ECE 307: Electricity and Magnetism Spring 2010 307 Chapter 7...ECE 307: Electricity and Magnetism Spring 2010 Instructor: J.D. Williams, Assistant Professor Electrical and Computer

ECE 307: Electricity and Magnetism

Spring 2010

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: [email protected]

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.

mailto:[email protected]

8/17/2012 2

Chapter 7: Magnetostatic Fields

• Topics Covered

– Biot-Savart’s Law

– Ampere’s Circuit Law

– Applications of Ampere’s Law

– Magnetic Flux Density

– Maxwell’s Eqns. For Scalar

Fields

– Magnetic Scalar and Vector

Potentials

– Derivation of Biot-Savart’s Law

and Ampere’s Law

• Homework:

All figures taken from primary textbook unless otherwise cited.

Introduction to Magnetic Fields • For the next two and a half chapters we focus our attention on magnetic

fields

• In electrostatics we studied divergent potential fields

• In magnetostatics, we will examine solenoid rotational fields

• Also whereas, electrostatic fields are generated by static charges,

magnetostatic fields are generated by static currents (charges that move

with constant velocity in a particular direction)

• There are several similarities between electrostatic and magnetostatic fields

• For example, as we had E and D for electrostatics, we now use B and H to

examine magnetic systems

• Our study of these fields allows us to evaluate and solve for a tremendous

number of electric and electromechanical devices.

• Furthermore this study, will provide the basis for formulating an universal

theory of Electromagnetic Fields that is utilized in almost every aspect of

electrical engineering

8/17/2012 3

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. 4

Electric Magnetic

v

E

L

enc

r

V

EDw

dt

dVCI

CVQ

SdD

r

dlV

VE

ED

mCS

D

mVl

VE

dQ

EQF

QSdD

ar

QQF

2

2

2

11

2

1

4

/

)/(

ˆ4

JA

HBw

dt

dILI

LI

SdB

R

lIdA

JVH

HB

mWbS

B

mAl

IH

lIduQ

BuQF

IldH

R

aIdlBd

E

m

enc

r

2

2

2

0

2

1

4

)0(,

/

)/(

4

ˆ

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

5

232 4

sin

44

ˆ

R

Idl

R

RlId

R

alIdHd R

Current Density

• One defines differential current based on the geometry of the current element being

investigated

• Evaluation of magnetic field intensity, H,

using these three current differentials is

6

dvJdSKlId

v

R

S

R

L

R

R

advJH

R

adSKH

R

alIdH

2

2

2

4

ˆ

4

ˆ

4

ˆ

aI

H

aI

H

aI

H

adI

H

adIH

ddz

z

letting

L

ˆ2

ˆ4

ˆcoscos4

ˆsin4

csc4

ˆcsc

csc

cot

12

3

22

2

2

1

H Field From a Strait Current

Carrying Filament • The H field is determined for a strait filament of current in a manner very similar to that

of the electric field determined from a line charge

7

Line from z = 0 to

L

z

z

L

z

adzIH

adzRld

azaR

adzld

R

RlIdH

2/322

3

4

ˆ

ˆ

ˆˆ

ˆ

4

Line from z = - to

H Field From a Ring of Current

Carrying Filament (1) • Again, the H field is determined for a ring filament of current in a manner very similar to

that of the electric field determined from a circular line charge

8

z

L

ahayxhR

adld

R

RldIH

ˆˆ0,,,0,0

ˆ

4 3

Dir not present in R due to symmetry

zz

z

z

z

adHadHHd

adahdh

IHd

adahd

h

d

aaa

Rld

ˆˆ

ˆˆ4

ˆˆ

0

00

ˆˆˆ

2

2/322

2

2/322

2

2

0

2/322

2

2/322

2

2

ˆ

4

ˆ

4

ˆˆ

h

aIH

h

adIH

h

adIHdadHHd

letting

z

z

zzz

H Field From a Ring of Current

Carrying Filament (1) • Again, the H field is determined for a ring filament of current in a manner very similar to

that of the electric field determined from a circular line charge

9

zz

z

adHadHHd

adahdh

IHd

ˆˆ

ˆˆ4

2

2/322

Line from z = - to

By symmetry, the terms sum to zero

12

2

2/3222

2/322

2

2/322

2

coscos2

sin2

sin2

sincsc

tan

22

1

1

nI

dnI

H

dnI

dH

da

zadadz

z

a

dzl

Nndzdl

za

ndzIa

za

dlIadH

z

z

z

H Field From Solenoid • A solenoid is a coil or wire passing current across it with uniform radius and a number of

loops, N. One can determine the field within a solenoid by summing each of the

respective magnetic fields due to each loop. Thus the total field anywhere in the

solenoid may be found as:

