PHYS 219
Spring semester 2016
Lecture 15: Examples of Induced emf;
Electric Generators
Ron Reifenberger
Birck Nanotechnology Center
Purdue University
Lecture 15
1
Faradays’ Law (1831) + Lenz’s Law (1835)
Electromagnetic Induction
  
 B  magnetic flux  B  A  | B | A cos (B, A)
Magnetic
flux
move magnet toward loop
loop with
area A
a b
Eab  
 B
t
2
1
To understand Lenz’s Law, you need to know the direction
of the magnetic field produced by current in a wire
Long straight wire
Wire loop
I
B
3
Lenz’s Law
There are many ways to state Lenz’s Law. Here
is one that makes sense to me:
An induced electric current is produced by a
changing magnetic field. The induced current
will flow in a direction such that it will create
its own induced magnetic field that opposes
the changing magnetic field that created it.
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2
Lenz’s Law – Example 1
I
A. Assume a metal loop in which the applied magnetic field (solid
arrows) passes upward through it
B. Assume the magnetic flux increases with time
C. An induced magnetic field is produced which opposes the change
in flux
D. Therefore, the induced magnetic field (dotted arrows) must be
downward
E. An induced current produces the induced magnetic field
F. Therefore the induced current must flow clockwise (CW) when
viewed from the top of coil
5
Lenz’s Law – Example 2 – the dropped magnet
The situation
when the North
pole just enters
the loop
loop
Be able to
distinguish the
applied B field
from the
induced B field
An induced current must flow in the loop to produce a magnetic
field (inside the loop) that opposes the change in flux.
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3
Lenz’s Law – Example 3 - Motion to and from a loop
(Focus on the Change in Flux w.Time)
a)
Iind
b)
Steady motion
away from loop
N
Iind
N
Steady motion
toward loop
loop
loop
initial value


 increasing
increasing time
 decreasing
initial
value
increasing time
The induced current always produces a magnetic
field that opposes (counteracts) the change in flux
7
Lenz’s Law – Example 4
Jumping Rings
Induced
magnetic
field
“PCA – ACR”
(see Lecture 12)
8
4
Review: take home message from Faraday’s Law
Any changing magnetic field that threads a loop will
i)
Iinduced
produce an induced current in a direction
such that the induced current will create an
induced magnetic field that opposes the
changing magnetic field that created it
(if the loop is closed),
OR
ii)
produce an induced voltage with a polarity to
generate an induced current that would create
an induced magnetic field to oppose the
changing magnetic field that created it
(if the loop is open).
Einduced
9
Example V: What is the induced emf if the magnetic field through a
six turn coil increases at a rate of 0.17 T/s?
E =-
ΔΦB
Δt
The negative sign
indicates that the
induced emf acts to
“oppose” the change in
magnetic flux that
causes it
E =
 
ΔΦB Δ(B  A) Δ BAcos θ 
=
=
Δt
Δt
Δt
Δ B 
=A
Δt
A = πR2 = π  0.53m  = 0.88 m2
2
θ=0
Note: It is often easier to take the
absolute value of Farady’s Law to
find the magnitude of the induced
emf and then use Lenz’s Law to
find the direction of the induced
current that results.
ΔB
= +0.17 T / s  given 
Δt
E =(0.88 m2 )(0.17 T / s) = 0.15 V
Since coil has six turns, E = 6 ×(0.15 V) = 0.90 V
a
Vab = 0.9 V
b
10
5
Example VI. Induced Voltage: a simple example
What is Va-Vb?
N=1
Uniform B
constant rate
of reduction
B
ro = 25 cm
1 T
a
time
0.001 s
b
|E| = N
ΔΦB
0 -1T
ΔB
2
= NA
= 1  π  0.25 
Δt
Δt
0.001s


