Download Electromagnetic Induction and Faraday`s Law

Slide 1 / 50
Electromagnetic Induction and
Faraday’s Law
Slide 2 / 50
Electromagnetic Induction and
Faraday’s Law
Induced EMF
Faraday’s Law of Induction
Lenz’s Law
EMF Induced in a Moving Conductor
Changing Magnetic Flux Produces an Electric Field
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Induced EMF
Almost 200 years ago, Faraday looked for evidence that a
magnetic field would induce an electric current with this
apparatus:
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Induced EMF
He found no evidence when the current was steady, but
did see a current induced when the switch was turned
on or off.
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Faraday’s Law of Induction; Lenz’s Law
The induced emf in a wire loop is proportional to the rate of
change of magnetic flux through the loop.
Magnetic flux is defined by: FB = B^A
Unit of magnetic flux: weber, Wb.
1 Wb = 1 T m2
F is the Greek letter "phi" and stands for "flux",
or flow.
FB represents magnetic flux, or the flow of
magnetic field through a surface.
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1
What is the flux through a loop
of wire if a magnetic field of 0.40
T is perpendicular to it and its
area is 5.0 m2? (always use 2
significant figures)
FB = B^A
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2
What is the flux through a loop of
wire if a magnetic field of 0.30 T is
perpendicular to it and its radius
is 2.0 m? (always use 2
significant figures)
FB = B^A
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In the below diagram, the magnetic field (blue) is
perpendicular to the plane of the loop of wire (orange) and
parallel to its normal (red) so the flux is just given by # B= BA.
If the flux passes directly though the loop of wire,
perpendicular to it, then it is a maximum and # B= BA.
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In this case, the magnetic field (blue) is parallel to the plane of
the loop of wire (orange) and perpendicular to its normal, so the
flux is just given by # B = 0.
If the magnetic field lines don't go through the loop of wire,
there is no flux.
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Faraday’s Law of Induction; Lenz’s Law
The magnetic flux is is proportional to the total number of
lines passing through the loop.
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3
What is the magnitude of the
magnetic flux through the loop of
wire shown below. The magnetic
field is 1.0 T and the area of the
loop of wire is 5.0 m2.
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4
What is the magnitude of the
magnetic flux through the loop of
wire shown below. The magnetic
field is 1.0 T and the area of the
loop of wire is 5.0 m2.
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Faraday’s Law of Induction; Lenz’s Law
Faraday’s law of induction:
E is the induced emf (electromovtive force) and is measured
in volts (V)
N is the number of loops of wire in the coil
D# Bo represents the change in the magnetic flux and is
measured in Webers (Wb)
Dt is the time interval during which the flux changed and is
measured in seconds (s)
the "-" sign has to do with the direction of the emf, and will be
discussed later
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Faraday’s Law of Induction; Lenz’s Law
The minus sign tells us the direction of the induced EMF.
Let's get back to that a little later, for right now let's
determine the magnitude of the induced EMF.
For instance, let's figure out the induced emf in a 8.0 m2
coil if it is in a perpendicular 0.40T magnetic field that
disappears over a time interval of 2.0s? (Let's assume
there's just a single loop of wire.)
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What is the induced emf in a 8.0 m2 coil (consisting of one loop) if it is
perpendicular to a 0.40T magnetic field that disappears over a time
interval of 2.0s?
N=1
A = 8.0 m2
B0 = 0.4T
Bf = 0
Dt = 2.0s
# Bo = B0A = (0.4T)(8.0 m2) = 3.2W
# Bf = B0A = (0)(8.0 m2) = 0
D#
E = -N B
Dt
(# Bf - # Bo)
E = -N
Dt
E = -1
(0 - 3.2Wb)
= 1.6V
2.0s
Alternatively...
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What is the induced emf in a 8.0 m2 coil (consisting of one loop) if it is
perpendicular to a 0.40T magnetic field that disappears over a time
interval of 2.0s?
