Module
M
d l 21:
21
Faraday’s
Faraday
s Law
1
Faraday’s
y Law
Fourth (Final) Maxwell
Maxwell’s
s Equation
Underpinning of Much Technology
2
Demonstration:
Falling
g Magnet
g
3
Magnet
g
Falling
g Through
g a Ring
g
Link to
t
movie
Falling magnet slows as it approaches a copper
ring which has been immersed in liquid nitrogen
nitrogen.
4
Demonstration:
Jumping
p g Rings
g
5
Jumping
p g Ring
g
An aluminum ring jumps into the air when the
solenoid beneath it is energized
6
What is Going
g On?
It looks as though the conducting loops have
current in them (they behave like magnetic
dipoles) even though they aren’t hooked up
7
D
Demonstration:
t ti
I d ti
Induction
Link to movie
8
Electromagnetic
g
Induction
9
Faraday’s
y Law of Induction
=−
dΦB
dt
A changing magnetic flux
induces an EMF
10
What is EMF?
=
G G
E ⋅ ds
Looks like potential. It’s a
“driving
driving force”
force for current
11
Magnetic
g
Flux Thru Wire Loop
p
Analogous to Electric Flux (Gauss’ Law)
(1) Uniform B
G G
Φ B = B⊥ A = B A c o s θ = B ⋅ A
(2) Non-Uniform B
G G
Φ B = ∫∫ B ⋅ dA
S
12
Faraday’s
y Law of Induction
=
G G
dΦB
E ⋅ ds = −
dt
A changing magnetic flux induces
an EMF,
EMF a curling E field
13
Faraday’s
y Law of Induction
dΦB
=−
dt
A changing magnetic flux
induces an EMF
14
Minus Sign?
g
Lenz’s Law
Induced EMF is in direction that opposes
the change in flux that caused it
15
Concept
p Question: Loop
p
The magnetic field through
a wire loop is pointed
upwards and increasing
with time. The induced
current in the coil is
1. Clockwise as seen from the top
2. Counterclockwise
16
Concept
p Question: Loop
p
The magnetic field through
a wire loop is pointed
upwards and decreasing
with time. The induced
current in the coil is
1. Clockwise as seen from the top
2. Counterclockwise
17
Ways
y to Induce EMF
ε
(
d
=−
BAcosθ
dt
)
Quantities which can vary with time:
• Magnitude of B
• Area A enclosed by the loop
• Angle
A l θ between
b
B and
d lloop normall
18
Ways
y to Induce EMF
ε
(
d
=−
BAcosθ
dt
)
Quantities which can vary with time:
• Magnitude of B
• Area A enclosed by the loop
• Angle
A l θ between
b
B and
d lloop normall
19
M
Magnet
tF
Falling
lli
Through
Th
h a Ring
Ri
Link to
t
movie
Falling magnet slows as it approaches a copper
ring
i which
hi h h
has b
been iimmersed
d iin liliquid
id nitrogen.
it
20
Concept Question: Loop in Uniform
Field
Bout
v
A rectangular wire loop is pulled thru a uniform B field
penetrating its top half, as shown. The induced
current and the force and torque on the loop are:
1. Current CW, Force Left, No Torque
2. Current CW, No Force, Torque Rotates CCW
3. Current CCW, Force Left, No Torque
4. Current CCW, No Force, Torque Rotates CCW
5. No current, force or torque
21
Concept Question: Faraday’s Law:
Loop
A coil moves up
f
from
underneath
d
th a
magnet with its
north
th pole
l pointing
i ti
upward. The
currentt in
i the
th coilil
and the force on the
coil:
il
1.
1
2.
3
3.
4.
Current clockwise; force up
Current counterclockwise; force up
Current clockwise; force down
Current counterclockwise; force down
22
Technology
Many Applications of
Faraday’s Law
23
Electric Guitar
24
Metal Detector
See Animation
S
A i ti off how
h
VLF metal
t l detectors
d t t
work:
k
http://home.howstuffworks.com/metal-detector2.htm
I d ti Stovetops
Induction
St
t
G
Ground
d Fault
F lt Interrupters
I t
t
(GFI)
25
Experiment 5:
Faraday’s
a aday s Law
a o
of Induction
duct o
26
Example: Magnitude of B
M
Magnet
tF
Falling
lli
Through
Th
h a Ring
Ri
Falling magnet approaches a copper ring
or Copper Ring approaches Magnet
27
Moving
g Towards Dipole
p
Move
ring
g
down
As ring approaches, what happens to flux?
Flux up increases
28
Part 1: Current & Flux
BLACK
I>0
RED
Current?
Flux?
t
Φ (t ) = − R ∫ I (t ' ) dt '
0
29
Concept Question
Predictions:
Fl & Current
Flux
C
t
30
Concept Q. : Flux Measurement
(A)
t
(B)
t
((D))
(C)
t
t
Moving from above to below and back
back, you will
measure a flux of:
1.
2.
3.
4.
A then A
C then C
A then C
C then A
5.
6.
5
6
7.
8.
7
8
5.
6.
7.
8.
