ECE 307: Electricity and Magnetism Spring 2010 307 Review.pdf · ECE 307: Electricity and Magnetism...

ECE 307: Electricity and MagnetismSpring 2010

Instructor: J.D. Williams, Assistant Professor Electrical and Computer EngineeringUniversity of Alabama in Huntsville

406 Optics Building, Huntsville, Al 35899Phone: (256) 824-2898, email: [email protected]

Course material posted on UAH Angel course management website

Textbook: M.N.O. Sadiku, Elements of Electromagnetics 4th ed. Oxford University Press, 2007.

Optional Reading:H.M. Shey, Div Grad Curl and all that: an informal text on vector calculus, 4th ed. Norton Press, 2005.

All figures taken from primary textbook unless otherwise cited.

mailto:[email protected]�

Course MaterialChapters 1-3. (Review Material)

Vectors Algebra, Coordinate Transformations, Vector CalculusChapter 4.

Coulomb’s Law, Electric Field Intensity, Charge Distribution, Electric Flux Density, Gauss’ Law, Electric Potential, E and V, Energy

Chapters 5-6.Properties of Materials, Currents, Continuity Equation, Poisson’s Equation, Laplace’s Equation, Resistance, Capacitance, Image Theory

Chapters 7-8.Biot-Savart Law, Ampere’s Law, Magnetic Flux Density, Maxwell’s Equations (Static), Magnetic Vector Potentials, Magnetic Forces, Magnetic Materials, Boundary Conditions

Chapter 9. (Partial)Faraday’s Law, Maxwell’s Equations (Time Varying), Optional: Time-Harmonic Fields, Plane Waves Poynting Vectors

• Grading: This course is graded on a curve based on the following % breakdowns• Homework (20%): Turned in weekly. Graded on attempted effort.• Exams (60%): 3 per Semester. Exam scores historically increase over the semester

– Exam 1: Chapters 4 and 5– Exam 2: Chapters 6 and 7– Exam 3: Chapters 8 and 9

• Final Comprehensive Exam (20%): Includes material from Chapters 4-10

ECE 307: Electricity and MagnetismChapters 1-3: Review of Mathematical Essentials

• Chapter 1: Vector Analysis– Scalars and Vectors– Unit Vector– Vector Addition and Subtraction– Position and Distance Vectors– Vector Multiplication– Components of a Vector

• Chapter 2: Coordinate Systems and Transformations– Cartesian Coordinates– Circular Cylindrical Coordinates– Spherical Coordinates– Constant Coordinate Surfaces

Note: Students will not be tested specifically on Chapters 1-3. However, the information contained within them will be used in almost every aspect of the course. Knowledge and skilled practice of these concepts will be required to complete all homework, and examinations throughout the semester

• Chapter 3: Vector Calculus– Differential Length, Area, and Volume– Line, Surface, and Volume Integrals– Del Operator– Gradient of a Scalar– Divergence of a Vector and the

Divergence Theorem– Curl of a Vector and Stokes’s

Theorem– Laplacian of a Scalar– Classification of Vector Fields

ECE 307: Electricity and MagnetismChapters 4-9: Short Overview

• Chapter 1-3: Mathematics Review• Chapter 4: Electrostatic Fields• Chapter 5: Electric Fields in Material Space• Chapter 6: Electrostatic Boundary Conditions• Chapter 7: Magnetostatic Fields• Chapter 8: Magnetic Forces, Materials, and Devices• Chapter 9: Maxwell’s Equations

Expect to be tested on any and all of the material presented in this overview prior to the end of the semester

8/11/2010 5

Point Charge Distributions and Coulomb’s Law

• The force, F, between two point charges Q1 and Q2 is:– Along the line joining them– Directly proportional to the product between them– Inversely proportional to the square of the distance between them

• The electric field, E, is equal to the force divided by the magnitude of the charge being acted upon.

– The equation above is easily calculated for a test charge, Q1, at the origin and a source charge, Q2, at a distance R away.

– The solution is slightly more complicated as we move our reference frame away from the two charge such that Q1 is referenced by the vector r1 and Q2 is referenced by the r2.

dvRrr

RQE

rR

QQEqF

v o

v

o

o

∫==

==

22

221

4ˆˆ

4

ˆ4

περ

πε

πε

Where , 2212 /10854.8 NmCo−⋅=ε

o

kπε41

=

8/11/2010 6

The Dielectric Constant• It is important to note that up to this point, we have not committed ourselves to the

cause of the polarization, P. We dealt only with its effects. We have stated that the polarization of a dielectric ordinary results from an electric field which lines up the atomic or molecular dipoles.

