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Magnetic Fields in MatterMagnetic Fields in Matter
Chapter 6Chapter 6
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Magnetization Magnetization
aIm
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Magnetization Magnetization
Bm
x sinmB
x sinIabB
x sin)sinIbB(a
x sinaFFrN
torque
o
90
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Magnetization Magnetization
Torques and Forces on Magnetic Torques and Forces on Magnetic DipolesDipolesrotationrotation
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Magnetization Magnetization
randomrandom
paramagnetismparamagnetism
DiamagnetismDiamagnetism
External B fieldExternal B field
FerromagnetismFerromagnetism
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Magnetization Magnetization
Force on magnetic dipoleCase of uniform field (B=constant):
0 B)d(I)Bd(IF
00
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Magnetization Magnetization
x
y
x )(cosB)R(IBdIF 2
Case of nonuniform field (B ≠ constant):
I
B
Fringing field
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Magnetization Magnetization
)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
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y
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x
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Magnetization Magnetization
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xxx
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Bm
}z
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}
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y
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010100
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y
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x
x mm )Bm(F
)]Ep(F[
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Magnetization Magnetization
Electric dipole
-
+
p
Magnetic dipole
S
N
m
Gilbert model
Im
Ampere model
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Magnetization Magnetization
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
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Magnetization Magnetization
R
vm
R
eFF e
ocE
2
2
2
4
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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
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Magnetization Magnetization
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
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Magnetization Magnetization
.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
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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
/
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Bound Currents Bound Currents
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
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Bound Currents Bound Currents
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(
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Bound Currents Bound Currents
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
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Bound Currents Bound Currents
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.
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Bound Currents Bound Currents
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
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Bound Currents Bound Currents
Exp 1:Exp 1:
bK
ˆsinR)R(vK
RR
),,r(
θθ
ˆ Msinr Mn MKb
MR
MB
03
2 236115 p@ ..Ex see
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Bound Currents Bound Currents
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
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M n
)z,,(P 00
23
Bound Currents Bound Currents
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
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Bound Currents Bound Currents
]]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
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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::
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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
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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
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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.
The Auxiliary Field H The Auxiliary Field H
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
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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
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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
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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
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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
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Linear and Nonlinear Media Linear and Nonlinear Media
tysceptibili magneticsu: HM mm
磁化率磁化率
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Linear and Nonlinear Media Linear and Nonlinear Media
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
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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
Linear and Nonlinear Media Linear and Nonlinear Media
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Linear and Nonlinear Media Linear and Nonlinear Media
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
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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
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Ferromagnetism Ferromagnetism
Ferromagnetic domains. (photo courtesy of R. W. DeBlois)Ferromagnetic domains. (photo courtesy of R. W. DeBlois)
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)MH(B
0
Ferromagnetism Ferromagnetism
Hysteresis loopHysteresis loop
saturation
remanence
coercive
c
a g
b
d
f
e
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Hysteresis loopHysteresis loop
hard
soft
Ferromagnetism Ferromagnetism
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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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The Auxiliary Field H The Auxiliary Field H