6 IENVE_Elect and Magn__Electromagnetic Induction_ 2014 vrs02a

Electricity and Magnetism
Electromagnetic Induction
Lectures by
Jarmo Lilja
Tampere University of Applied
Sciences
2014

Electromagnetic Induction
• Basic concepts
Power sources and emf
Faraday’s and Lenz’s Laws
Magnetic flux
21-Feb-14 TAMK University of Applied Sciences 2

Electromagnetic Induction
• Introduction
Magnetically induced currents
N
B
N
B
S
Magntic field B
increases
S
Magnetic field B
degreases
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Electromagnetic Induction
Applications
• AC generator for emf production
• Mechanical energy transformation to electrical energy
T
Period
A
w
ˆE
Voltage amplitude
N
S
ˆE
V
Rotation of coil
Production of emf
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Electromagnetic Induction
• Metal detectors
induced currents
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Electromagnetic Induction
• Wind turbine
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Electromagnetic Induction
• Basics
Emf = voltage which maintains
the electrical current of circuit
Electrical power supplies could be
for example
batteries,accumulators
But how it is possible to convert
mechanical energy into electrical
energy?
Rotating steam turbine
• Faraday’s and Lenz’s law
Experiment in figures below
• The bar magnet is moved
towards to coil or away from it
Electrical current is induced into
coil
• The direction of I depends on
wether the flux density B is
increasing or degreasing with
time
Generator
N
B
N
B
emf
Magnetically induced currents
Principles developed 150 years ago
• FARADAY’S LAW
S
Magnetic
flux density
B increases
S
Magnetic
flux
density B
degreases
21.2.2014 Jarmo Lilja TAMK Fysiikan laitos 7

Electromagnetic Induction
New quantity is needed:
magnetic flux F
Magnetic flux F is defined as
F
B
=
B d A ,
A
_
_
• where B = magnetic flux
density
• d  = surface area vector on
surface of conductor loop
See figure where magnetic flux
lines are intersects the surface
• The orientation of surface
respect to magnetic flux density
is indicated by the surface
vector d (pependicular to the
surface) .
21.2.2014 Jarmo Lilja TAMK Fysiikan laitos 8

•
Electromagnetic Induction
Properties of magnetic
flux F :
• Proportional to number of
intersected flux lines in
surface A
• The unit is weber = 1T .
m 2 =V . s.
The connection between
time varying flux F and
induced emf E is
Faraday’s law
E
=
dF
dt
B
• i.e the induced emf of the
conductor loop is a time
derivative of magnetic flux
• If flux F is not changing
• then E = 0.
• Lenz’s law
• The direction of induced
current
• It is trying to
resist the change
of magnetic flux
density
The new magnetic field
of induced current will
resist the external
magnetic field to
increase or weaken
• Based on the equation the
esential question is
• what are the different
ways to change magnetic
flux with time
21.2.2014 Jarmo Lilja TAMK Fysiikan laitos 9

Electromagnetic Induction
Let us look at the flux
quantity
Scalar product for vectors
could be written as
• where q = angle between
B and dA
Magnetic flux F will change
with time if if any of the
quantities B , dA or q will
change with time
_ _
B = B (t),
F B
= B d A ,
dA = dA(t) or
• q = q(t)
A
B . dA = B . dA cos q,
dF
E = N
dt
Time
dependent
flux
A coil with N turns
• the Faraday’s induction law could
be written as
B
F(t)
21.2.2014 Jarmo Lilja TAMK Fysiikan laitos 10

Electromagnetic Induction
Esim.
A coil has 500 turns and its
crossection is 20 cm 2 . The coil
is moved into magnetic field,
where the flux density is 0,30 T
in time period 0,02 s. Calculate
the induced emf of the coil. The
surface of coil is all the time
perpendicular to magnetic field
Solution.
N = 500; A = 20 . 10 -4 m 2
∆t = 0,020 s ; E = ?
• Let us calculate the change
of flux dF
• The change is caused by
the change of magnetic flux
density dB
F = B
A = B
A
B
=
=
A B
20 10
= 610
4
4
Vs
0,30
21.2.2014 Jarmo Lilja TAMK Fysiikan laitos
m
2
T
E
• Induced emf is
=
=
=
A
d F
N
d t
- 500
-15V
V
B 1 = 0
B
F
N
t
-4
610
Vs
0,020s
change
ΔB=B 2 -B 1
A
B
B 2 = 0,3 T
11

