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Chapter 29 Electromagnetic Induction
• Consider Faraday’s
Law
• Consider Lenz’s Law
• Study motional emf
• Explore induced electric
fields
• Eddy currents
• Displacement current
• Introduction to
Maxwell’s equations
• Superconductivity and
Magnetism
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Induced current Experiments
– A changing magnetic flux causes an induced current.
– The induced emf is the corresponding emf causing the current.
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Faraday’s Law of Induction
The induced emf is proportional to the rate
of change of magnetic flux ,  B ,through
the coil.
•B, the magnetic field can change with
time.
•A, the area can change with time
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Electromagnetic Induction (Chapter 29, Sec 2)
Example 29-1
EMF and Current Induced in a Loop
dB
 0.020T / s
dt
A = 120 cm2
R (loop resistance) = 5 
Determine induced voltage (VEMF = Vab).
Determine induced current (I).
Verify current direction using Lenz’s Law.
Figure 29-4
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Magnetic flux through an area element
– Figure 29.3 below shows how to calculate the magnetic
flux through an element of area.
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Faraday’s Law Area and Magnetic Flux Direction
Regardless of what moves, knowing the magnetic
flux around a conducting entity will allow
determination of current induced.

B
  B  d A   BdA cos 
For a uniform B field

B
Faraday’s Law of Induction
 B  A  BA cos 
d
   B
dt
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Direction of the induced emf
• Follow the text discussion on the direction of the induced emf,
using Figure 29.6 below.
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Magnitude and direction of an induced emf
• Read Problem-Solving Strategy 29.1.
• Follow Example 29.2 using Figure 29.7 below.
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A simple alternator
• Follow Example 29.3 using
Figures 29.8 (below) and
29.9 (right).
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A Simple Alternator (generator)
 = t, where  is the angular velocity of the loop.
 B  BA cos   BA cos t (B is the total magnetic flux penetrating area A )
d B
  Vab  
 BA sin t
dt
d B
 NBA sin t
For an N-turn coil, Vab   N
dt
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DC generator and the average back emf
• The back emf in a motor is the absolute value of the alternator emf.
  Vab   N
dB
 N BA sin t
dt
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A conducting rod moving in a uniform magnetic field
dA
dA  Lvdt
 
 B   B  dA   B cos dA
cos   1
d B  B cos dA  BdA
d B  BdA  BLvdt
d B
 BLv  VEMF
dt
Figure 29-10
Faraday’s Law of Induction
The induced emf () in a closed loop equals the negative of the time rate of
change of magnetic flux through the loop.
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Work and power in the slidewire generator
• Follow Example 29.6 using Figure 29.12 below.
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Changing the Coil Shape
Change the shape of the coil and you change the shape
of the magnetic flux. The rate at which you change the
shape is the rate of change of the magnetic flux
This induces a current in the coil to oppose the change
in flux.
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Lenz’s Law
•The direction of any magnetic induction effect is such as to oppose the
cause of the effect.
•The “cause” can be a change flux, or the motion of a conductor.
•In all cases the induced current tries to prevent the status quo by opposing
motion or change of flux.
•The induced current due to the
change in B is clockwise as seen
from above the loop. (RHR
opposite direction of fingers)
•The Binduced that it caused is
downward, opposite the change
in the upward field B.
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Lenz’s Law I – Counter clockwise induced current
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Lenz’s Law II – clockwise induced current
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Motional electromotive force
• The motional electromotive force across the ends of a rod
moving perpendicular to a magnetic field is  = vBL. Figure
29.15 below shows the direction of the induced current.
• Follow the general form of motional emf in the text.
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Motional Electromotive Force
•As the rod moves the magnetic field causes positive
free charge to go to move to one end and negative
charge to the other.
•Equilibrium is established when the magnetic force
equals the electric force.
B
E