• In many substances, experimental evidence shows that the polarization is proportional to the electric field, provided that E is not too strong. These substances are said to have a linear, isotropic dielectric constant

• This proportionality constant is called the electric susceptibility, χe. The convention is to extract the permittivity of free space from the electric susceptibility to make the units dimensionless. Thus we have

• From the previous slide

• If the electric field is too strong, then it begins to strip electrons completely from molecules leading conduction effects. This is called dielectric breakdown.

• The maximum strength of the electric that a dielectric can tolerate prior to which breakdown occurs is called the dielectric strength.

EP eo

χε=

ED

ED

EPED

ro

eoo

ε

εε

χεε

=

=

+=+= )1( The dielectric constant (or relative permittivity) of the material, εr, is the ratio of the permittivity to that of free space

8/11/2010 7

Gauss’ Law• Gauss’ Law: The electric flux, Ψ, through any closed surface is equal to the

total charge enclosed by that surface, thus Ψ=Qenc

Integral Form

Applying the divergence theorem, we have

yielding the differential form

This is the first of the 4 Maxwell Equations which clearly states that the volume charge density is equal to the divergence of the electric flux density

tBE

D v

∂∂

−=×∇

=⋅∇

ϕ

( ) dvdvDSdDv

vvs

∫∫∫ =⋅∇=⋅ ϕ

vD ϕ=⋅∇

encv

vss

QdvSdDSdE ==⋅=⋅=Ψ ∫∫∫ ϕε

tDJH

B

∂∂

−=×∇

=⋅∇

0Maxwell’s

EquationsIn matter

8/11/2010 8

Gauss’ Law(2)• Gauss’ law is simply an alternative statement of Coulomb’s law.• Gauss’ law provides an easy means of finding E or D for symmetrical charge

distributions• Applications:

r

enc

rrenc

senc

r

r

ar

QD

rDddrDQ

ddraaDQ

SdDQ

ddraSd

aDD

2

222

0 0

22

0 0

2

4

4sin

sin

sin

π

πφθθ

φθθ

φθθ

ππ

ππ

=

==

⋅=

⋅=

=

=

∫ ∫

∫ ∫

∫

Point Charge

Figure from: M.N.O. Sadiku, Elements of Electromagnetics 4th ed. Oxford University Press, 2007.

8/11/2010 9

Electric Potential• We define the electric potential (or the irrotational scalar field), V, describing any

electrostatic vector field, E, as the magnitude of the difference between E at two points a and b and some standard (common) reference point o.

• Now, the Gradient theorem states that

∫∫∫

∫

⋅−=⋅−⋅−=−

⋅−=

b

a

o

a

b

o

p

o

ldEldEldEaVbV

ldEpV

)()(

)(

EdldV

EdldV

ldEdV

−=

−=

⋅−=

max

cosθ

Note the crucial role that independence of path plays: If E was dependent on path, then the definition of V would be nonsense because the path would alter the value of V(p)

negative by convention

VE −∇=

0=∂∂

−=×∇tBE

Maxwell’sEquationFor Static Fields

8/11/2010 10

Using Gauss’ Law With Dielectrics

AQdldDV

QDAdaD

QSdD

aEED

aEEE

encS

zsrtrott

zo

st

εε

ϕεεεε

εϕ

−=⋅−=

==

=⋅

===

=+=

∫

∫

∫

21

zo

s aE

εϕ21 =

zo

s aE

εϕ22 =

Concentric Conducting Spheres with radius a, b (b>a) with a dielectric fill

Two flat conductive plates of area, A, filled with dielectric

d

z

a

bεr

εr

−=

−=

⋅−=

∝

∫

∫

baQ

drr

Q

ldDV

RaQE

b

a ro

o

R

114

4

4

2

2

πε

επε

ε

πε

If capacitance, C=Q/V, then what is the value for each example?

