Preparing the lecture we applied figures from:
• Nondestructive Testing Resource Center www.ndt-ed.org
• Lectures of Dr. Ali R. Koymen, University of Texas, Arlington USA
www.uta.edu./physics/main/faculty/koymen/
• Lectures of Prof. John G. Cramer, University of Washington, Seattle
USA, faculty.washington.edu/jcramer/
• Lectures of Prof. Alan Murray, University of Edinburgh UK,
http://www.see.ed.ac.uk/~afm/?http://oldeee.see.ed.ac.uk/~afm/
• Lectures of Prof. Horst Wahl, Florida State University, Tallahassee
USA, http://www.hep.fsu.edu/~wahl/
• Lectures of G.L. Pollack and D.R. Stump, Michigan State University,
USA, http://www.pa.msu.edu/
• Lectures of Professor Joachim Raeder, University of New Hampshire
USA, www.physics.unh.edu/phys408/
W. Borys and K. Zubko
Military University of Technology, Institute of Applied Physics, Warsaw Poland
Faraday's Law
by W. Borys and K. Zubko
(electro)magnetic induction - indukcja (elektro)magnetyczna
[repelling; attracting] force - siła [odpychania; przyciągania]
[N; S] pole of a magnet - biegun [pn; płd] magnesu
[electric; magnetic (E-, B-)] field - pole [elektryczne; magnetyczne]
electric field intensity E - natężenie pola elektrycznego E
[tangent; perpendicular] to the curve - [styczny; prostopadły] do krzywej
electromotive force (emf) - siła elektromotoryczna
magnetic flux - strumień pola magnetycznego
rate of change - szybkość zmian
X to Y ratio = stosunek X/Y
voltage - napięcie elektryczne
current intensity I - natężenie prądu I
electric circuit - obwód elektryczny
current [increase; decrease (= decay)] - [wzrost; zanik] prądu
time derivative of a function - pochodna funkcji po czasie
equation - równanie
length = długość
sense of a vector = zwrot wektora
[scalar; vector] product = iloczyn [skalarny; wektorowy]
infinitely small = nieskończenie mały
line integral - całka liniowa, cyrkulacja
closed surface integral - całka po powierzchni zamkniętej
[coil; turn of winding] - zwój, pętla
[mutual; self-] inductance - indukcyjność [wzajemna; własna]
eddy currents - prądy wirowe
ELECTROMAGNETIC INDUCTION
•
•
•
•
•
•
•
•
•
Review of some magnetic phenomena
Motional Electromotive Force (emf)
Faraday’s Law of Eectromagnetic Induction
Lenz’s Law
Induced Electric Fields
Mutual Inductance
Self - Inductance
Energy in Inductor
LR Circuit
• Eddy Currents
• Electromagnetic Waves-introduction
Magnetic field around a permanent magnet.
B
Interaction of two permanent bar magnets.
Magnetic field around a straight conductor carrying a steady
current I.
Magnitude of B is directly proportional to the current I value and inversely
proportional to the distance from the conductor.
Properties of the magnetic force F