 
FB  q(v xB)
At equilibrium
qE  qvB sin 
E  vBsin 
FE  FB
FB  qvB sin 


FE  qE
FE  qE
for  = 90°
  Vab  EL  vBL
(29.6)
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A slidewire generator and a dynamo
• Follow Example 29.9 for the slidewire generator.
• Follow Example 29.10 for the Faraday disk dynamo, using
Figure 29.16 below.
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Changing Magnetic fields will Induced electric fields
• The windings of a long solenoid carrying a current I
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Eddy currents
• Induced currents produce
eddy currents that oppose the
induced currents (and the
motion of the disk)
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Using eddy currents
• Figure 29.20 below illustrates an airport metal detector and a
portable metal detector, both of which use eddy currents in their
design.
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Displacement current and Maxwell’s equations
• A varying electric field will give rise to a magnetic field.
• Explains electromagnetic waves.
Displacement current iD
between plates
Proof of iD is the
measure magnetic field
between plates
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Maxwell’s equations
• Maxwell’s equations consist of
Gauss’s law for the electric field
Gauss’s law for the magnetic field
Ampere’s law
Faraday’s law.
• Follow the text discussion for the mathematical form
of these four fundamental laws.
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Superconductivity
• When a superconductor is
cooled below its critical
temperature, it loses all
electrical resistance.
• Follow the text discussion
using Figures 29.23
(below) and 29.24 (right).
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Q29.1
A circular loop of wire is in a region of
spatially uniform magnetic field. The
magnetic field is directed into the
plane of the figure. If the magnetic
field magnitude is constant,
A. the induced emf is clockwise.
B. the induced emf is counterclockwise.
C. the induced emf is zero.
D. The answer depends on the strength of the field.
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Q29.2
A circular loop of wire is in a region of
spatially uniform magnetic field. The
magnetic field is directed into the
plane of the figure. If the magnetic
field magnitude is decreasing,
A. the induced emf is clockwise.
B. the induced emf is counterclockwise.
C. the induced emf is zero.
D. The answer depends on the strength of the field.
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Q29.3
A circular loop of wire is placed next
to a long straight wire. The current I in
the long straight wire is increasing.
What current does this induce in the
circular loop?
A. a clockwise current
B. a counterclockwise current
C. zero current
D. not enough information given to decide
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Q29.4
A flexible loop of wire lies in a
uniform magnetic field of magnitude
B directed into the plane of the
picture. The loop is pulled as shown,
reducing its area. The induced
current
A. flows downward through resistor R and is proportional to B.
B. flows upward through resistor R and is proportional to B.
C. flows downward through resistor R and is proportional to B2.
D. flows upward through resistor R and is proportional to B2.
E. none of the above
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Q29.5
The rectangular loop of wire is being moved
to the right at constant velocity. A constant
current I flows in the long straight wire in
the direction shown. The current induced in
the loop is
A. clockwise and proportional to I.
B. counterclockwise and proportional to I.
C. clockwise and proportional to I2.
D. counterclockwise and proportional to I2.
E. zero.
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Q29.6
The loop of wire is being moved to the right
at constant velocity. A constant current I
flows in the long straight wire in the
direction shown. The current induced in the
loop is
A. clockwise and proportional to I.
B. counterclockwise and proportional to I.
C. clockwise and proportional to I2.
D. counterclockwise and proportional to I2.
E. zero.
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Q29.7
The rectangular loop of wire is being moved
to the right at constant velocity. A constant
current I flows in the long wire in the
direction shown. What are the directions of
the magnetic forces on the left-hand (L) and
right-hand (R) sides of the loop?
A. L: to the left; R: to the left
B. L: to the left; R: to the right
C. L: to the right; R: to the left
D. L: to the right; R: to the right
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Q29.8
The drawing shows the uniform magnetic field
inside a long, straight solenoid. The field is
directed into the plane of the drawing, and is
increasing.
What is the direction of the electric force on a
positive point charge placed at point a?
A. to the left
B. to the right
C. straight up
D. straight down
E. misleading question — the electric force at this point is zero
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Q29.9
The drawing shows the uniform magnetic field
inside a long, straight solenoid. The field is
directed into the plane of the drawing, and is
increasing.
What is the direction of the electric force on a
positive point charge placed at point b?
A. to the left
B. to the right
C. straight up
D. straight down
E. misleading question — the electric force at this point is zero
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Q29.10
The drawing shows the uniform magnetic field
inside a long, straight solenoid. The field is
directed into the plane of the drawing, and is
increasing.
What is the direction of the electric force on a
positive point charge placed at point c (at the
center of the solenoid)?
A. to the left
B. to the right
C. straight up
D. straight down
E. misleading question — the electric force at this point is zero
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