-++

-


8/11/2010 11

Continuity Equation: Electric Current

• Remembering that all charge is conserved, the time rate of decrease of charge within a given volume must be equal to the net outward flow through the surface of the volume

• Thus, the current out of a closed surface is

• For steady state problems, the derivative of charge with respect to time equals zero, and thus gradient of current density at the surface is zero, showing that there can be no net accumulation of charge.

tJ

dvt

dvJSdJ

dvtdt

dQSdJI

v

v

v

vS

v

venclosed

S

∂∂

−=⋅∇

∂∂

−=⋅∇=⋅

∂∂

−=−=⋅=

∫∫∫

∫∫

ϕ

ϕ

ϕ

Applying Stokes Theorem

Continuity Equation

8/11/2010 12

Electrical Resistance• Consider a conductor whose ends are maintained at a potential difference ( i.e. the

electric field within the conductor is nonzero and a field is passed through the material.)• Note that there is no static equilibrium in this system. The conductor is being fed

energy by the application of the electric field (bias potential)• As electrons move within the material to set up induction fields, they scatter and are

therefore damped. This damping is quantified as the resistance, R, of the material.• For this example assume:

– a uniform cross sectional area S, and length l.– The direction of the electric field, E, produced is the same as the direction of flow of positive

charges (or the same as the current, I).

σρ

σ

σσ

1====

===

=

Sl

Sl

IVR

lVE

SIJ

lVE

c∫

∫⋅

⋅==

s

v

SdE

ldE

IVR

σ


8/11/2010 13

Capacitance• Capacitance is the ratio of the magnitude of charge on two separated plates

to the potential difference between them

• Note that The negative sign is dropped in the definition above because we are interested in the absolute value of the voltage drop

• Capacitance is obtained by one of two methods– Assuming Q, and determine V in terms of Q– Assuming V, and determine Q in terms of V

• If we use method 1, take the following steps– Choose a suitable coordinate system– Let the two conducting plates carry charges +Q and –Q– Determine E using Coulomb’s or Gauss’s Law and find the magnitude of

the voltage, V, via integration– Obtain C=Q/V

∫ ⋅−= ldEV ∫

∫⋅

⋅==

ldE

SdEVQC

ε


8/11/2010 14

Capacitance vs. Resistance

σε

ε

σ

=

⋅

⋅==

⋅

⋅==

∫∫∫∫

RC

ldE

SdEVQR

SdE

ldEIVR

Parallel Plates

Isolated Spherea

RaC

baR

ba

C

Lab

R

abLC

SdR

dSC

πσπε

πσπε

πσπε

σε

41,4

4

11

,11

4

2

ln,

ln

2

,

==

−

=

−

=

=

=

==

Between 2 Spheres

Coaxial Cylinders

8/11/2010 15

Summary Diagram of Electrostatics

VE −∇=

∫ ⋅−= ldEV

0=×∇

−=⋅∇

E

Eo

v

εϕ

ϕ

EV

o

vVεϕ

−=∇2

∫−= νϕπε

dr

V v

o41

∫= νϕπε

draE rv

o24

1

Figure is recopied from Griffiths, Introduction to Electrodynamics,3rd ed., Benjamin Cummings, 1999.

Biot-Savart’s Law

• The differential magnetic field intensity, dH, produced at a point P, by the differential current element, Idl, is proportional to the product Idl and the sine of the angle between the element and the line joining P to the element and is inversely proportional to the square of the distance, R, between P and the element

16

232 4sin

44ˆ

RIdl

RRlId

RalIdHd R

πα

ππ=

×=

×=


• Ampere’s law: The line integral of H around a closed path is the same as the net current, Ienc, enclosed by the path

– Similar to Gauss’ law since Ampere’s law is easily used to determine H when the current distribution is symmetrical

– Ampere’s law ALWAYS holds, even if the current distribution is NOT symmetrical, however the equation is only used effectively for symmetric cases

– Like Gauss and Coulomb’s Laws, Ampere’s aw is a special case of the Biot-Savart law and can be derived directly from it.

• Applying Stokes’s theorem provides alternative solution methods

∫ =⋅ encIldH

Ampere’s Circuit Law

17

( )

JH

SdJI

SdHldHI

Senc

SLenc

=×∇

⇒

⋅=

⋅×∇=⋅=

∫

∫∫

Maxwell’s 3rd Eqn. for Static Fields

Definition of Current provided in Chapter 5

8/11/2010 18

• Lets now examine time dependent fields from the perspective on Ampere’s Law.