10

zz

z

al

NIanIH

al

if

lNn

where

anI

H

ˆˆ

/

ˆcoscos2

12

Line from z = - to

Derivation:

• 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 typically used for symmetric cases

– Like Gauss and Coulomb’s Laws, Ampere’s law 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

11

JH

SdJI

SdHldHI

S

enc

SL

enc

Maxwell’s 3rd Eqn.

Definition of Current provided in Chapter 5

• A simple application of Ampere’s law can be used to easily derive the magnetic field

intensity from an infinite line current

aI

H

HdHadaHI

ldHIenc

ˆ2

2ˆˆ

Applications of Ampere’s

Circuit Law

12

• Consider an infinite sheet of current in the z=0 plane with a uniform current

density, K=Kyay

bHbHabHa

ldHldH

zaH

zaHH

bKldHI

x

x

yenc

000

2

1

3

2

4

3

1

4

0

0

200

0,ˆ

0,ˆ

Ampere’s Circuit Law:

Infinite Sheet of Current

13

0,ˆ2

1

0,ˆ2

1

zaK

zaKH

KH

from

xy

xy

yo

Ampere’s law and integral summation

naKH ˆ2

1

Thus, for an

infinite sheet

Apply Ampere’s

law

• Consider 2 infinite sheets of current in the z=0 and z=4 planes with a uniform

current density, K=-10ax A/m and K=10ax A/m respectively

• At a point between the two parallel plates, P (1,1,1) where 0 < (z = 1) < 4

• At a point outside of the plates, P(0,-3,10) where (z = 10) > 4 > 0


Infinite Parallel Plate Capacitor

14

mAaHHH

mAaaaaKH

mAaaaaKH

y

yzxn

yzxn

/ˆ10

/ˆ5ˆˆ102

1ˆ

2

1

/ˆ5ˆˆ102

1ˆ

2

1

40

44

00

mAHHH

mAaaaaKH

mAaaaaKH

yzxn

yzxn

/0

/ˆ5ˆˆ102

1ˆ

2

1

/ˆ5ˆˆ102

1ˆ

2

1

40

44

00

• One can use Ampere’s law to directly show the shielding of magnetic fields

using coaxial wires


Infinitely Long Coaxial Cable

15


using coaxial wires

2

2

2

2

22

002

2

2

2

ˆ

ˆ

0

1

a

IH

a

IIHdlH

a

Idd

a

ISdJI

addSd

aa

IJ

SdJldHI

a

enc

L

enc

z

z

enc



16


using coaxial wires

2

2

1

IH

IHdlH

IldHI

ba

enc

L

enc



17


using coaxial wires

0

21

2

21

2

2

22

2

22

22

22

ldHI

tb

btt

bIH

btt

bII

ddbtb

III

abtb

IJ

SdJII

IHldHI

tbb

enc

enc

enc

z

enc

enc



18


using coaxial wires

tb

tbbabtt

bI

baaI

aaa

I

H

,0

,ˆ2

12

,ˆ2

0,ˆ2

2

22

2



19

• A toroid is a solenoid turned in on itself like a donut

l

NINIH

aaNI

H

NIHldHI

o

approx

oo

enc

2

,2

2ˆ


Toroid

20

• 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

mH

HB

/104 7

0

0

Magnetic Flux Density

21

ldB

Iencˆ

0

SS

SdHSdB

0

where

0

B

dvBSdBSS

Definition of a solenoidal field

and Maxwell’s 4th eqn.

• Unlike electrostatic flux however, magnetic flux always follows a closed path and

fold in on themselves. This simple statement has profound consequences. In

electrostatics, we can easily define a point charge in which electric fields emanate

to infinity. However, the solenoidal nature of the magnetic field requires magnetic

flux to travel from a positive (north) to a negative (south) pole and it is not possible

to have a single magnetic pole at any time.

