Download 3.4 Faraday`s Law

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.
1
ECT1026 Field Theory
2007/2008
In the previous lectures
Electric Current  Magnetic Field
How to determine the magnetic field?
Biot-Savart Law
Long Straight Wire
Pie-shaped Wire Loop
Circular Loop Wire
Ampere’s Law
Long Straight Wire
Long Solenoid
Toroid
Magnetic Field  Electric Current
?
2
ECT1026 Field Theory
3.5 Faraday’s Law
2007/2008
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
3
ECT1026 Field Theory
3.5 Faraday’s Law
2007/2008
First Experiments
Conducting
loop
Sensitive
current meter
Since there is no battery or
other source of emf included,
there is no current in the
circuit
Move a bar magnet toward
the loop, a current suddenly
appears in the circuit
The current disappears
when the bar magnet stops
If we then move the bar magnet away, a
current again suddenly appears, but now
in the opposite direction
4
ECT1026 Field Theory
3.5 Faraday’s Law
2007/2008
Discovering of the First Experiments
1. 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.
5
ECT1026 Field Theory
2007/2008
3.5 Faraday’s Law
An Experiment - Situation A
Constant flux
though the loop
Current in the
coil produces a
magnetic field B
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
6
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.
7
ECT1026 Field Theory
2007/2008
3.5 Faraday’s Law
An Experiment - Situation C: Reconnect Battery
Sudden change
of magnetic flux
through the loop
Deflection of
Galvanometer needle
in the opposite
direction
Magnetic field
becomes nonzero
Current in the
coil produces a
magnetic field B
Link: http://micro.magnet.fsu.edu/electromag/java/faraday/index.html
8
ECT1026 Field Theory
3.5 Faraday’s Law
2007/2008
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
9
ECT1026 Field Theory
3.5 Faraday’s Law
2007/2008
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.
10
ECT1026 Field Theory
3.5 Faraday’s Law
2007/2008
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.
11
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
12
ECT1026 Field Theory
3.5 Faraday’s Law
2007/2008
Minus Sign  Lenz’s Law
The induced voltage acts to produce an opposing flux
13
ECT1026 Field Theory
3.5 Faraday’s Law
2007/2008
Minus Sign  Lenz’s Law
The induced voltage acts to produce an opposing flux
14
ECT1026 Field Theory
3.5 Faraday’s Law
2007/2008
Minus Sign  Lenz’s Law
The induced voltage acts to produce an opposing flux
15
ECT1026 Field Theory
3.5 Faraday’s Law
2007/2008
If the closed path is taken by an N-turn filamentary conductors
Magnetic flux F?
The magnetic flux F linking a surface S is defined as the
total magnetic flux density B passing through S:
(Wb)

16
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
17
ECT1026 Field Theory
3.5 Faraday’s Law
2007/2008
Faraday’s Law (B and E fields)
– Stationary Loop in a Time-Varying Magnetic Field
– Moving Conductor in a Static Magnetic Field
18
ECT1026 Field Theory
3.5 Faraday’s Law
2007/2008
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


19
ECT1026 Field Theory
3.5 Faraday’s Law
2007/2008
If B is not time-varying, i.e.
OR
Maxwell’s Eqn of Electrostatic
20
ECT1026 Field Theory
3.5 Faraday’s Law
2007/2008
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:
21
ECT1026 Field Theory
3.5 Faraday’s Law
2007/2008
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
22
ECT1026 Field Theory
3.5 Faraday’s Law
2007/2008
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

Voltage induced: motional emf,
Induces a voltage difference
between ends 1 and 2
End 2 being at the higher potential
23
ECT1026 Field Theory
3.5 Faraday’s Law
2007/2008
For the conducting wire:


24
ECT1026 Field Theory
3.5 Faraday’s Law
2007/2008
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
25
ECT1026 Field Theory
2007/2008
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
Middle Finger
Direction of Induced
emf or Current Flow
Thumb
Direction of
Conductor Motion
26
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).
27
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.
28
ECT1026 Field Theory
3.5 Faraday’s Law
2007/2008
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.
29
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
x̂
30
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:
x̂
31
ECT1026 Field Theory
3.5 Faraday’s Law
2007/2008
Example 3.5-1:
At y2 = 2.5 m
Induced voltage V43 is:
Current in the circuit is:
32
ECT1026 Field Theory
3.5 Faraday’s Law
2007/2008
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.
33
ECT1026 Field Theory
2007/2008
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

34