( )

( )( )

tDJH

tDJ

tDD

ttJJ

JJH

JJHt

J

JH

JH

d

vd

d

d

v

∂∂

+=×∇

∂∂

=

∂∂⋅∇=⋅∇

∂∂

=∂∂

=⋅−∇=⋅∇

⋅∇+⋅∇==×∇⋅∇

+=×∇

≠∂∂

−=⋅∇

⋅∇==×∇⋅∇

=×∇

ρ

ρ

0

0

0

Maxwell’s 3rd equation for time varying fields (Ampere’s Circuit Law)

This one relates Magnetic Field Intensity to conduction and displacement current densities

Displacement Current

This vector identity for the cross product is mathematicallyvalid. However, it requires that the continuity eqn. equalszero, which is not valid from an electrostatics standpoint!

Thus, lets add an additional current density term to balance the electrostatic field requirement

We can now define the displacement current density as the time derivative of the displacement vector

• Magnetic Flux density, B, is the magnetic equivalent of the electric flux density, D. As such, one can define

• Similarly, Ampere’s Law is

• And the Magnetic flux through a surface is

• The magnetic flux through an enclosed system is

mHHB

/104 70

0−×=

=

πµ

µ

Magnetic Flux Density

19

∫ ⋅= ldBIencˆ

0µ

∫∫ ⋅=⋅=SS

SdHSdB

0µψ

where

( )0=⋅∇

⋅∇=⋅= ∫∫B

dvBSdBSS

ψ

Definition of a solenoidal fieldand Maxwell’s 4th eqn.

8/11/2010 20

Faraday’s Law (1)• We have introduced several methods of examining magnetic fields in terms of forces,

energy, and inductances.• Magnetic fields appear to be a direct result of charge moving through a system and

demonstrate extremely similar field solutions for multipoles, and boundary condition problems.

• So is it not logical to attempt to model a magnetic field in terms of an electric one? This is the question asked by Michael Faraday and Joseph Henry in 1831. The result is Faraday’s Law for induced emf

• Induced electromotive force (emf) (in volts) in any closed circuit is equal to the time rate of change of magnetic flux by the circuit

where, as before, λ is the flux linkage, Ψ is the magnetic flux, N is the number of turns in the inductor, and t represents a time interval. The negative sign shows that the induced voltage acts to oppose the flux producing it.

• The statement in blue above is known as Lenz’s Law: the induced voltage acts to oppose the flux producing it.

• Examples of emf generated electric fields: electric generators, batteries, thermocouples, fuel cells, photovoltaic cells, transformers.

dtdN

dtdVemf

Ψ−=−=

λ

8/11/2010 21

Faraday’s Law (2)• To elaborate on emf, lets consider a battery circuit.

• The electrochemical action within results and in emf produced electric field, Ef

• Acuminated charges at the terminals provide an electrostatic field Ee that also exist that counteracts the emf generated potential

• The total emf generated in by a time dependent motion or magnetic field is

• Note the following important facts • An electrostatic field cannot maintain a steady current in a close circuit since

• An emf-produced field is nonconservative• Except in electrostatics, voltage and potential differences are usually not

equivalent

∫ ==⋅L

e IRldE 0

IRldEldEldE

EEEP

Nf

Lf

L

ef

=⋅=+⋅=⋅

+=

∫∫∫

0

( )

( )BudtBdE

ldBudtBdldEV

m

LSLemf

××∇+−=×∇

⋅×+−=⋅= ∫∫∫


• We now know that closed magnetic circuit carrying current I produces a magnetic field with flux

• We define the flux linkage between a circuit with N identical turns as

• As long as the medium the flux passes through is linear (isotropic) then then flux linkage is proportional to the current I producing it and can be written as

Where L is a constant of proportionality called the inductance of the circuit. A circuit that contains inductance is said to be an inductor.