– There are NO magnetic monopoles, stipulating that an isolated magnetic

charge DOES NOT EXIST

– The minimum field requirement for magnetics is a dipole.

Magnetic Flux Density

22

Maxwell’s Eqns. for Static Fields

8/17/2012 23

Differential Form Integral Form Remarks

Gauss’s Law

Nonexistence of the

Magnetic Monopole

Conservative nature

of the Electric Field

Ampere’s Law JH

E

B

D v

0

0

0 SdBS

SdJldHSL

0L

ldE

dvSdD v

S

• In Chapter 4-6, we discussed several electrostatic problems that were more easily

solved using the electric potential to define the electric field intensity, E.

• The same approaches also reduce the difficulty in examining magnetic field

problems as well as coupled field problems that will be discussed in the 2nd course

on electromagnetic fields.

• Recalling from chapter three that a solenoidal field can be described by its scalar

and vector potentials, we can define a magnetic field using the following

requirements.

• Just as , we can define a magnetic scalar potential Vm related to H when

the current density is zero as

Magnetic Scalar & Vector Potential

8/17/2012 24

0

0

A

V

VE

0,0

0

0,

2

JV

VHJ

JVH

m

m

m

• The requirement for a solenoidal field (and Maxwell’s 4th law of electrostatics) stipulates

• And we can therefore define a magnetic vector potential, A, as

• Just as we defined the Electric Potential as

• We can define the Magnetic Vector Potential as

8/17/2012 25

for Line Current

for Surface Current

for Volume Current

0 B

AB

r

dQV

04

L

L

L

R

dvJA

R

dSKA

R

lIdA

4

4

4

0

0

0


• One can also derive these expressions directly from the magnetic field

• Where R is the distance vector from the line element dl’ at the source to the field point

(x,y,z)

• Yielding


8/17/2012 26

L

R

RlIdB

3

0 '

4

23

222

ˆ1

''''

R

a

R

R

R

zzyyxxrrR

R

0' ld

R

ldld

RRld

FfFfFf

identitytheApplying

RlIdB

L

''

11'

__

1'

4

0

However, Del operates on(x,y,z) and dl’ is a

function of (x’,y’,z’) thus

L

L

R

lIdA

R

lIdB

R

ld

Rld

4

4

'

'1'

0

0

• Applying Stokes’s Theorem provides some rather useful practical relations, including

but not limited to the total magnetic flux through an area, S, enclosed by a contour, L.

8/17/2012 27

L

Lss

ldA

ldASdASdB


• Recall the basic vector identity

Derivation of Biot-Savart’s Law

8/17/2012 28

L

R

R

L

LL

R

aldIB

R

a

R

R

R

zzyyxxrrR

R

ldld

R

IB

R

ldI

R

lIdB

AAA

2

0

23

222

0

00

2

ˆ'

4

ˆ1

''''

''

1

4

'

44

'

• Derivation of the Vector Poisson’s Eqn. for magnetic fields and current density

• Derivation of Ampere’s Law

Derivation of Ampere’s Law

8/17/2012 29

JHA

A

AAB

00

2

2

0

IldH

ISdJldH

JAA

SdASdHldH

L

SL

SSL

0

2

0

1

END

8/17/2012 30

zyx

yx

yxyx

z

z

aaaI

HHH

aaI

H

aaaaaa

aI

H

aI

H

aa

aI

H

HHH

ˆ5

31

4

1ˆ

25

3ˆ

25

4

4

ˆ5

3ˆ

5

4

20

516943

ˆ5

3ˆ

5

4ˆ

5

4ˆ

5

3ˆˆcoscos

ˆcoscos4

ˆ5

31

16

41640

ˆˆ

5/3cos

1cos

ˆcoscos4

21

2

22

12

122

1

22

1

2

121

21

H Field From an L Shaped

Current Carrying Filament • The H field is determined for a strait filament of current in a manner very similar to that

of the electric field determined from a line charge

31

L

z

z

L

z

adzIH

adzRld

azaR

adzld

R

RlIdH

2/322

3

4

ˆ

ˆ

ˆˆ

ˆ

4

Documents

ECE 307: Electricity and Magnetism Spring 2010 307 Chapter 7...ECE 307: Electricity and Magnetism Spring 2010 Instructor: J.D. Williams, Assistant Professor Electrical and Computer