ECE 307: Electricity and Magnetism

Fall 2012

Instructor: J.D. Williams, Assistant Professor

Electrical and Computer Engineering

University of Alabama in Huntsville

406 Optics Building, Huntsville, Al 35899

Phone: (256) 824-2898, email: john.williams@uah.edu

Course material posted on UAH Angel course management website

Textbook:

M.N.O. Sadiku, Elements of Electromagnetics 5th ed. Oxford University Press, 2009.

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.

8/17/2012 2

Chapter 9: Maxwell’s Equations

• Topics Covered

– Faraday’s Law

– Transformer and Motional

Electromotive Forces

– Displacement Current

– Magnetization in Materials

– Maxwell’s Equations in Final

Form

– Time Varying Potentials

(Optional)

– Time Harmonic Fields (Optional)

• Homework: 3, 7, 9, 12, 13, 16,

18, 21, 22, 30, 33

All figures taken from primary textbook unless otherwise cited.

8/17/2012 3

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, is the flux linkage, 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.

dt

dN

dt

dVemf

8/17/2012 4

Faraday’s Law (2)

• To elaborate on emf, lets consider a battery circuit.

• The electrochemical action within the battery 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 the between the two open terminals in the battery is therefore

• 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

IRldEldEV

P

N

e

P

N

femf

L

e IRldE 0

P

N

f

L

f

L

ef

ldEldEldE

EEE

0

8/17/2012 5

Transformer and Motional

Electromotive Forces (1)

• For a single circuit of 1 turn

• The variation of flux with time may be caused by three ways

1. Having a stationary loop in a time-varying B field

2. Having a time-varying loop in a static B field

3. Having a time-varying loop in a time-varying B field

• A stationary loop in a time-varying B field

SL

emf

emf

SdBdt

dldEV

dt

d

dt

dV

dt

BdE

SdBdt

dSdEldEV

SSL

emf

One of Maxwell’s for time varying fields

8/17/2012 6

• A time-varying loop in a static B field

BuE

ldBuSdE

TheoremsStokesby

uBlV

IlBF

BIlF

ldBuldEV

BuQ

FE

fieldEmotionalain

BuQF

m

LL

m

emf

m

m

LL

emf

m

m

_'_

____

Some care must be used when applying this equation

1. The integral of presented is zero in the portion of the

loop where u=0. Thus dl is taken along the portion

of the lop that is cutting the field where u is not

equal to zero

2. The direction of the induced field is the same as that

of Em. The limits of the integral are selected in the

direction opposite of the induced current, thereby

satisfying Lenz’s Law

Transformer and Motional

Electromotive Forces (2)

8/17/2012 7

• A time-varying loop in a time-varying B field

Budt

BdE

ldBuSddt

BdldEV

m

LSL

emf

One of Maxwell’s for time varying fields

Transformer and Motional

Electromotive Forces (3)