• One can equate the inductance to the magnetic flux of the circuit as

where L is measured in units of Henrys (H) = Wb/A• The magnetic energy (in Joules) stored by the inductor is expressed as

Inductors and Inductance

8/11/2010 22

Ψ= Nλ

∫ ⋅=Ψ SdB

LI=λ

IN

IL Ψ

==λ

2

21 LIWm =


• As we eluded to before, you should think of an inductor as a conductor shaped in such a way as to store magnetic energy

• Typical examples include toroids, solenoids, coaxial transmission lines, and parallel-wire transmission lines

• One can determine the inductance for a given geometry using the following technique– Choose a suitable coordinate system– Let the inductor carry current, I– Determine B from Biot-Savart’s or Amperes Law and calculate the magnetic flux– Find L as a function of the flux times the number of turns over the current carried

• Mutual inductance may be calculated by a similar approach– Determine the internal inductance, Lin for the flux generated by the first inductor– Determine the external inductance, Lext produced by the flux external of the first inductor– The sum of the internal and external inductance equals the individual inductances plus the

mutual inductance between the elements

• For circuit theory, we can also right the inductance as which provides a very useful equation when quickly mapping out electronic circuits

Inductors and Inductance (2)

8/11/2010 23

∫ ⋅=Ψ1

212S

SdB

2

121

2

1212 I

NI

M Ψ==

λ

µσσεµε

σεµε

1=⇒=⇒

=

=

extext

ext

LR

LR

RC

CL

Lorentz Force Law• Recall that the force on a charged particle is simply F=qE• If the particle moves however, then an additional force is imposed from the charge

displacement of velocity, u, quantified by the magnetic field, B. The combined force is called the Lorentz Law:

• Recall from Newton’s Law that

• The kinetic energy of a charged particlein an electric field is therefore

8/11/2010 24

dtudmamBuEqF

==×+= )(

2

21

)(

umKE

dtm

qEu

dtm

qEu

dtm

qEu

dtudmEqF

zz

yy

xx

=

=

=

=

==

∫

∫

∫

dtuldtldu

ii ∫=

=

The location of the particle can also be found as

• Since we know that magnetic fields produce forces on nearby current elements, and that those magnetic fields can be generated by an isolated or coupled set of current carrying circuits, then it is only reasonable that such circuits may induce fields and magnetization between them

• We can calculate the individual flux linkage between the two components as

• Likewise we can determine a mutual inductance between the circuits that is equal from circuit 12 as it is from circuit 21 as

• Individual inductances are

• The total magnetic energy in the circuit is

Inductors and Inductance

8/11/2010 25

∫ ⋅=Ψ1

212S

SdB

2

121

2

1212 I

NI

M Ψ==

λ

1

11

1

111 I

NI

L Ψ==

λ

2

22

2

222 I

NI

L Ψ==

λ

2112222

2111221 2

121 IIMILILWWWWm ±+=++=


Forces Due to a Magnetic Field• Recall that the force on a charged particle is simply F=qE• If the particle moves however, then an additional force is imposed from the charge

displacement of velocity, u, quantified by the magnetic field, B. The combined force is called the Lorentz Law:

• Recall from Newton’s Law that

• The kinetic energy of a charged particlein an magnetic field is therefore

8/11/201026

dtudmamBuEqF

==×+= )(

2

21

)()(

)()(

)()(

)(

umKE

dtm

BvBvqdt

mBvqu

dtm

BvBvqdtm

Bvqu

dtm

BvBvqdt

mBvqu

dtudmBvqF

xyyxzz

xzzxyy

zyzyxx

=

−=

×=

−−=

×=

−=

×=

=×=

∫∫

∫∫

∫∫

dtldu

=

The location of the particle can also be found as

For B, u, and a in orthogonal directions,One can deduce a coordinate system in which

0)(

ˆ)(

)(

33

212

2

11

=×

=

=×

=

==×

=

∫

∫

∫∫∫

dtm

Bvqu

uudtm

Bvqu

dtvdtm

Bvqdt

mBvqu

ω

mBq

=ωCyclotron Resonance Frequency

dtul ii ∫=

Magnetic Torque and Moment

• Now that we have examined the force on a current carrying loop. Let’s examine the Torque applied to it

• Torque, T, on the loop is the vector product of the force, F, and the moment arm, r.

8/11/2010 27BmT

aISm

IBST

yieldingSlw

but

IBlwT

IBlF

Buniformforand

n

×=

=

=

=

=

=

ˆ

sin

sin

___

0

α

α

0ˆˆ

sin

00

0

0

=−=

×+×=×=

=

×=

∫∫∫ FFBadzBadzIBlIdF

where

FrT

FrT

lz

l

zL

αWhere we can now define a quantity m as the magnetic dipole moment with units A/m2 which is the product of the current and area of the loop in the direction normal the surface area defined by the loop


• A bar magnet or small filament loop is generally referred to as a magnetic dipole• Assume a bar magnetic of length, l, generates a uniform magnetic field, B, and a dipole

moment, |m|=Qml• Torque, T, on the loop is the vector product of the force, F, and the moment arm, r.