=  0.196m 2  1000 T
1 click
s

 = 196V
11
Motional emf
an application of the Lorentz force
law F=qvBsin(Θ)
+q
metallic
bar
v
B
Length L
ΔV
An induced emf (E)
appears across wire
When velocity (v) is not
perpendicular to B, define
Θ=  (v,B):
E  V  BLv sin()
E
B
v
-q
qtest E  qtest vB
|E|
 vB
L
|E| V  B Lv
E
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6
Example VII: the Space Shuttle
B
-
+
L~24 m
v350 m/s
B5×10-5 T (earth’s B field)
(Assume v perpendicular to B)
ΔV= LvB= 24 m• 350 m/s • 5×10-5 T
 0.4 V
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Area Changes with Time
Assume the
metal rails
and lightbulb
have a
combined
resistance R
B
I
1 electron
F
v
L
x
0 (because L=constant)
| E |= ΔV = -
Δ  Lx 
ΔΦ
ΔA
 x ΔL L Δx 
 B 

= B
= B
Δt
Δt
Δt
Δt 
 Δt
Δx
= BLv
Δt
ΔV BLv
I =
=
R
R
= BL
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7
Example VIII
I
Bexternal
1 electron
v
F
If L=0.5 m, v=10 m/s, R=1.5Ω and Bexternal=1.0 T, what is
the current?
E  V  LvB  0.5m  10m / s  1.0T  5 V
I
5V
V LvBexternal


 3.3 A
1.5
R
R
15
Example IX. EMF produced by Rotating Loop
Rotating Loop (N=1)
Stationary Loop
Bex
f
=BexAcos()
If loop rotates
(t) = o+2πft=o+ωt
f = No. rev./s or Hz
|E  t | = N
ΔΦ
= NBexAω sin  ωt 
Δt
where ω = 2πf = angular frequency in rad s
16
8
Compare DC vs. AC
17
Electric Generators
Converting Mechanical Work into Electrical Power
Z-T. Gramme Dynamo, Belgium (1869)
Produced direct current with the use of a
commutator. Used in the first commercial
power plants in Paris.
Steam Generator
9
Details of Electric Generator
Convert rotational energy
into electrical energy
ΔΦB
Δt
ΦB = BAcos θ = BAcos 2πft 
E =-
ΔΦB
 -sin 2πft 
Δt
E = Vmax sin 2πft 
where Vmax = NBAω
f
Slip ring
contacts (ac)
Brush
Commutator
Split-ring
Commutator (dc)
f
f
Brush
Brush
An old but very well done video on generators and motors:
https://www.youtube.com/watch?v=LisefA_YuVg
Brush
Loop (not shown)
Loop (not shown)
19
ac Voltages and ac Currents Become a Reality
 Edison vs. Westinghouse & Tesla (1882-1903)
 Chicago World’s Fair, 1893:




$399,000 winning bid (~ 10 M$ in today’s $’s)
100,000 incandescent lamps
27 million visitors
Museum of Science and Industry, Chicago
 ac stands for alternating current
 Accepted source of power becomes a device
producing an electric potential that varies with time
 Frequency and peak voltage become associated with electric
potential: Standard frequency in US is 60 Hz; Standard peak
voltage in US is 165 V.
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10
A New Industry is Born
1st edition circa 1905
1960’s
2000’s
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Up Next – ac Voltages, ac Currents, and
Transformers
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11
Appendix
Right Hand Rules we have introduced:
 Force on moving charge q in B-field
 Current Loop – direction of B-field at center of loop
 Current in long straight wire - direction of B-field that curls
around the wire
 Induced current (Lenz’s Law) – direction of current induced
by changing magnetic flux
Make sure you can distinguish between the
different situations
23
Why is Lenz’s Law Confusing?
There are four cases to consider for one loop!
Bapplied
conducting
loop
Bapplied
conducting
loop
If Bapplied
INCREASES
with time
If Bapplied
DECREASES
with time
If Bapplied
INCREASES
with time
Iinduced
Iinduced
Iinduced
Iinduced
CCW
CW
CW
CCW
Don’t memorize it – think it through
If Bapplied
DECREASES
with time
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12
If you like computer simulations, check out
Important to
realize: The
changing
magnetic
field could
also be
produced by a
2nd coil
https://phet.colorado.edu/sims/faradays-law/faradays-law_en.html
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