Magnetic Fields in Matter Chapter 6. 2 Magnetization.

Magnetic Fields in MatterMagnetic Fields in Matter

Chapter 6Chapter 6

2

Magnetization Magnetization

aIm

3


Bm

x sinmB

x sinIabB

x sin)sinIbB(a

x sinaFFrN

torque

o

90

4


Torques and Forces on Magnetic Torques and Forces on Magnetic DipolesDipolesrotationrotation

5


randomrandom

paramagnetismparamagnetism

DiamagnetismDiamagnetism

External B fieldExternal B field

FerromagnetismFerromagnetism

6


Force on magnetic dipoleCase of uniform field (B=constant):

0 B)d(I)Bd(IF

00

7


x

y

x )(cosB)R(IBdIF 2

Case of nonuniform field (B ≠ constant):

I

B

Fringing field

8


)Bm(F

}z

By

y

Bz{I

}y

B)z (

z

B)y ({I

)]}z,,(B)z,,(B[)z dz(

)],y,(B),y,(B[)y dy({I

)}z,,(B)z dz(),y,(B)y dy(

)z,,(B)z dz(),y,(B)y dy[(I

BIdFd

2

000

000

000

000

I

y

z

x

9


x

xxx

xzyx

zyxzyx

Bm

}z

Bz

y

By

x

Bx{m

}z

Bz

z

Bx

y

Bx

y

By{m

}

z

B

z

B

z

B

zyx

y

B

y

B

y

B

zyx

{m

}z

By

y

Bz{IFd

010100

2

I

y

z

x

x mm )Bm(F

)]Ep(F[

10


Electric dipole

-

+

p

Magnetic dipole

S

N

m

Gilbert model

Im

Ampere model

11


Effect of magnetic field on atomic orbits

z evR)z R)(R

ev(SIm

moment dipole orbitalR

ev

T

eI

2

1

2

2

2

12


R

vm

R

eFF e

ocE

2

2

2

4

1

e

e

ee

eo

cBE

m

eRBv

vv

v

m

eRB

)vv)(vv(R

m

R

vvmBve

R

vmBve

R

e

FFF

2

2

4

1

22

2

2

2

only electric forceonly electric force

add magnetic fieldadd magnetic field

)vvvvv( 0

13


z evRm

m

eRBv

e

2

1

2

when B is turn on,the electron speeds up.when B is turn on,the electron speeds up.

Bm

RezB

m

RezR)v(em

ee

442

1 2222

14


.volume unit per moment dipole magneticM

20

4 R

Rm)r(A

d

R

R)r(M)r(A

20

4

R

x

y

z

M

r

r

)z,y,x(P

15

Bound Currents Bound Currents

d )]

R()r(M[d

R

R)r(M)r(A

1

440

20

23

2322

222

2

2

1

11

R

R

R

R

)zz()yy()xx(

]z)zz(y)yy(x)xx[()(

)zz()yy()xx()z

zy

yx

x(

R

/

16


d ]R

)r(M[d )]r(M[

R

d )]R

()r(M[)r(A

1

4

1

4

0

0

21 p @ )v.(eq ,)f(A)A(f)Af(

]ad)r(M[R

d )]r(M[R

1

4

1

400

17


advd )v(

])adv([cd )]v[(c

)adv(cd )]v(c[

)adv(cd )]c(v)v(c[

ad)cv(d )cv(

theorem divergenceby

constantc where, cvv let

advd )v(

:proof

07151

21

p @ )..(eq ,

)AB(C)BA(C)AC(B)CB(A

p @ )iv.(eq ,)B(A)A(B)BA(

18


current bound surface: n MK

current bound volume: MJ

adR

)r(K d

R

)r(J

]ad)r(M[R

d )]r(M[R

)r(A

b

b

S

b

V

b

44

1

4

1

4

00

00

S

b

V

b adR

R)r(K d

R

R)r(J)r(B

20

20

44

--Using Biot-Savart law--Using Biot-Savart law

19


Exp 1: Exp 1: Find the magnetic field Find the magnetic field at the center at the center of a uniformly of a uniformly magnetized sphere.magnetized sphere.