Torque and Dipole Properties of a Bar Magnet

8/11/2010 28

ISlQ

ISBlBQT

BQF

BlQBmT

m

m

m

m

=⇒

==

=

×=×=

Thus, because the bar magnet represents a magnetic dipole moment equal in magnitude to the dipole moment of a current loop, a bar magnet can also be taken as a magnetic dipole

Therefore the field at a reasonable distance away from any bar magnet is mathematically identical to that of a dipole.


Maxwell’s Eqns. for Static Fields

8/11/2010 29

Differential Form Integral Form Remarks

Gauss’s Law

Nonexistence of the Magnetic Monopole

Conservative nature of the Electric Field

Ampere’s LawJH

E

B

D v

=×∇

=×∇

=⋅∇

=⋅∇

0

0

ρ

0=⋅∫ SdBS

SdJldHSL

⋅=⋅ ∫∫

0=⋅∫L

ldE

∫∫ =⋅ dvSdD vS

ρ

8/11/2010 30

• It was James Clark Maxwell that put all of this together and reduced electromagnetic field theory to 4 simple equations. It was only through this clarification that the discovery of electromagnetic waves were discovered and the theory of light was developed.

• The equations Maxwell is credited with to completely describe any electromagnetic field (either statically or dynamically) are written as:

Maxwell’s Time Dependent Equations

Differential Form Integral Form Remarks

Gauss’s Law

Nonexistence of the Magnetic Monopole

Faraday’s Law

Ampere’s Circuit Law

tDJH

tBE

B

D v

∂∂

+=×∇

∂∂

−=×∇

=⋅∇

=⋅∇

0

ρ

0=⋅∫ SdBS

SdtDJldH

SL

⋅

∂∂

+=⋅ ∫∫

∫∫ ⋅∂∂

−=⋅SL

SdBt

ldE

∫∫ =⋅ dvSdD vS

ρ

Analogy Between Electric and Magnetic Fields

• Basic Laws

• Force Law• Source Element• Field intensity• Flux density• Relationship Between Fields

• Potentials

• Flux

• Energy Density• Poisson’s Eqn.

31

Electric Magnetic

( )

ερ

ψ

ψπερ

ε

ψ

πε

v

E

L

enc

r

V

EDw

dtdVCI

CVQ

SdDr

dlV

VEED

mCS

D

mVlVE

dQEQF

QSdD

ar

QQF

−=∇

⋅=

=

==

⋅=

=

−∇=

=

=

=

=

=⋅

=

∫∫

∫

2

2

211

21

4

/

)/(

ˆ4

( )

JA

HBw

dtdILI

LI

SdBRlIdA

JVH

HB

mWbS

B

mAlIH

lIduQBuQF

IldHR

aIdlBd

E

m

enc

r

µ

ψ

ψπ

µ

µ

ψ

πµ

−=∇

⋅=

=

=

⋅=

=

=−∇=

=

=

=

=

×=

=⋅

×=

∫∫

∫

2

2

20

21

4

)0(,

/

)/(

4ˆ

8/11/2010 32

Electromagnetic Work and Power

( )

( )

( )

( ) ( ) ∫∫∫

∫∫

−⋅+⋅∂∂

−=⋅×=

⋅+⋅==

+=

=⋅==∆∆

=

=⋅==∆∆

=

→∆

→∆

vS

T

meT

mvm

eve

dvEdvBHEDt

SdHEP

dvBHEDdvwW

www

BBHHv

Ww

DDEEv

Ww

2

22

0

22

0

21

21

221

21lim

221

21lim

σ

µµ

εε Electric energy density

Magnetic energy density

Electromagnetic energy

Total electromagnetic power = rate of decrease in stored energy – ohmic power dissipated

Force from a Parallel Plate Capacitor

8/11/2010 33

( )

2

2

2

2

2222

2

)(2

2)(

221

21

21

xAV

xxCV

x

VxC

xWF

CVAxxVdv

xVdvEW

eex

vve

ε

εεε

−=

∂∂

=∂

∂

=∂∂

=

=

=

== ∫∫

+Q -Q

Fex

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Electrostatic Boundary Conditions• Electrostatic boundary conditions for E and D crossing any material interface must

match the following conditions developed using Guass’s law and conservation of the electric field

• Two different dielectrics characterized by ε1 and ε2.

