lesson 12 magnetostatics in materials - national tsing hua
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Lesson 12Magnetostatics in Materials
楊尚達 Shang-Da YangInstitute of Photonics TechnologiesDepartment of Electrical EngineeringNational Tsing Hua University, Taiwan
Sec. 12-1Static Magnetic Field in Materials
1. Classical models of induced magnetic dipoles
2. Magnetization vectors3. Magnetization currents4. Magnetic field intensities
Induced magnetic dipole-1
Any material has many small magnetic dipoles (current loops) arising from (1) orbiting electrons, (2) spinning nucleus and electrons
A material bulk made up of a large number of randomly oriented molecules typically has no macroscopic dipole moment in the absence of external magnetic field
What’s spin? (1)
Spin of elementary particles cannot be explained by assuming they are made up of even smaller particles rotating about a center
Spin is about angular momentum of elementary particles, quantized, cannot be altered
What’s spin? (2)
A particle with charge q, mass m, spin S has an intrinsic magnetic dipole moment:
Smqg
2=μ
2≈g
Angular momentum of an electron (spin-1/2) measured along any direction can only take on values of 2h±
Photon (spin-1): h± ,0
Induced magnetic dipole due to orbiting electron-1
00 ωφrau vv =
EeFe
vv−=
204 r
qaE r πεvv
=
rma er20ω
v=
Induced magnetic dipole due to orbiting electron-2
BueFm
vvv×−=
ωφrau vv =
rma er2ωv=
↑==πω
π 22e
rueI
⇒> ,0ωω
( )2rIam z π⋅Δ−=Δ vv
πωω
2)( 0−
=ΔeI
diamagnetism
Strategy of analysis
It is too tedious to directly superpose the elementary fields:
Instead, we define magnetization vector as:
the kth dipole moment inside a differential volume Δv
⇒ Magnetization current , Magnetic field intensity H
v
[ ]θθπμ
θ sincos24
)( 30 aaRmrB R
vvvv +≈
vm
M k
v Δ≡ ∑
→Δ
vv
0lim
mJv
Magnetization surface current density-1
On the air-material interface, is dis-continuous, net magnetization current exists
Mv
Magnetization surface current density-2
Consider a differential volume
with rectangular current loop of
dxdydzV =Δ
,dydzS =Δ I
IdydzaSIaM xxvvv
=Δ=Δ
dxIa
dxdydzIdydza
VMM xx
vvv
v==
ΔΔ
=
Magnetic dipole moment:
Magnetization vector:
Magnetization surface current density-3
By inspection, the surface current density:dx
IaVMM x
vv
v=
ΔΔ
=
⎟⎠⎞
⎜⎝⎛=
mA
dxIaJ zms
vv
( )mA nms aMJ vvv×=⇒
To generalize the result,
,dxIaa
dxIaaM zyxn
vvvvv=×=×
Magnetization volume current density-1
In the interior of a magnetic material, net magnetization current exists where is inhomogeneous:
Mv
Magnetization volume current density-2
Consider two m-dipoles with x-dependent, z-direction magnetization vectors: , )(xMaz
v )( dxxMaz +v
( )
dzxMxIdz
xIdzSSxIxM
)()(
)()(
=⇒
=⋅ΔΔ
=
where
Similarly,
dzdxxMdxxI )()( +=+
Magnetization volume current density-3
Net current passing through the interfacing surface bounded by C is: ( ) ( )
[ ]dzdxxMxMdxxIxI
)()( +−=+−
The current density is:
dxdzdxxIxIaJ ym
)()( +−= v
v
xMa
dxdxxMxMa
zy
y
∂∂
−=
+−=
v
v )()(
Magnetization volume current density-4
xMa
xMx
aaa
MMMzyx
aaaM
zy
z
zyx
zyx
zyx
∂∂
−=∂∂=
∂∂∂∂∂∂=×∇
v
vvv
vvv
v
)(0000
To generalize the result:
( )2mA MJm
vv×∇=⇒
vdrrR
rJrAV
′′′
= ∫ ′ 0
),()(
4)( vv
vvvv
πμ
Comment-2
Equivalent current densities:
can be substituted into the formulas:
to evaluate the contribution of m-field due to magnetized materials.