N=1
A = 8.0 m2
B0 = 0.4T
Bf = 0
Dt = 2.0s
D#
E = -N B
Dt
E = -N
DBA
Dt
E = -NA DB
Dt
E = -NA
Bf-B0
Dt
E = -1(8.0 m2)
(0 - 0.40T)
= 1.6V
2.0s
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The second approach works because anything that is not
changing can be put in front of the delta symbol. For instance, if
A is constant while B is changing:
E = -N
D# B
Dt
E = -N
D(BA)
Dt
(BA)f - (BA)0
E = -N
Dt
(BfAf) - (B0A0)
E = -N
Dt
In this case Af = A0 = A this becomes:
(BfA) - (B0A)
E = -N
Dt
Since A is in both terms, it can be factored
out
A(Bf - B0)
E = -N
Dt
But [Bf - B0] is just DB
E = -NA
DB
Dt
The same approach allows me to factor out B if it
is not changing while A is changing
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5
What is the magnitude of the
induced emf in a single loop 2.0m2
coil if it is perpendicular to a 0.50T
magnetic field which is turned off
over a time interval of 4.0s?
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6
What is the magnitude of the
induced emf in a ten loop coil of
wire whose area is 2.0m2 if it is
perpendicular to a magnetic field
which is increased from 0.30T to
1.5T over a time interval of 4.0s?
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Faraday’s Law of Induction; Lenz’s Law
Magnetic flux will change if the area of the loop changes:
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Faraday’s Law of Induction; Lenz’s Law
Magnetic flux will change if the angle between the
loop and the field changes:
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7
A 4.0m2 single loop coil of wire is
initially perpendicular to a 0.60T
magnetic field. It is then rotated so
that it become parallel to that
magnetic field 2.0s later. What is
the magnitude of the induced emf?
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8
A coil of wire, consisting of 50 loops, is
perpendicular to a 1.2T magnetic field.
The area of the coil is increased from
0.40m2 to 1.2m2 over a time of 5.0s.
What is the magnitude of the induced
emf in the coil?
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Lenz’s Law
The minus sign tells us that the direction of the induced emf is
such that the resulting current produces a magnetic field that
resists the change of flux through the loop.
For instance, if the external field gets weaker, the current tries to
replace the "missing" external field.
If the external field gets stronger, the induced current tries to
opposes the "extra" external field.
Only worry about the field within the loop, ignore the field
outside it.
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Lenz’s Law
Initial External
Field (red)
. . . .
. . . . . .
. . . . . .
. . . .
Final External
Field (red)
Plus Field due
to Induced
Current (blue)
. . . .
. . . . . .
. . . . . .
. . . .
The current flows CCW to
create a field out of the
board (dots) to oppose the
change in the external
field
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Lenz’s Law
Initial External
Field (red)
Final External
Field (red)
. .
. . .
. . .
. .
. .
. . .
. . .
. .
Plus Field due
to Induced
Current (blue)
. . . .
. . . . . .
. . . . . .
. . . .
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
The current flows CW to
create a field into the
board (blue "x"s) to
cancel the new external
field (red dots)
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Lenz’s Law
Initial External
Field (red)
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
Final External
Field (red)
. .
. . .
. . .
. .
. .
. . .
. . .
. .
Plus Field due
to Induced
Current (blue)
. . . .
. . . . . .
. . . . . .
. . . .
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x x
x x
x
x
x
x
x
The current flows CW to
create a field into the
board (blue "x"s) to cancel
the new external field (red
dots) and replace the old
external field (red "x"s)
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Faraday’s Law of Induction; Lenz’s Law
Problem Solving: Lenz’s Law
Determine whether the magnetic flux is increasing, decreasing,
or unchanged.
The magnetic field due to the induced current points in the
opposite direction to the original field if the flux is increasing; in
the same direction if it is decreasing; and is zero if the flux is not
changing.
Use the right-hand rule to determine the direction of the current.
Remember that the external field and the field due to the induced
current are different.
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9
A magnetic field is pointing
straight up through a coil of wire.
The field is suddenly turned off.
What is the direction of the
induced current in the wire?
