chung-ang university field & wave electromagnetics ch 6. static magnetic fields

Chung-Ang University Field & Wave Electromagnetics

CH 6. Static Magnetic Fields


6-1 IntroductionStatic Electric Field (N), (6-3)eF qE

Magnetic Force mF

(6 2)

0 F/m ( )

3vv

0

-120

ρ= ρ (C/m ):total volume charge density

ε

ε = 8.854 10 permitivity

E

E

Electric Force / ( / / )eE F q N C V m

2

,

or ( / )

(6.1)f f

D E electric displacement

electric flux density C m

D = ρ ρ : free charge density

2 ( / , )B Wb m Teslas T

uq

mF

B

BmF

(N).mdF Idl B

(N) (6-4)mF qu B

Idl

CD Pickup Control

Fleming’s Left hand rule

“Hunt for Red October”(magneto-hydro-dynamics)

Particle accelerator (cyclotron) CRT monitor

Magnetic Flux Density

1 10000 gauss

The earth magnetic field 0.5 gauss

T


Nicola Tesla – extra-terrestrial?

Have you ever heard of Nikola Tesla? You should get to know him. Maybe he was a brother - specialized in energy. He invented the Tesla Coil and was instrumental in discovering ways of propagating energy (electrical) wirelessly and over wires in the most efficient manner. A book on him has pictures and a lot of information on his many brilliant devices. There seems to be a lot of speculation that either he was a walk- in, arrived on our planet as a child (left on a doorstep), was in touch with higher beings from Mars, etc. There are several theories to explain his brilliance - all involving ETs in one way or another. Nikola Tesla was born in 1856 in Smiljan Lika, Croatia. He was the son of a Serbian Orthodox clergyman. Tesla studied engineering at the Austrian Polytechnic School. He worked as an electrical engineer in Budapest and later emigrated to the United States in 1884 to work at the Edison Machine Works. He died in New York City on January 7, 1943. During his lifetime, Tesla invented fluorescent lighting, the Tesla induction motor, the Tesla coil, and developed the alternating current (AC) electrical supply system that included a motor and transformer, and 3-phase electricity. – How come? Surprised!Tesla is now credited with inventing modern radio as well; since the Supreme Court overturned Guglielmo Marconi's patent in 1943 in favor of Nikola Tesla's earlier patents. Believe or not?!– He invented a converter that converts cosmic energy to electrical energy and he mounted the converter on a car to drive electric car!


Nicola Tesla – extra-terrestrial?

high voltage discharge experiments, Colorado Springs Laboratory. Dec 31, 1899


General Three-Beam Optical Pickup Organization


6-1 Introduction

Elecromagnetic Force F

eF

mF

BuqEq

(N) (6-5)q E u B

Lorentz’s Force EquationThere is no constant such as !!

☞ are the fundamental quantities, not or !!

,

BE

& HD

With the help of the four Maxwell equations, the equation of continuity, and the Lorentz force equation,

we can now explain all of the electromagnetic phenomena!

f

DH J

t

��

fD ��

0 B

t

BE

Maxwell equations( Ch. 7)

the equation of continuity(Ch. 5)

Jt

the Lorentz force equation(Ch.6)

F q E u B


6-2 Fundamental Postulates of Magnetostatics in Free Space

(6-6)0 B

0 (6-7) TB J

2

70

(A/m ) : total volume current density

4 10 H/m (permeability)

TJ

(6-9)0 S

B ds

“There are no magnetic flow sources, (no magnetic monopole)

and the magnetic flux lines always close upon themselves”

The Law of conservation of magnetic flux

0 F/m ( )

3TT

0

-120

ρ= ρ (C/m ):total volume charge density

ε

ε = 8.854 10 permitivity

E

E

0v S

Bdv B ds

0( . / )S

cf E ds Q

N

SS

NS

N

S

NS

N

Magnetic poles cannot be isolated.

SN

SN

SN

SN


6-2 Fundamental Postulates of Magnetostatics in Free Space

The magnetic flux lines follow closed paths from one end of a magnet to the other end outside the magnet,and then continue inside the magnet back to the first end.