,nms aMJ vvv×= MJ m
vv×∇=
ABvv
×∇=
Example 12-1: Permanent magnet (1)
Consider a uniformly magnetized cylinder of radius b, length L, and . Find the on-axis
0MaM zvv
=
Bv
On the side wall:
00 MaaMaaMJ rznms φvvvvvv
=×=×=
On the top & bottom walls:( ) 00 =±×=×= zznms aMaaMJ vvvvv
In the interior:( ) 00 =×∇=×∇= MaMJ zmvvv
Example 12-1: Permanent magnet (2)
( )[ ] 2/320 12
),0,0(−
+= bzbIazB z
μvv2/32
00 12
−
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ ′−
+′
=b
zzb
zdMaBd z
μvv
Example 12-1: Permanent magnet (3)
Total magnetic flux density:
⎥⎥⎦
⎤
⎢⎢⎣
⎡
+−
−−
+== ∫
=′
=′ 222200
0 )(2 bLzLz
bzzMaBdB z
Lz
z
μvvv
Magnetic field intensity - Definition (1)
Total M-field is created by free & magnetization currents:
JBrv
0μ=×∇ ( )mJJBvvv
+=×∇ 0μ
Mv
×∇
( ),00 MJBvvv
×∇+=×∇ μμ
( ) ,00 JMBvvv
μμ =×∇−×∇⇒
JMB vvv
=⎟⎟⎠
⎞⎜⎜⎝
⎛−×∇⇒
0μ
Magnetic field intensity - Definition (2)
⇒(A/m)
…Ampere’s law
only free current
JMB vvv
=⎟⎟⎠
⎞⎜⎜⎝
⎛−×∇
0μ
JHvv
=×∇
MBHv
vv
−=0μ
IldHC
=⋅∫ vv
…Magnetic field intensity
Comment-1
Total M-field is the summation of fields due to free current & magnetization
( )Bv
~( )H
v~ ( )M
v+~
,00 MHBvvv
μμ +=⇒ MBHv
vv
−=0μ
Comment-2
Total E-field is the summation of fields due to free charge & polarization
( )Ev
~( )D
v~ ( )P
v−~
,0 PDEvvv
−=⇒ ε PEDvvv
+= 0ε
0=Hv
0HhHvv
=02HhH
vv=
Magnetic field intensity - Usefulness (1)
For linear, homogeneous, and isotropicmagnetic materials, the magnetization vector is proportional to the external magnetic field :
SIhm ⋅Δ−=vv0=mv
Susceptibility, independent of magnitude, position, and direction of H
vHM m
vvχ=
SIhm ⋅Δ−= 2vv
Magnetic field intensity - Usefulness (2)
⇒
…Permeability of the medium
A single constant replaces the tedious induced m-dipoles, magnetization vector, equivalent magnetization currents in determining the total magnetic field.
μ
,0
MBHv
vv
−=μ
( )MHBvvv
+=⇒ 0μ
( ) ( )HHH mm
vvvχμχμ +=+= 100
HBvv
μ=
( )mχμμ += 10
Example 12-2: Physical meanings of H, M, B
: free current: magnetization current
: total current
Hv
Mv
0μBv
Example 12-3: Magnetic circuit (1)
A current I0 flows in N turns of wire wound around a toroidal core of permeability μ, mean radius r0, cross-sectional radius a, narrow air gap of length lg. Find , both in the core and the air gap.
Bv
Hv
Example 12-3: Magnetic circuit (2)
Assume the flux has no leakage and nor fringing effect in the air gap, ⇒ total flux (thus ) is constant throughout the loop: B
v
BaBB gf φvvv
==
IldHC
=⋅∫ vv
HBvv
μ=
00
NIlBlBgf =+⇒
μμ
ferromagnetic core
gap
Example 12-3: Magnetic circuit (3)
,00
NIlBlBgf =+
μμ 0
0
μμ gf llNIB+
=⇒
,μ
ff
BH
vv
=⇒0μg
g
BH
vv
=
gf BB =
,0μμ >>Q gf HH <<⇒
A small can induce a strong in the ferromagnetic material, providing a majority of magnetic flux in the ferromagnetic core.