2211

21

εε nn

nn

EEDD=

=

∫ =⋅⇒=⋅∇S

encv QSdDD

ϕ ∫ ⋅==×∇ ldEE

0

tt

tt

EE

DD

21

2

2

1

1

=

=εε


• Magnetic boundary conditions for B and H crossing any material interface must match the following conditions developed using Guass’s law for magnetic fields and Ampere’s circuit law

Magnetic Boundary Conditions

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0=⋅∫ SdB

∫ =⋅ IldH

2

2

1

1

21

µµtt

tt

BB

HH

=

=

∫ =⋅ IldH

nn

nn

HH

BB

2211

21

µµ =

=


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Maxwell’s Time Dependent Equations:Identity Map


• In general we use the magnetic susceptibility (or relative permeability) to classify materials in terms of their magnetic property

• A material is said to be nonmagnetic if there is no bound current density or zero susceptibility. Otherwise it is magnetic

• Magnetic materials may be grouped into three classes, diamagnetic, paramagnetic, and ferromagnetic

• For many practice purposes, diamagnetic and paramagnetic materials exhibit little to no magnetic susceptibility. What magnetic properties these materials do have follows a linear response over a large range of applied fields

• Ferromagnetic materials kept below the Curie temperature exhibit very large nonlinear magnetic susceptibility and are used for conventional magnetic device applications

Classification of Magnetic Materials (1)

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• Diamagnetism– Occurs when the magnetic fields in the material due to individual electron moments

cancels each other out. Thus the permanent magnetic moment of each atom is zero.– Such materials are very weakly affected by magnetic fields.– Diamagnetic materials include Copper, Bismuth, silicon, diamond, and sodium chloride

(table salt)– In general this effect is temperature independent. Thus, for example, there is no

technique for magnetizing copper– Superconductors exhibit perfect diamagnetism. The effect is so strong that magnetic

fields applied across a superconductor do not penetrate more than a few atomic layers, resulting in B=0 within the material

• Paramagnetism– Materials whose atoms exhibit a slight non-zero magnetic moment– Paramangetism is temperature dependent– Most materials (air, tungsten, potassium, monell) exhibit paramagnetic effects that provide

slight magnetization in the presence of large fields at low temperatures


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• Ferromagnetism– Occurs in atoms with a relatively large magnetic moment– Examples: Cobalt, Iron, Nickel, various alloys based on these three– Capable of being magnetized very strongly by a magnetic field– Retain a considerable amount of their magnetization when removed from the field– Lose their ferromagnetic properties and become linear paramagnetic materials (non magnetic) when the

temperature is raised above a critical temperature called the Curie temperature.– Their magnetization is nonlinear. Thus the constitutive relation B=µ0µrH does not hold because µr

depends directly on B and cannot be represented by a single value.

– Ferromagnetic shielding• Ferromagnetic materials can be used to “focus” and guide the flow of incident magnetic fields• By placing a ferromagnetic material completely around a device, one can shield said device from

an external field. This shielding occurs b/c the ferromagnet acts as a magnetic waveguide, that transmits the field around the shape of the structure and not within it.


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• Ferromagnetism - B-H Curve– The magnetization of a ferromagnet in an external applied field, H, is presented below. – As H is increased, the magnetic field, B, within the material increases significantly and then begins to

saturate to a value Bmax saturate as |H| approaches Hmax

– As the applied field, H, is removed, the ferromagnetic material retains some degree of its magnetization until the point at which the applied field H is completely reversed at which time the magnetic field inside the material saturates to the –Bmax

– The applied field is then increased again to generate the complete Hysteresis curve– Two other defining values are indicative of every B-H magnetization (Hysteresis) curve.

• When the applied field is maxed and then again reduced to a zero value. The magnetic field within the material remains at some positive value Br referred to as the permanent flux density.

• The value upon which B become zero under an applied H value is called the coercive field intensity, Hc

• Materials with small coercive field intensity values are said to be soft magnetic materials and do not retain significant magnetization upon the removal of the field

• Hard magnets (permanent magnets) have very largecoercive field intensity values


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ECE 307: Electricity and Magnetism Spring 2010 307 Review.pdf · ECE 307: Electricity and Magnetism...

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