007/2008
3.5 Faraday’s Law
Magnetic Field can produce an electric current in a closed loop, if the magnetic flux linking the surface area of the loop changes with time.
This mechanism is called “Electromagnetic Induction”
The electric Current Produced  Induced Current

4. Sensitive current meter
ECT1026 Field Theory
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3.5 Faraday’s Law
First Experiments
Move a bar magnet toward the loop, a current suddenly appears in the circuit
Conducting loop
The current disappears when the bar magnet stops
Sensitive current meter
If we then move the bar magnet away, a current again suddenly appears, but now in the opposite direction
Since there is no battery or other source of emf included, there is no current in the circuit

5. Discovering of the First Experiments
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3.5 Faraday’s Law
Discovering of the First Experiments
A current appears only if there is relative motion between the loop and the magnet
2. Faster motion produces a greater current
3. If moving the magnet’s N-pole towards the loop causes clockwise current, then moving the N-pole away causes counterclockwise.

6. Current in the coil produces a magnetic field B
ECT1026 Field Theory
2007/2008
3.5 Faraday’s Law
An Experiment - Situation A
Current in the coil produces a magnetic field B
Constant flux though the loop
DC current I, in coil produces a constant magnetic field, in turn produces a constant flux though the loop
Constant flux, no current is induced in the loop. No current detected by Galvanometer

7. Magnetic field drops to zero
ECT1026 Field Theory
2007/2008
3.5 Faraday’s Law
An Experiment - Situation B: Disconnect battery suddenly
Magnetic field drops to zero
Deflection of Galvanometer needle
Sudden change of magnetic flux to zero causes a
momentarily deflection of Galvanometer needle.

8. An Experiment - Situation C: Reconnect Battery
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2007/2008
3.5 Faraday’s Law
An Experiment - Situation C: Reconnect Battery
Current in the coil produces a magnetic field B
Sudden change of magnetic flux through the loop
Magnetic field becomes non-zero
Deflection of Galvanometer needle in the opposite direction
Link:

9. Conclusions from the experiment
ECT1026 Field Theory
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3.5 Faraday’s Law
Conclusions from the experiment
Current induced in the closed loop when magnetic flux changes, and direction of current depends on whether flux is increasing or decreasing
If the loop is turned or moved closer or away from the coil, the physical movement changes the magnetic flux linking its surface, produces a current in the loop, even though B has not changed
In Technical Terms
Time-varying magnetic field produces an electromotive force (emf) which establish a current in the closed circuit

10. Electromotive force (emf) can be obtained through the following ways:
ECT1026 Field Theory
2007/2008
3.5 Faraday’s Law
Electromotive force (emf) can be obtained through the following ways:
1. A time-varying flux linking a stationary closed path. (i.e. Transformer)
2. Relative motion between a steady flux and a close path. (i.e. D.C. Generator)
3. A combination of the two above, both flux changing and conductor moving simultaneously. A closed path may consists of a conductor, a capacitor or an imaginary line in space, etc.

11. ECT1026 Field Theory
2007/2008
3.5 Faraday’s Law
Faraday summarized this electromagnetic phenomenon into two laws ,which are called the Faraday’s law
Faraday’s First Law
When the flux magnet linked to a circuit changes, an electromotive force (emf) will be induced.

12. ECT1026 Field Theory
2007/2008
3.5 Faraday’s Law
Faraday’s Second Law
The magnetic of emf induced is equal to the time rate of change of the linked magnetic flux F.
(volts)
Minus Sign  Lenz’s Law
Indicates that the emf induced is in such a direction as to produces a current whose flux, if added to the original flux, would reduce the magnitude of the emf

13. The induced voltage acts to produce an opposing flux
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2007/2008
3.5 Faraday’s Law
Minus Sign  Lenz’s Law
The induced voltage acts to produce an opposing flux

14. The induced voltage acts to produce an opposing flux
ECT1026 Field Theory
2007/2008
3.5 Faraday’s Law
Minus Sign  Lenz’s Law
The induced voltage acts to produce an opposing flux

15. The induced voltage acts to produce an opposing flux
ECT1026 Field Theory
2007/2008
3.5 Faraday’s Law
Minus Sign  Lenz’s Law
The induced voltage acts to produce an opposing flux

16. If the closed path is taken by an N-turn filamentary conductors
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3.5 Faraday’s Law
If the closed path is taken by an N-turn filamentary conductors
Magnetic flux ?
The magnetic flux  linking a surface S is defined as the total magnetic flux density B passing through S:
(Wb) 

17. ECT1026 Field Theory
2007/2008
3.5 Faraday’s Law
From Chapter 2 Electrostatics – Part B (pg 4-5)
For a closed loop with contour C, the emf is defined by:
Take N = 1 
In Electrostatics – an electric field intensity E due to static charge distribution must lead to zero potential difference about a closed path.
Here – the line integral leads to a potential difference with time-varying magnetic fields, the results is an emf or a voltage

18. Faraday’s Law (B and E fields)
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3.5 Faraday’s Law
Faraday’s Law (B and E fields)
Stationary Loop in a Time-Varying Magnetic Field
Moving Conductor in a Static Magnetic Field