8/17/2012 8

• Conducting element is stationary and the

magnetic field varies with time

• Assume the bar is held stationary at y =0.08 m

and B = 4cos(106t)az mWb/m2

• Assume the length between the two conducting

rails the bar slides along is 0.06 m

Transformer and Motional

Electromotive Forces: Example1

dt

BdE

Sddt

BdV

m

S

emf

Vt

t

txy

dxdyt

Sdatdt

dSd

dt

BdV

S

S

z

S

emf

)10sin(2.19

)10sin()10)(10)(4(06.008.0

)10sin()10)(10)(4(

)10sin()10)(10)(4(

ˆ))10cos(004.0(

6

663

663

663

6

8/17/2012 9

• Conductor moves at a velocity u = 20ay m/s in

constant magnetic field B=4az mWb/m2

• Assume the length between the two

conducting rails the bar slides along is 0.06 m

mV

xdx

adxaaV

BuE

ldBuldEV

x

L

zyemf

m

LL

emf

8.406.008.0

08.008.0

ˆˆ004.0ˆ20

Transformer and Motional

Electromotive Forces: Example 2

8/17/2012 10

• Conductor moves at a velocity u = 20ay m/s in time

varying magnetic field B=4cos(106t-y)az mWb/m2

• Assume the length between the two conducting

rails the bar slides along is 0.06 m

Transformer and Motional

Electromotive Forces: Example 3

L

xzy

z

S

zemf

LSL

emf

adxayta

adxdyaytdt

dV

ldBuSddt

BdldEV

ˆˆ)10cos()4)(10(ˆ20

ˆˆ)10cos()4)(10(

63

63

)10cos(240)10cos(240

)10cos(4000)10cos(4000

)10cos()4(10)10cos()10(8)4)(10(

)10cos()4)(10(20

)10cos()4(10)10cos()4)(10(

)10cos()4)(10(20

)10cos()4(10)10cos()4)(10(

)10cos()4)(10(20

ˆˆ)10sin(10)4)(10(

66

66

63623

63

6363

63

6363

63

663

tytV

txytxV

xtxytV

xyt

xtxytV

dxyt

xtxytV

dxyt

adxdyaytV

emf

emf

emf

emf

emf

z

S

zemf

8/17/2012 11

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

t

DJH

t

DJ

t

DD

ttJJ

JJH

JJH

tJ

JH

JH

d

vd

d

d

v

0

0

0

Another of Maxwell’s for time varying fields

This one relates Magnetic Field Intensity to conduction

and displacement current densities

Displacement Current (1)

This vector identity for the cross product is mathematically

valid. However, it requires that the continuity eqn. equals

zero, 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

8/17/2012 12

• Using our understanding of conduction and displacement current density. Let’s test this

theory on the simple case of a capacitive element in a simple electronic circuit.

Idt

dQSdD

tSdJldH

ISdJldH

IISdJldH

Sdt

DSdJI

t

DJH

SS

d

L

enc

SL

enc

SL

d

22

2

1

0

If J =0 on the second surface then Jd must be

generated on the second surface to create a time

displaced current equal to current on surface 1

Displacement Current (2)

Ampere’s circuit law to a closed path provides the following eqn.

for current on the first side of the capacitive element

However surface 2 is the opposite side of the capacitor and has no

conduction current allowing for no enclosed current at surface 2

Based on the equation for displacement current density, we can

define the displacement current in a circuit as shown

8/17/2012 13

• Show that Ienc on surface 1 and dQ/dt on surface 2of the capacitor are both equal to C(dV/dt)

dt

dVC

dt

dV

d

S

dt

dES

dt

dDS

dt

dS

dt

dQI

surfacefrom

dt

dVC

dt

dV

d

SSJI

dt

dV

dt

DJ

d

VED

sc

dd

d

1__

Displacement Current (3)

8/17/2012 14

• 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

t

DJH

t

BE

B

D v

0

0 SdBS

Sdt

DJldH

SL

SL

SdBt

ldE

dvSdD v

S

8/17/2012 15

• A few other key equations that are routinely used are listed over the next couple of slides

Maxwell’s Time Dependent Equations (2)

0ˆ

ˆ

ˆ

0ˆ

12

21

21

21

n

sn

n

n

aBB

aDD

KaHH

aEE

Continuity Equation

Compatibility Equations

Boundary Conditions for Perfect Conductor

Boundary Conditions

Equilibrium Equations

0E

m

m

Jt

BE

B

t

DJH

D v

m = free magnetic density

0J

0H

0nB

0tE

tJ v

Lorentz Force Law

BuEQF

8/17/2012 16

Maxwell’s Time Dependent Equations:

Identity Map

8/17/2012 17

Time Varying Potentials

Jt

AA

t

VV

ApotentialsforConditionLorentzApply

t

VA

gchoobyconditionsfieldvectortheLimit

t

A

t

VJAA

yields

AAA

identityvectortheApplying

t

A

t

VJA

t

AV

tJ

t

EJA

dt

DdJABH

LawCircuitsAmpereApplying

v

2

22

2

22

2

22

2

2

2

0:____

:sin______

:

:___

11

:__'_

At

VE

t

AVE

Vt

AE

t

AE

Att

BE

LawsFaradayApplying

AB

AfromBofDefinition

R

dvJA

R

dvV

potentialsField

v

v

v

v

2

0

:_'_

:____

4

4

:_

8/17/2012 18

Wave Equation

00

00

2

22

2

22

2

22

2

22

1

1

___

u

cn

c

u

t

BB

t

EE

yieldsspacefreeIn

Jt

AA

t

VV v

Refractive index

Speed of the wave in a medium

Speed of light in a vacuum