F  q(v  B)
F  q  v  B  sin 
F  q v B


2
Magnetic flux
 
 B   B  dS
S
 B   B  ds  cos 
S
 B   Wb
1Wb  1T  m 2
How is Electricity Produced?
• Friction: “static electricity” from rubbing (walking across a
carpet)
• Pressure: piezoelectricity from squeezing crystals together
(quartz watch)
• Heat: voltage produced at junction of dissimilar metals
(thermocouple)
• Light: voltage produced from light striking photocell (solar
power)
• Chemical: voltage produced from chemical reaction (wet or dry
cell battery)
• Magnetism: voltage produced using electromotive induction
(AC or DC generator).
Basic Terminology
• Electromotive Force (
 ,E, V)
– known as emf, potential difference, or voltage
– unit is volt [V]
– „force” which causes electrons to move from one
location to another
– operates like a pump that moves charges
(predominantly electrons) through “pressure” (=
voltage)
Separating Charge and EMF
Separating Charge and EMF
E  vlB
Motional emf
Apply the Lorentz Force
quation:
F  qE  qvB  0
qE  qvB
E  vB
E    vB
  Bv
Faraday’s Law
Consider the loop shown:
d m d
dx
 Bl x  Bl
dt
dt
dt
dx
E  Blv  Bl
dt
d m
Therefore, E 
dt
CONCLUSION: to produce emf one should make ANY
change in a magnetic flux with time!
FARADAY’S LAW
• Changing magnetic flux produces an emf
(or changing B-Field produces E-Field)
• The rate of change of magnetic flux is
required
Changing Flux due to moving
permanent magnet
Polarity of the Induced Emf
The polarity (direction) of the induced emf is
determined by Lenz’s law.
LENZ’S Law
The direction of the
emf induced by
changing flux will
produce a current that
generates a magnetic
field opposing the flux
change that produced it.
Lenz’s Law
B, H
N
S
Iinduced
V+, V-
Lenz’s Law: emf appears and current flows that creates a
magnetic field that opposes the change – in this case an
decrease – hence the negative sign in Faraday’s Law.
Lenz’s Law
B, H
N
S
Iinduced
V-, V+
Lenz’s Law: emf appears and current flows that creates a
magnetic field that opposes the change – in this case an
increase – hence the negative sign in Faraday’s Law.
Faraday’s Law for a Single Loop
d
E 
dt
Faraday’s Law for a coil having N turns
d
E    N
dt
Lenz's Law
Claim: Direction of induced current must be so as to
oppose the change; otherwise conservation of
energy would be violated.
• Why???
– If current reinforced the change, then the
change would get bigger and that would in
turn induce a larger current which would
increase the change, etc..
– No perpetual motion machine!
Conclusion: Lenz’s law results from energy
conservation principle.
Induced Current – quantitative
xxxxxx
Suppose we pull with velocity x x x x x x
v a coil of resistance R through x x x x x x
a region of constant magnetic
xxxxxx
field
x
I
w
We must supply energy to produce the current
and to move the loop (until it is completely out
of the B-field region). The work we do is
exactly equal to the energy dissipated in the
resistor, i.e.
W=I2Rt
v
Nature of a changing flux
 B   B  dA   B cos  dA
• How can we induce emf?
- B can change with time
- A can change with time
-  can change with time
Generators
Applications of Magnetic Induction
• AC Generator
Water turns wheel
 rotates magnet
 changes flux
 induces emf
 drives current
Single-Phase Generator
Three Phase Generator
Three Phase Voltage
1.5000
1.0000
0.5000
0.0000
1
-0.5000
-1.0000
-1.5000
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Sine
Sine + 120
Sine + 240
Some Other Applications of
Magnetic Induction
The Magnetic Playback Head of a
Tape Deck
• Tape / Hard Drive etc
– Tiny coil responds to change in flux as the magnetic
domains go by (encoding 0’s or 1’s).
– Credit Card Reader
– Must swipe card
 generates changing flux
– Faster swipe  bigger signal
Electric Guitar
Mutual induction
Mutual induction
•
A changing flux in one element induces an
emf in another
 total2  N 2  21  M 21i1
 total1  N112  M 12i2
d 21
di1
2   N 2
  M 21
dt
dt
d12
di2
1   N1
  M 12
dt
dt
N 2  21
M 21 
i1
N112
M 12 
i2
Measurement of induced emf in coil C
I1  I 0 sin t
U 2  U  f ( I 0 , , n2  n)
U () =  cos( t) const
11,000
10,000
9,000
U [mV]
8,000
7,000
y = 0,0655x - 1,4864
6,000
5,000
R2 = 0,9637
4,000
3,000
2,000
1,000
0,000
20,0
40,0
60,0
80,0
100,0
f [kHz]
120,0
140,0
160,0
180,0
Transformers
Transformers
A transformer is a device for increasing or decreasing an ac
voltage.
The changing magnetic flux produced by the current in the
primary coil induces an emf in the secondary coil.
At the far right is the symbol for a transformer.
Transformer Equations
Using Faraday’s law we can write expressions for the
primary and secondary voltages as follows:

VS   N S .
t

VP   N P
.
t
Dividing the above equations we get,
VS
NS

.
VP
NP
Assuming that there is no power loss, we can write,
VS I S  V P I P .
VS I P N S


.
VP I S N P
Power Loss in Transmission Lines
Transformers play a key role in the transmission of electric power.
PLoss  I R
2
Self-induction
Self-inductance (L)
The alternating current in the coil generates an alternating
magnetic field that induces an emf in the same circuit.
The effect in which a changing current in a circuit induces
an emf in the same circuit is referred to as self-induction.
Definition and Units
total  N  Li
N
L
i
d
  N
dt
di
  L
dt
Unit of L is henry (H):
volt-second/meter
N
n
l

