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 elektromotorycznamagnetic flux - strumień pola magnetycznego

rate of change - szybkość zmian X to Y ratio = stosunek X/Y voltage - napięcie elektryczne

current intensity I - natężenie prądu Ielectric circuit - obwód elektryczny

current [increase; decrease (= decay)] - [wzrost; zanik] prądutime derivative of a function - pochodna funkcji po czasie

equation - równanielength = 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 INDUCTIONELECTROMAGNETIC INDUCTION

• Review of some magnetic phenomenaReview of some magnetic phenomena• Motional Electromotive Force (emf)Motional Electromotive Force (emf)• Faraday’s Law of Eectromagnetic InductionFaraday’s Law of Eectromagnetic Induction• Lenz’s LawLenz’s Law• Induced Electric FieldsInduced Electric Fields• Mutual Mutual IInductancenductance• Self Self - I- Inductancenductance• Energy in Energy in IInductornductor• LR LR CCircuitircuit

• Eddy CurrentsEddy Currents• Electromagnetic Waves-introductionElectromagnetic 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

sinBvqF

)( BvqF

BvqF 2

Magnetic flux

S

B SdB

cos S

B dsB

211 mTWb

WbB

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

vlBE

Motional emfMotional emfApply the Lorentz Force Apply the Lorentz Force quation:quation:

0qvBqEF

vBE

vBE

vB

qvBqE

Faraday’s Law

md d dxBl x Bl

dt dt dt

dxBlv Bl

dt E

mTherefore, d

dt

E

CONCLUSION: to produce emf one should make ANY change in a magnetic flux with time!

Consider the loop shown:

FARADAY’S LAWFARADAY’S LAW

• Changing magnetic flux produces an emfChanging magnetic flux produces an emf

(o(or r cchanging B-Field produces E-Fieldhanging B-Field produces E-Field))

• The rate of change of magnetic flux is The rate of change of magnetic flux is requiredrequired

Changing Flux due to moving Changing Flux due to moving permanent magnetpermanent 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 The direction of the emf induced by emf induced by changing flux will changing flux will produce a current that produce a current that generates a magnetic generates a magnetic field opposing the flux field opposing the flux change that produced itchange that produced it..

Lenz’s Law

B, H

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.

B, H

N S

V+, V-

Iinduced

Lenz’s Law

B, H

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.

B, H

N S

V-, V+

Iinduced

Faraday’s Law for a Single Loop

dt

dE

Faraday’s Law for a coil having N turnsFaraday’s Law for a coil having N turns

dt

dNE

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

Suppose we pull with velocity v a coil of resistance R through a region of constant magnetic field

vw

x

Ix x x x x x

x x x x x x

x x x x x x

x x x x x x

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

Nature of a changing fluxNature of a changing flux

• How can we induce emf?How can we induce emf?

B B dA

cosB dA

- - BB can change can change with time with time

- - AA can change can change with time with time

-- can changecan change with time 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

0.5000

1.0000

1.5000

1 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 inductMutual inductionion

Mutual inductMutual inductionion• A changing flux in one element induces an A changing flux in one element induces an

emf in anotheremf in another

2121211

1212122

iMN

iMN

total

total

dt

diM

dt

dN 1

2121

22

dt

diM

dt

dN 2

1212

11

1

21221 i

NM

2

12112 i

NM

Measurement of induced emf in coil C

),,( 202 nnIfUU

tII sin01

U () = cos( t) const

y = 0,0655x - 1,4864

R2 = 0,9637

0,000

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

9,000

10,000

11,000

20,0 40,0 60,0 80,0 100,0 120,0 140,0 160,0 180,0

f [kHz]

U [mV

]

Transformers

 Transformers

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.

A transformer is a device for increasing or decreasing an ac voltage.

Transformer EquationsUsing Faraday’s law we can write expressions for the primary and secondary voltages as follows:

.t

NV SS

.t

NV PP

.P

S

P

S

N

N

V

V

Dividing the above equations we get,

Assuming that there is no power loss, we can write,

.PPSS IVIV

.P

S

S

P

P

S

N

N

I

I

V

V

Power Loss in Transmission Lines

RIPLoss2

Transformers play a key role in the transmission of electric power.

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 UnitsDefinition and Units

• Unit Unit of L of L is henry (H)is henry (H):: volt-second/metervolt-second/meter

dt

dN

dt

diL

dt

dilAn

dt

dinNA

o

o

2

dt

diL

LiNtotal

VnAnL oo22

i

NL

l

Nn HnIB 00

Inductors and self inductance Land Back EMF-voltage

Changing flux induces emf in same Changing flux induces emf in same element that carries currentelement that carries current

A A “back” emf “back” emf is generated by a is generated by a changing currentchanging current

emf opposes the change causing it emf opposes the change causing it (Lenz’s Law)(Lenz’s Law)

dt

diL

LR circuitLR circuit

t

eR

I

R

L

At t=0 the At t=0 the switch is just openswitch is just open. . Apply Kirchhoff”s Loop RuleApply Kirchhoff”s Loop Rule

0 IRL

0IRdt

dIL

LR circuitLR circuit

L

t

e1R

i

R

LL

Apply Kirchhoff’s Loop RuleApply Kirchhoff’s Loop Rule

0iR L

Li

L

R

dt

di

At At tt = 0, = 0, ii = 0, and = 0, and switch is just closedswitch is just closed

Energy in an inductorEnergy in an inductor

idt

diLIP

it

diLiPdt0

0

2Li2

1W 2Li

2

1WU

Induced electric fieldsInduced 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 fieldsInduced electric fieldsLet’s calculate the value of work Let’s calculate the value of work one has to do to moving a charge one has to do to moving a charge along the circular path s:along the circular path s:

0

l

l

ldE

ldEqldFW

dt

dldE

l

Induced fields

dt

BdErot

dt

dldE

0 ldE

0Erot

E

ReminderReminder: in electrostatics: : in electrostatics:

ConclusionsConclusions

• The electric field produced by static charge is conservative:The electric field produced by static charge is conservative:

- Zero- Zero work must be done over a closed path (circuit)work must be done over a closed path (circuit)

• The The electric field due to an emf is NOT conservativeelectric field due to an emf is NOT conservative

– Net work must be done over a closed path (circuit)Net work must be done over a closed path (circuit)

• Therefore, the closed path integral of E is non-zeroTherefore, the closed path integral of E is non-zero

– Charges will accelerate parallel to E.Charges will accelerate parallel to E.

Eddy Currents

Eddy CurrentsEddy 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 FieldFrom Test Coil

Magnetic Field From

Eddy Currents

Eddy Currents

Crack

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.

DemoE-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

Maglev trains today can travel up to 310 mphMay eventually use superconducting loops to produce B-field No power dissipation in resistance of wires!

N

S

rails“eddy” current

Summary

• “Let there be light!!!”

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). 

Faraday's law of induction describes the production of an electric field by a changing magnetic field.

dt

BdErot

dt

dldE

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

Production of Electromagnetic Waves

The Electromagnetic Spectrum

In 1831 Joseph Henry discovered magnetic induction.

The History of Induction

Joseph Henry

(1797-1878)

Michael Faraday(1791-1867)

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!

The authors appreciate helpful discussion withProf. Mieczysław DEMIANIUK

while preparing the lecture.