19. Since the loop is stationary, d/dt operates on B(t) only
ECT1026 Field Theory
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3.5 Faraday’s Law
Stationary Loop in a Time-Varying Magnetic Field
A single-turn (N =1), conducting loop is placed in a time-varying magnetic field B(t).
Since the loop is stationary, d/dt operates on B(t) only 
Applying Stoke’s theorem to the closed line integral  

20. Maxwell’s Eqn of Electrostatic
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3.5 Faraday’s Law
If B is not time-varying, i.e. OR
Maxwell’s Eqn of Electrostatic

21. Moving Conductor in a Static Magnetic Field
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3.5 Faraday’s Law
Moving Conductor in a Static Magnetic Field
A wire with length l moving across a static magnetic field
at a constant velocity u (points to x).
The conducting wire contains free electron.
Magnetic force Fm acting on any charged particle “q” moving with velocity u is:

22. Em is in a direction perpendicular to the plane containing u and B
ECT1026 Field Theory
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3.5 Faraday’s Law
This Fm is equivalent to the electrical force that would be exerted o the particle by an electric field Em given by:
Em is in a direction perpendicular to the plane containing u and B
The electric field Em generated by the motion of the charged particle is called a motional electric field

23. For the wire, Em is along -y
ECT1026 Field Theory
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3.5 Faraday’s Law
For the wire, Em is along -y ^
Magnetic force acting on the electrons in the wire causes them to move in the direction of -Em
i.e. towards the end labeled 1
Induces a voltage difference between ends 1 and 2 
Voltage induced: motional emf,
End 2 being at the higher potential

24. For the conducting wire:
ECT1026 Field Theory
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3.5 Faraday’s Law
For the conducting wire:  

25. ECT1026 Field Theory
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3.5 Faraday’s Law
In general, if any segment of a closed circuit with contour C moves with a velocity u across a static magnetic field B, then the induced motional emf is:
Only those segments of the circuit that cross magnetic field lines contribute to

26. Fleming Right Hand Rule
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3.5 Faraday’s Law
Fleming Right Hand Rule
Direction of Induced e.m.f, Magnetic Flux, Conductor Motion
Fore Finger
Direction of Field Flux
Thumb
Direction of Conductor Motion
Middle Finger
Direction of Induced emf or Current Flow

27. Fleming's right hand rule (for generators)
Fleming's right hand rule shows the direction of induced current flow when a conductor moves in a magnetic field.
The right hand is held with the thumb, first finger and second finger mutually at right angles, as shown in the diagram
The Thumb represents the direction of Motion of the conductor.
The First finger represents the direction of the Field.
The Second finger represents the direction of the induced or generated Current (in the classical direction, from positive to negative).

28. Fleming's left hand rule (for electric motors)
Fleming's left hand rule shows the direction of the thrust on a conductor carrying a current in a magnetic field.
The left hand is held with the thumb, index finger and middle finger mutually at right angles.
The First finger represents the direction of the Field.
The Second finger represents the direction of the Current (in the classical direction, from positive to negative).
The Thumb represents the direction of the Thrust or resultant Motion.

29. Application of Faraday’s Law
ECT1026 Field Theory
2007/2008
3.5 Faraday’s Law
Application of Faraday’s Law
Example 3.5-1:
The rectangular loop shown in the figure is situated in the x-y plane and moves away from the origin at a velocity (m/s) in a
magnetic field given by:
(T)
If R = 5 , find the current I at the instant that the loop sides are at
y1 = 2m and y2= 2.5m .
The loop resistance may be ignored.

30. 3.5 Faraday’s Law Example 3.5-1: The induced voltage V12 is given by:
ECT1026 Field Theory
2007/2008
3.5 Faraday’s Law
Example 3.5-1:
The induced voltage V12 is given by:
Since
is along
Voltage are induced across only the sides oriented along
i.e. sides (1-2) and (3-4)
B decreases exponentially with y

31. 3.5 Faraday’s Law Example 3.5-1: Induced voltage V12 is:
ECT1026 Field Theory
2007/2008
3.5 Faraday’s Law
Example 3.5-1:
The induced voltage V12 is given by:
B decreases exponentially with y
At y1 = 2 m
Induced voltage V12 is:

32. 3.5 Faraday’s Law Example 3.5-1: Induced voltage V43 is: At y2 = 2.5 m
ECT1026 Field Theory
2007/2008
3.5 Faraday’s Law
Example 3.5-1:
At y2 = 2.5 m
Induced voltage V43 is:
Current in the circuit is:

33. Example 3.5-2: AC Generator
ECT1026 Field Theory
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3.5 Faraday’s Law
Example 3.5-2: AC Generator
The Faraday’s Law is the principle at work in an electric generator.
The essential design is a conducting coil rotating in the magnetic field
of a fixed magnet.

34. Example 3.5-2: AC Generator
ECT1026 Field Theory
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3.5 Faraday’s Law
Example 3.5-2: AC Generator
For constant angular velocity, the magnetic flux through the coil area A is:
Conducting Coil B 