20


Exp 1:Exp 1:

ˆ Msinr Mn MK

MJ

constantM, zMM where

adR

)r(K d

R

)r(J)r(A

b

b

S

b

V

b

0

4400

bK

ˆad

R

sinM)r(A

S 4

0

21


Exp 1:Exp 1:

bK

ˆsinR)R(vK

RR

),,r(

θθ

ˆ Msinr Mn MKb

MR

MB

03

2 236115 p@ ..Ex see

22


Exp 1:Exp 1:

Mz M

z dsinM

B

z bdb

)sinb)(sinM(

z b

)sinb(dIBd

00

0

30

3

20

3

20

3

2

3

22

2

2

z )za(

aI)z(B

/ 2322

20

2

d

a

R

z

222 zab, sinba

bK

b

)bd(sinMdKdI b

21865 p@ ..Ex see

M n

)z,,(P 00

23


Find the magnetic field of point Find the magnetic field of point PP..

MszMnMK

MJ

b

b

00

0

constantM, zMM 00

L

dI

zd

z

z]]R)Lz[(

Lz

)Rz(

z[

M

z]R)zz[(

)zz(M

z]R)zz[(

zdMRB

z]R)zz[(

zdMRz

]R)zz[(

dIRBd

//

L

/

L

/

//

2122212200

02122

00

0 23220

20

23220

20

2322

20

2

2

2

22

zdMzdKdI let b 0

24


]]R)Lz[(

Lz

)Rz(

z[

R

]R)zz[(

)zz(

R

]Ru[

u

Rcos

R

dsinR

cscR

dcscR

]Ru[

du

]R)zz[(

zd

//

L

/

Lz

z/

Lz

z /

L

/

212221222

021222

212222

2

33

2

2322

0 2322

1

1

11

1

2

1

2

1

2

1

LzuLz,zuz

zdduuzz

0

222222

2

1 cscR)(cotRRu

dcscRducotRu

)z,,(P 00

dI

zd

z

L

uu

RR

25

Physical Interpretation of Bound Currents Physical Interpretation of Bound Currents

MtI MatIa

MVIam

nMKb

Mt

IK current surface b

When the magnetization is When the magnetization is uniformuniform::

26

When the magnetization is When the magnetization is nonuniformnonuniform::

Physical Interpretation of Bound Currents Physical Interpretation of Bound Currents

dydzy

Mdz)]y(M)dyy(M[I

:directionx in

zzzx

y

Mda/I)J( :density current volume z

xxb

dzdyz

Mdy)]z(M)dzz(M[I

:directiony in

yyyx

z

Mda/I)J( :density current volume y

xxb

z

M

y

M)J( yzxb

MJb

0 MJb

27

The Auxiliary Field H The Auxiliary Field H

M

fJ

fb JJJ

fenc

f

f

f

f

IdH

adJad)H(

JH

J)MB

(

)M(JJ)B(

law s'Ampere

0

0

1

MB

H

0

28

The Auxiliary Field H

Exp 2: Exp 2: A long copper rod of radius A long copper rod of radius RR carries a uniformly carries a uniformly distributed (free) distributed (free) current current II. Find . Find HH inside and outside the rod. inside and outside the rod.