fH Mv
Example 12-3: Magnetic circuit (4)
( ) ( ) gf
m
gf RRSlSlNIBS
+≡
+==Φ
V
0
0
μμTotal flux:
2aS π=
0NIm =V …Magnetomotive force (mmf)
SlRi
ii μ= (i =f, g) …Reluctance
Example 12-3: Magnetic circuit (5)
gf
m
RR +=Φ
V
Analogy between magnetic & electric circuits:
gf RRVI+
=
∑∑ Φ=k
kkj
jj RIN
Multiple magnetic sources and reluctances:
Comment
IldHC
=⋅∫ vv
HBvv
μ=... ,0
0
==+⇒ BNIlBlBgf μμ
For ferromagnetic materials, μ depends on the magnitude (nonlinear) and history (hysteresis) of H, ⇒modification is required to find B
Tangential BC-1
⇒
,
IldHC
=⋅∫vv
)( 21 wHwHldH
abcda
vvvvvvΔ−⋅+Δ⋅=⋅⇒ ∫
wJwHwH sntt Δ=Δ⋅−Δ⋅= 21
sntt JHH =− 21
Component of in the ab-direction
iHv
In general,
( ) sn JHHavvvv =−× 212
Component of along thumbav
sJv
Comment
The projections of and on the interface are generally not collinear.
1Hv
2Hv
( ) sn JHHavvvv =−× 212,21 sntt JHH =−
Comment-1
Only free surface current density counts in:
If the two interfacing media are non-conducting:
( ) sn JHHavvvv =−× 212
tts HHJ 21 ,0 =⇒=v
Comment-3
tt EE 21 =
( ) sn DDa ρ=−⋅ 212
vvv
( ) sn JHHavvvv =−× 212
nn BB 21 =v.s.
BCs in electrostatics: BCs in magnetostatics:
Sec. 12-3Properties of Magnetic Materials
1. Diamagnetic materials2. Paramagnetic materials3. Quantum view of ferromagnetism4. Hysteresis curve
Diamagnetic materials
,HM m
vvχ= 0<mχ
( ),10 mχμμ += 0μμ <
⇒
Present in all materials, usually weakand masked in paramagnetic/ferromagnetic materials. Disappears when the external field is off.
( )510~ −−mχ
Diamagnetic materials-examples
pyrolytic graphite
permanent neodymium magnet
Frog flying in strong M-field(D=32 mm, B=16T)
Paramagnetic materials
,HM m
vvχ= 0>mχ
( ),10 mχμμ += 0μμ >
⇒
( )510~ −mχUsually very weak
reduced by thermal vibration (randomizing the dipole moments). Disappears when the external field is off.
Ferromagnetic materials-1
Non-classical, can only be explained by quantum mechanical view.
Ferromagnetism is determined by both the chemical makeup and the crystalline structure. E.g.
1. Heusler alloys: ferromagnetic, but constituents are not ferromagnetic.
2. Stainless steel: not ferromagnetic, but composed almost exclusively of ferromagnetic metals.
Ferromagnetic materials-2
Main source: the spin of the electrons, Pauli’sexclusion principle (quantum mechanical).
For atoms with fully filled shell (electrons are paired with up/down spins), no net dipole moment exists.
Ferromagnetic materials-3
For atoms with partially filled shell (unpairedelectrons/spins exist), net dipole moment arise.
E.g. Lanthanide elements can carry up to 7 unpaired electrons in the 4f-orbitals
Quantum numbers:n =1, 2, .., ~energy
l =0(s), 1(p), 2(d), 3(f), ..(n-1), .., ~angular momentum, orbital shape, # of nodal planes
ml = -l, …, l, ~axis
ms = 1/2, -1/2, ~spin
Ferromagnetic materials-4
Unpaired dipoles tend to align in parallel to external M-field (classical model), ⇒paramagnetism
Unpaired dipoles tend to align spontaneously, (quantum mechanical effect), ⇒ferromagnetism
Ferromagnetic materials-5
Exchange interaction:
Two electrons (fermions) from adjacent atoms with parallel spins will have lower system energy (more stable) than those with opposite spins.
021 <⋅− SJS
Energy difference due to spin-spin interaction:
>0, if parallel spins>0, if pure Coulomb interaction
Ferromagnetic materials-6
In iron, exchange (spin-spin) interaction is 1000 times stronger than dipole-dipole interaction, ⇒spontaneous alignment, magnetic domains.
At long distance, exchange interaction is overtaken by tendency of dipoles to anti-align, ⇒many randomly oriented domains.
Ferromagnetic materials-7
Placed in strong external M-field, domains will be aligned with the field.
The alignment remains after the field is turned off because the domain walls are pinned on defects in the lattice, ⇒ permanent magnets.
Domains are disorganized (demagnetized) if above Curie temperature (thermal energy > exchange interaction energy).
Hysteresis curve-1
1. Reversible magnetization: if external M-field is weak (up to P1), domain wall movement is reversible, ⇒ B-H curve is a function