0B

0SsdB

0 TB J

SSsdJsdB

0 0 TCB dl I

Path C is the contour bounding the surface S,

I is the total current through S.

Ampere’s Circuital Law

the circulation of the magnetic flux density in free space around any closed

path is equal to times the total current flowing through the surface

bounded by the path.0

0 TCB dl I

Ex.6-1, -2, -3, p228

(6-10)


0 TB J

Ex.6-1

b

02

01 2

ˆ 2

ˆ 2

IB a r b

rrI

B a r bb

1 ?B

2 ?B

I2

ˆz

IJ a

b

1 ?B

2 ?B 1

0 0

r za a a

Br r z

rB

1 0B J

2 0B

0 TB J

Function of position!!!

Function of position!!!

0 TCB dl I

2

0ˆ

CB dl a B

0ˆ Ta rd I

02 TB r I

x

y2

2

TI I r b

rI r b

b


Magnetic Flux Density Inside a Closely Wound Toroidal Coil (Ex. 6-2)

0 TCB dl I

1. In order to calculate B, contour C should be taken such that B is constant on the contour!

B=0

B=02. Determine coordinate system and the direction

of B.3. Integrate and calculate B.

r

NIBB

2

ˆˆ 0

(for

)()

2 0

abrab

NIrBdBC

)()(

0

abrabr

B

orfor

r

B

b - a b+ a


nIBnLIBL 00

Infinitely long solenoid can be considered as a part of toroidal coil of infinite radius.

Or as a special case of toroid

- no magnetic field outside- B field inside must be parallel to the

axis.

nIIb

NB 00 2

Magnetic Flux Density Inside an Infinitely Long Solenoid (Ex. 6-3)

1. In order to calculate B, contour C should be taken such that B is constant on the contour!

2. Determine coordinate system and the direction of B.

3. Integrate and calculate B.0 0

0

TCB dl I

B

n turns/m


6-3 Vector Magnetic Potential

(6-15) B A

A vector field is determined to within an additive constant if both its divergence and its curl are specified every where.

How to find a divergence of A

0B J A

So, JA

0

AAA

AAA

)(

)(2

2

Laplacian of A

A

Vector Magnetic Potential

0 B

We know that 0)( A

☞ Helmholtz’s theorem

( 0, ( ) 0 )E V E V

2V V

2 2 2

2 2 2V

x y z

2V

Poisson’s Equation

2 0V

Laplace’s Equation

ˆ( xax

ˆya

y

ˆ )za

z

ˆ( xa

x

ˆyay

ˆ )za V

z

Can we determine A with this equation?



yyyyxx AaAaAaA 2222 ˆˆˆ

The Laplacian of a vector field is another vector field whose components are

the Laplacian of the corresponding components of .

A

A

JAA

02)( Therefore,

With the purpose of simplifying above equation to the greatest extent possible,

0 (6-20)A

Coulomb condition (gauge)

20 (6-21)A J

vector Poisson’s equation

In Cartesian coordinates,

zzyyxx JAJAJA 02

02

02 , ,

and becomesJAA

02)(

Now, we have 0A

B A

2V

2 A



Poisson’s equation in electrostatics

0

2

V

vdR

VV

04

1

So, we have the solution for JA

02

0 (Wb/m) (6-23)4 V

JA dv

R

2 2 20 0 0 , , x x y y z zA J A J A J

4

0 vdR

JA

V

xx

4

0 vdR

JA

V

yy

40 vd

R

JA

V

zz

This enables us to find the vector magnetic potential from the volume

current density .

A

J

( , , )x y z

( , , )x y z

Jdv

A

ˆ ( )xR a x x

ˆ ( )za z z ˆ ( )ya y y

R

Source point

Field point


6-3 Vector Magnetic PotentialMagnetic Flux

Vector potential relates to the magnetic flux through a given area S that is

bounded by contour C in a simple way;

A

(6-24)S

B ds

sdAS

(6-25)

CA dl

Thus, vector magnetic potential does have physical significance in that its line

integral around any closed path equals the total magnetic flux passing through the

area enclosed by the path.