B then B
D then D
B then D
D then B
31
Concept Q.: Current Measurement
((A))
(B)
t
(D)
(C)
t
NOTE: CCW
i positive!
is
iti !
Moving
g from above to below and back,, you
y will
measure a current of:
1 A then A 5
1.
5. B then B
2. C then C 6. D then D
3 A th
3.
then C 7
7. B th
then D
4. C then A 8. D then B
5.
6.
5
6
7
7
t
t
32
Part 2: Force Direction
Force when
Move Down?
Move Up?
p
Test with
aluminum
l i
sleeve
33
Concept Question: Flux Behavior
(1)
(2)
t
1.
2.
3.
4.
1
2
3
4
t
(4)
(3)
NOTE: Magnet
“Upside Down”
t
t
Moving from below to above, you would measure a
p
by
y which p
plot above ((taking
g
flux best represented
upward flux as positive)?
34
Concept Q.: Current Behavior
(1)
(2)
t
t
(4)
(3)
NOTE: Magnet
“Upside Down”
1.
2.
3.
4.
1
2
3
4
t
t
Moving from above to below, you would measure a
p
by
y which p
plot above ((taking
g
current best represented
counterclockwise current as positive)?
35
Concept Question Confirming
Predictions?
Flux & Current
36
Concept Question Question:
Wrap-Up
Faraday’s Law
37
Concept Q.: Loop Below Magnet
A conducting loop is below a magnet and moving
d
downwards.
d Thi
This iinduces
d
a currentt as pictured.
i t d Th
The
I ds x B force on the coil is
0%
0%
0%
1. Up
2 Down
2.
3. Zero
38
Ways
y to Induce EMF
ε
(
d
=−
BAcosθ
dt
)
Quantities which can vary with time:
• Magnitude of B
• Area A enclosed by the loop
• Angle
A l θ between
b
B and
d lloop normall
39
The last of the Maxwell
Maxwell’s
s
Equations
q
(Kind
(
of))
40
Maxwell’s
Maxwell
s Equations
Creating Electric Fields
G G Qin
(Gauss's Law)
w
∫∫ E ⋅ dA =
S
ε0
G G
dΦB
⋅
=
−
E
d
s
vC∫
dt
(Faraday'ss Law)
(Faraday
Creating
G G Magnetic Fields
((Magnetic
g
Gauss's Law))
w
∫∫ B ⋅ dA = 0
S
G G
B
⋅
d
s
=
μ
I
0
enc
v∫
C
(Ampere's Law)
42
Faraday’s
y Law of Induction
dΦB
=−
dt
Changing magnetic flux induces an EMF
Lenz: Induction opposes change
43
Faraday s Law
Faraday’s
Problem Solving
g Session
43
Ways
y to Induce EMF
ε
(
d
=−
BAcosθ
dt
)
Quantities which can vary with time:
• Magnitude of B e.g. Falling Magnet
• Area A enclosed by the loop
• Angle
A l θ between
b
B and
d lloop normall
44
Problem: Changing
g g Area
Conducting rod pulled along two conducting rails in a
uniform magnetic field B at constant velocity v
l
1. Direction of induced
current?
2 Direction of resultant
2.
force?
3 Magnitude of EMF?
3.
4. Magnitude of current?
5. Power externally
supplied to move at
constant v?
45
Ways
y to Induce EMF
ε
(
d
=−
BAcosθ
dt
)
Quantities which can vary with time:
• Magnitude of B e.g. Moving Coil & Dipole
• Area A enclosed e.g. Sliding bar
• Angle
A l θ between
b
B and
d loop
l
normall
46
Changing
g g Angle
g
r r
ΦB = B ⋅ A = BA
A
r r
ΦB = B ⋅ A = 0
47
Motors & Generators
48
Concept Question Question:
Generator
49
Concept Question: Generator
A square coil rotates in a
magnetic
ti fifield
ld di
directed
t d to
t
the right. At the time
shown,
h
th
the currentt in
i th
the
square, when looking
d
down
ffrom th
the top
t off the
th
square loop, will be
1.
2
2.
3.
4
4.
Clockwise
C
Counterclockwise
t l k i
Neither, the current is zero
I don’t
d ’ kknow
50
Problem: Generator
Square loop
S
l
((side
id L) spins
i with
ith angular
l ffrequency ω
in a field of strength B. It is hooked to a load R.
1) W
Write
i an expression
i ffor current I(t)
(t) assuming
i the
h
loop is vertical at time t = 0.
2) How much work from generator per revolution?
3)) To make it twice as hard to turn, what do yyou
do to R?
51
Concept Question Question:
Wrap-Up
Faraday’s Law
52
Concept Question: Circuit
A circuit in the form of a
rectangular
t
l piece
i
off wire
i iis
pulled away from a long
wire carrying current I in
the direction shown in the
sketch. The induced
current in the rectangular
circuit is
1.
2
2.
3.
Clockwise
Counterclockwise
Neither, the current is zero
53
MIT OpenCourseWare
http://ocw.mit.edu
8.02SC Physics II: Electricity and Magnetism
Fall 2010
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