A
Out of the page
B
Into the page
C
Clockwise
D
Counterclockwise
E
There is no induced current
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10
A magnetic field is pointing
straight up through a coil of wire.
The field is suddenly doubled in
magnitude. What is the direction
of the induced current in the
wire?
A
Out of the page
B
Into the page
C
Clockwise
D
Counterclockwise
E
There is no induced current
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11
A coil of wire is sitting on a table
top. A magnet is held above it
with it's north pole pointed
downwards. What is the direction
of the induced current?
A
Out of the page
B
Into the page
C
Clockwise
D
Counterclockwise
E
There is no induced current
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12
A coil of wire is sitting on a table
top. A magnet is held above it
with it's north pole pointed
downwards. The magnet is
released and falls towards the
coil. What is the direction of the
induced current?
A
Out of the page
B
Into the page
C
Clockwise
D
Counterclockwise
E
There is no induced current
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EMF Induced in a Moving Conductor
This image shows another way the magnetic flux can change:
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EMF Induced in a Moving Conductor
The induced current is in a direction that tends to slow the moving
bar – it will take an external force to keep it moving.
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EMF Induced in a Moving Conductor
The induced emf has magnitude
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EMF Induced in a Moving Conductor
Another perspective
SF = ma
Magnetic Force
Electric Force
FB - FE = 0
qvB - qE = 0
vB = E
vB = V/d
V/d = vB
V is E
E = Blv
but E = V/d
but d is just l and
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13
What is the voltage between the
ends of a 100m long metal rod
which is traveling at a velocity
of 400 m/s perpendicularly
through a 5 x 10-4T magnetic
field?
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Changing Magnetic Flux Produces an Electric Field
A changing magnetic flux induces an electric field; this is a
generalization of Faraday’s law. The electric field will exist
regardless of whether there are any conductors around.
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Electric Generators
A generator is the opposite of a motor – it transforms mechanical
energy into electrical energy. This is an ac generator:
The axle is rotated by an
external force such as falling
water or steam. The brushes are
in constant electrical contact
with the slip rings.
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Electric Generators
A dc generator is similar,
except that it has a split-ring
commutator instead of slip
rings.
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Transformers and Transmission of Power
A transformer consists of two coils, either interwoven or linked
by an iron core. A changing emf in one induces an emf in the
other.
The ratio of the emfs is equal to the ratio of the number of
turns in each coil:
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Transformers and Transmission of Power
This is a step-up
transformer – the
emf in the
secondary coil is
larger than the emf
in the primary:
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Transformers and Transmission of Power
Energy must be conserved; therefore, in the absence of losses, the
ratio of the currents must be the inverse of the ratio of turns:
P in = P out
IinVin = IoutVout
IprimaryVprimary = Is e condaryVs e condary
Is e condary
Vprimary
=
Iprimary
Vs e condary
Is e condary
Nprimary
=
Iprimary
Ns e condary
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Transformers and Transmission of Power
Transformers work only if the current is changing;
this is one reason why electricity is transmitted as ac.
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Applications of Induction: Sound Systems,
Computer Memory, Seismograph, GFCI
This microphone works by induction; the vibrating membrane
induces an emf in the coil
Slide 46 / 50
Applications of Induction: Sound Systems,
Computer Memory, Seismograph, GFCI
Differently magnetized
areas on an audio tape or
disk induce signals in the
read/write heads.
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21.8 Applications of Induction: Sound Systems,
Computer Memory, Seismograph, GFCI
A seismograph has a fixed
coil and a magnet hung on
a spring (or vice versa),
and records the current
induced when the earth
shakes.
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Applications of Induction: Sound Systems,
Computer Memory, Seismograph, GFCI
A ground fault circuit interrupter (GFCI) will interrupt the current
to a circuit that has shorted out in a very short time, preventing
electrocution.
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Summary of Chapter 21
· Magnetic flux:
· Changing magnetic flux induces emf:
· Induced emf produces current that
opposes original flux change
· Changing magnetic field produces an electric field
Slide 50 / 50