A


6-4 The Biot-Savart LawWe are interested in determining the magnetic field due to

a current-carrying circuit.For a thin wire with cross-sectional area S, we have

Jdv JSdl Idl

4

0 vdR

JA

V

0

C (Wb/m)

4

I dlA

R

Magnetic flux density is then

0 0 (6-28)4 4C C

I Idl dlB A

R R

Unprimed curl operation implies differentiations with respect to the space

coordinates of the field point, and the integral operation is with respect

to the primed source coordinates.

?B


6-4 The Biot-Savart Law

0 1 1

4 C

IB dl d

R R

we use the following identity GfGfGf

)()(

So, Magnetic flux density is equal 0

1

2 2 2 21x x y y z z

R

Rza

Rya

Rxa

R zyx

1ˆ

1ˆ

1ˆ

1

23

222

)(ˆ)(ˆ)(ˆ

zzyyxx

zzayyaxxa zyx

23

1ˆ

Ra

R

RR

0

4 C

I dB A

R

ˆ ( )xR a x x

ˆ ( )za z z ˆ ( )ya y y

02

(6-32)ˆ

4

R

C

I d aB

R

Biot-Savart Law

Sometimes it is convenient to write above equation in two steps:

CBdB

02

(6-33b)ˆ

with 4

RI d adB

R

0

3(6-33c)or with

4

I d RdB

R

Ex. 4,5,6 p236


AB

from findingBy (a)

'

0 '

4 C R

dIA

LrL

LrLIz

rzzI

zrz

dzIzA

L

L

L

L

22

220

220

22

0

ln4

ˆ

''ln4

ˆ'

'

4ˆ

symmetry lcylindrica from 0,ˆ1ˆ)ˆ(

zzzz

A

r

AA

rrAzAB

)(2

ˆ ,2

ˆln4

ˆ 0

22

0

22

220 Lr

r

IB

rLr

IL

LrL

LrLI

rB

B from a Current-Carrying Straight Wire (Ex. 6-4) (I)

ˆzd a dz 2 2R z r


B from a Current-Carrying Straight Wire (Ex. 6-4) (II)

ˆ rR a r��

(b) By applying Biot-Savart law

0

3/ 22 2

0

2 2

'ˆ

4 '

ˆ 2

L

L

B d B

I rdza

z r

ILa

r L r

��

ˆ 'za z

ˆ' ' zd R a dz �� ˆ( ra r ˆ ', a rdzˆ ') za z

03

(6-33c) 4

I d RdB

R

Which method do you like,

using = ,B A��

or using Biot-Savart law?

It is easier to use = -E V

to find E. Why?


Magnetic flux density at the center of the square loop is equal to four times that caused by a single side of length L.Using the result of Ex. (6-4),

w

Iz

w

IzB

00 22

ˆ42

ˆ

B at the Center of a Square Loop (Ex. 6-5)

222

ˆ22

0 w/, r w/LrLr

ILB

for

2)2(2

ˆ)2()2()2(2

)2(ˆ2

0

22

0

w/

I

w/w/w/

w/IB

Converting the direction to z and multiplying 4,

However, it takes considerable efforts to calculate B other than center!


22

,ˆˆ

,'ˆ'

bzR

brzzR

bdd

Cylindrical symmetry : only consider z-component

2/322

20

2

0 2/322

20

2ˆ

'ˆ

4 bz

Ibz

bz

dbz

IB

B at a Point on the Axis of a Circular Loop (Ex. 6-6)

Apply Biot-Savart law to the circular loop

'ˆ'ˆ

)ˆˆ('ˆ

'

2

dbzdbzr

brzzdb

Rd

03

(6-33c) 4

I d RdB

R

2

0r̂d

ˆ2 r

chung-ang university field & wave electromagnetics ch 6. static magnetic fields

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