B  0nI  0 H
di
dt
di
  n 2lA  o
dt
di
 L
dt
  NA o n
L   o n A   o n V
2
2
Inductors and self inductance L
and Back EMF-voltage
di
  L
dt
Changing flux induces emf in same
element that carries current
A “back” emf is generated by a
changing current
emf opposes the change causing it
(Lenz’s Law)
LR circuit
At t=0 the switch is just open. Apply Kirchhoff”s Loop Rule
 L  IR  0
dI
 L  IR  0
dt
I

R

e
t

L

R
LR circuit
At t = 0, i = 0, and switch is just closed
Apply Kirchhoff’s Loop Rule
  iR   L  0
di
R


i
dt
L
L
L
t



i  1  e  L
R





L

R
Energy in an inductor
di
P  I   L i
dt
t
i
0
0
 Pdt    Li di
1 2
W   Li
2
1 2
U  W  Li
2
Induced electric fields
Induced fields
Let us discuss two ways of production of electric field:
(1) A Coulomb electric field that is created by positive or negative
charges;
(2) A non-Coulomb electric field that is created by a changing
magnetic field.
Induced electric fields
Let’s calculate the value of work
one has to do to moving a charge
along the circular path s:
 
 
W   F  dl  q0  E  dl
 
   E  dl
l
l
 
d
l E  dl   dt
Induced fields
Reminder: in electrostatics:
E
 
E

d
l

0


rotE  0
 
d
 E  dl   dt


dB
rotE  
dt
Conclusions
• The electric field produced by static charge is conservative:
- Zero work must be done over a closed path (circuit)
• The electric field due to an emf is NOT conservative
– Net work must be done over a closed path (circuit)
• Therefore, the closed path integral of E is non-zero
– Charges will accelerate parallel to E.
Eddy Currents
Eddy Currents
Eddy currents are induced electric currents that flow in a
circular path
Eddy Currents
A magnetic braking system.
Generation of Eddy Currents (cont.)
Eddy currents flowing in the material will generate their
own “secondary” magnetic field which will oppose the
coil’s “primary” magnetic field.
Crack Detection
Crack detection is one of the primary uses of eddy current inspection.
Cracks cause a disruption in the circular flow of the eddy currents
and weaken their intensity.
Magnetic Field
From Test Coil
Magnetic Field
From
Eddy Currents
Crack
Eddy Currents
Material Thickness Measurement
Eddy current inspection is often used in the aviation
industries to detect material loss due to corrosion and
erosion.
Material Thickness Measurement
Eddy current inspection is used extensively to inspect
tubing at power generation and petrochemical facilities
for corrosion and erosion.
Metal Detectors
Metal detectors like those used at airports can detect any metal objects,
not just magnetic materials like iron. They operate by induced currents.
Demo
E-M Cannon
v
~
side view
More Applications of Eddy Currents
• Magnetic Levitation (Maglev) Trains
– Induced surface (“eddy”) currents produce field in opposite
direction
 Repels magnet
 Levitates train
S
N
“eddy” current
rails
Maglev trains today can travel up to 310 mph
May eventually use superconducting loops to produce B-field
 No power dissipation in resistance of wires!
Summary
Faraday's law of induction describes the production of an
electric field by a changing magnetic field.

 

d
dB
E

d
l


rot E  

dt
dt
James MAXWELL concluded that a changing magnetic field (B)
will produce a changing electric field (E) and the changing E will
produce a changing B. The net result of the interaction of the
changing E and B fields is the production of a wave which has both an
electric and a magnetic component and travels through empty space.
This wave is referred to as an electromagnetic wave (EM).
• “Let there be light!!!”
Production of Electromagnetic Waves
The speed of EM waves in a vacuum is given by
v = 1/(0 o)
where 0 is the permittivity of free space 0 = 8.85x10-12 C2/N m2 and
o is the permeablity of free space o = 4 x10-7 T m/A.
v = 3.00x108 m/s speed of light in vacuum
The Electromagnetic Spectrum
The History of Induction
In
1831 Joseph Henry discovered magnetic induction.
Joseph Henry
(1797-1878)
Michael Faraday's ideas about conservation of
energy led him to believe that since an electric
current could cause a magnetic field, a magnetic
field should be able to produce an electric current.
He demonstrated this principle of induction in
1831.
So the whole thing started 176 years ago!
Michael Faraday
(1791-1867)
The authors appreciate helpful discussion with
Prof. Mieczysław DEMIANIUK
while preparing the lecture.