Electromagnetic Induction
• Rotating conductor loop. AC
generator
Let us consider a simple
conductor loop adjusted in
a uniform magnetic field
The loop has central axis
as shown in figure
Suppose that the loop is
rotating with a angular
velocity w
Angular velosity w = Dq/Dt
and hence Dq =w Dt
• Suppose that t = 0,
when q =0
• Then at time t= t it
is q = wt
What is the voltage V
across the end of the
loop? Now V = emf !
Based on Faraday’s
induction law
•
N
E =
E =
=
A
w
S
V
dΦB
- N
dt
d( BA cos t)
-
dt
BAd(
cos t)
-
dt
and then
E = BA
sint
21.2.2014 Jarmo Lilja TAMK Fysiikan laitos 12

Electromagnetic Induction
If the coil has N turns we
can get an important
equation
Example. Draw the emf E(t)
with given quantities shown in
the table below
E
=
NBA sin
t
Substituting w = 2 p f, an
hence
T
Time
period
Amplitude
E
= NB A
sin(2
f t)
ˆE
We can get sine function
• sinusoidal alternating
voltage
• The maximum voltage or
amplitude of voltage is
ˆE
E ˆ
=
N B A
• Frequency f = 1/T where T
= period
21.2.2014 Jarmo Lilja TAMK Fysiikan laitos 13

Electromagnetic Induction
• Inductance
Let us take a look at a
circuit where
• emf E
• resistance R
• Coil (with inductance L) ja
• switch K
When the switch is closed
• Increasing current I will produce
into the coil
• Increasing magnetic flux
• Variable flux will produce
• voltage
• Direction of voltage (polarity))
• tries to prevent the change
of current
• Phenomenon: self inductance
I
No inductance
E
R
With inductance
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t

Electromagnetic Induction
Self inductance
• The coil has N turns
• It has the same magnetic
flux F in every loops
• Based on Faraday’s law
induced voltage into coil is
E =
dF
N
dt
• This could be written (N=
constant)
E
=
d(N F)
dt
• The magnetic flux density B
and also magnetic flux F is
directly proportional to
electric current I and hence
N F
where L is selfinductance
or shortly inductance
• unit [L] = H = henry
• Induced emf is now E
• And also
=
LΙ
d( NF
)
E =
dt
d( LI )
=
dt
E
=
dI
L
dt
The polarity (”sign”) of
the voltage is such that it
will resist of the change
of current
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Electromagnetic Induction
Mutual inductance
• Two coils near by each other
• Let us denote
• 1-coil:
In the figure there is
increasing current I 1
N 1
N 2
• Alternating current I 1
• Number of turns N 1
• Number of turns N 2
• Magnetic flux F 2
• Alternating current in coil 1 will
induce in coil 1 a voltage E 2 into
the coil 2
dI
E M 1
2
=
dt
where M = mutual
inductance
• [M] = H = henry
F 1
I 1
E 2
E 1
• 2-coil:
• Magnetic flux F 1
• Alternating current I 2
21.2.2014 Jarmo Lilja TAMK Fysiikan laitos 16

Electromagnetic Induction
• Coil 2 will induce into coil
a voltage E 1 , respectively
dI
E M 2
1
=
dt
•
•N 1
•N 2
•F
2
•I 2
•E 1
•E 2
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Electromagnetic Induction
Exersice. Derive the inductance
of ring solenoid
• Inside
B =
=
• Magnetic flux
F
H
N I
• Definition of inductance
o
o
= B A
=
L
o
N I
A
N F
I
• Substituting F into L we get
inductance of ring solenoid
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L =
D
A
2
N A
Example. The average diameter
of ring solenoid is 5,0 cm, and the
the area of crossection is 4,9
cm 2 , number of turns is 1500 and
it has an air core
(a) Calculate the inductance of
the ring solenoid
(b) What is the self induced
voltage of the ring solenoid when
the electrical current is
degreasing at the rate of 0,50
kA/s?
o
N

Electromagnetic Induction
• Applications
Metal detectors
• Induced current into metal=
eddy currents
Induction melting
• Eddy currents will
increase the temperarure
into the melting point
Current signal I o is
flowing in detector coil
producing a magnetic
field B o
The magnetic field of
coil Bo will induse eddy
current into the metal .
The eddy current will
produce an opposite
magnetic field B and
this field will induse
electrical current I´
into the coil and it has
opposite directon than
the original signal
current I o which could
be detected.
Magnetic dampers
• Used in measuring devices
• Based on eddy currents
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Electromagnetic Induction
• Applications
Transformers
• Preventing the heating effect by eddy currents the iron core has
a laminated structure
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