Rs, ˆ s

I

Rs, ˆR

Is

H

Rs, I

Rs, R

sI

sH

Rs, I

Rs, )s)(R

I(

)ˆsd()ˆH(

IdH law s'Ampere by fenc

2

2

2

2

2

2

22

I

29

Exp 2:Exp 2: The Auxiliary Field H The Auxiliary Field H

Rs, ˆ s

I

Rs, ˆ )MR

Is(

Rs, ˆ 0)s

I(

Rs, ˆ )MR

Is(

)MH(B

MBH

Rs, ˆ s

I

Rs, ˆ R

Is

H

0

0

2

2

2

2

12

2

0

20

0

20

0

0

2

30

A Deceptive Parallel A Deceptive Parallel

JB

0

fJH

M)MB(H

0

1

Ampere law in vacuum :Ampere law in vacuum :

Ampere law in magnetic materials :Ampere law in magnetic materials : HB

0

Whereas ,the divergence of is not, in general, zero.Whereas ,the divergence of is not, in general, zero.0 B

H

31

Boundary condition Boundary condition

)MM(HH

a)MM(a)HH(

ad)MH(adB

belowabovebelowabove

belowabovebelowabove

0

00

belowabove BB

a

n

nbelowH

aboveH

belowM

aboveM

32

nKHH

KHH

KHH

IdH

fbelowabove

f//below

//above

f//below

//above

fenc

Boundary condition Boundary condition

)nK(BB belowabove

0

n

bf KKK

//aboveH

//belowH

33

Linear and Nonlinear Media Linear and Nonlinear Media

tysceptibili magneticsu: HM mm

磁化率磁化率

34


materials cdiamagneti,

materials icparamagnet,

J)H(MJ

typermeabili: )(

HB

HHH)()MH(B

HM

m

m

fmmb

mr

rm

m

0

0

1

1

00

000

磁導率磁導率

35

For linear isotropic homogeneous mediaFor linear isotropic homogeneous media

Electric Magnetic

)1( er )1( mr

PED 0

EP e

0

Er

0

E

MHB

00

HM m

Hr

0

H


36


Exp 3: Exp 3: An infinite solenoid (An infinite solenoid (nn turns per unit length, current turns per unit length, current II) is ) is filled with linearfilled with linear material of susceptibility material of susceptibility χ χ mm. Find the magnetic field . Find the magnetic field inside the solenoid.inside the solenoid.

z InH InH

IdH

law s'Ampere by

fenc

ˆ nI)sH(nMK

current surface bound the

z nI)(H)(HB

mmb

mm

11 00

37

Ferromagnetism Ferromagnetism

Ferromagnetic Materials : Fe, Co, Ni, Gd, Dy

domain

The domains range from about 10-12 to 10-8 m3 in volume and contain 1017 to 1021 atoms.

domain wall

38


Ferromagnetic domains. (photo courtesy of R. W. DeBlois)Ferromagnetic domains. (photo courtesy of R. W. DeBlois)

39

)MH(B

0


Hysteresis loopHysteresis loop

saturation

remanence

coercive

c

a g

b

d

f

e

40

Hysteresis loopHysteresis loop

hard

soft


56

In diamagnetic materials (bismuth 鉍 ): spin moments tend to be dominant and produce fields that oppose the external field. Thus, the internal magnetic field is reduced slightly compared to the external field.

If diople moments dominate slightly, then the internal field is increase slightly over the external field and the material is paramagnetic (tungsten 鎢 ).

Large diople moments are produced in certain regions or domains for the ferromagnetic materials (iron). A random domain alignment exists for virgin ferromagnetic material. When an external field is applied and then removed, a net alignment occurs given permanent magnetization and hysteresis effect . Alloys of some of the ferromagnetic materials are also ferromagnetic (alnico磁性合金 )

In ferrimagnetic materials, adjacent atoms develop unequal, but oppositely directed moments, allowing a rather larger response to external fields. From the point of view of engineering applications, the ferrites are very important ferrimagnetic materials. Ferries possess a very high resistance, and hence give very little eddy current loss at higher frequencies when used as transformer cores.

The magnetic tape used for audio and video recording is a superparamagnetic material and is composed of an array of small ferromagnetic particles.

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Magnetic Fields in Matter Chapter 6. 2 Magnetization.

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Transcript of Magnetic Fields in Matter Chapter 6. 2 Magnetization.