lecture note #6a chapter 6. magnetic fields in matter...
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Inha University 1
Chapter 6. Magnetic Fields in MatterLecture Note #6A
6.1 Magnetization
6.2 The Field of a Magnetized Object
6.3 The Auxiliary Field H
6.4 Linear and Nonlinear Media
• 자성의 근원: 전자의 궤도 운동에 의한 자기 모멘트와 전자 스핀
- 외부 자기장이 가해지면 회전력 발생 → 자기 모멘트 변화
• 자기장이 가해지면 원자 내 전자 궤도 운동에 의한 자기모멘트 변화 발생 +
-R
spin angular
momentumorbital angular
momentumzevRnRInIAm ˆ
2
1ˆ ˆ 2
Bm
RezR
m
eRBezRvem
ee
4ˆ
22
1ˆ
2
1 2
- 상자성 (paramagnet), 반자성 (diamagnet) , 강자성 (ferromagnet)
- 자화도 (magnetization) = 단위 부피당 자기 모멘트V
mM
Inha University 2
일반물리 복습 (pages 2 ~ 9)
▣원자의 자기모멘트
◈전자의 궤도운동에 의한 자기모멘트
◈전자의 스핀 모멘트
◈원자핵의 자기 모멘트
▣자성 물질 : 자기 감수율에 따라 다음과 같이 분류
◈상자성 (paramagnet)
◈강자성 (ferromagnet)
◈반자성 (diamagnet)
+
-R
spin angular
momentumorbital angular
momentum
Inha University 3
원자의 자기모멘트
▣원운동하는 전자의 궤도운동에 의한 자기 모멘트
◈원운동하는 입자의 각운동량 :
◈한바퀴 도는 주기 :
◈전류 :
◈자기 모멘트 :
◈각운동량으로 다시 표현하면 :
▣전자의 스핀 자기 모멘트 :
▣양자론에 의하면, 자기모멘트는 보어자자수 (Bohr magnetron)의 정수 배만 가
지며, 전자의 스핀 자기모멘트는 1 보어자자수이다.
v/2T r
vqrmL
r
qqi
2
v
T
2
v
2
v 2 rqr
r
qiAe
rv
q
qm
q
em
qL
2
e
sm
eS
Lm
e
2
bllZZ mmm
eL
m
e -
2-
2- gnetron Bohr Ma 1027.9
2
24-
TJ
m
eb
zBB ˆ
2
1zS
(보어자자수)lml ...., ,2 ,1 ,0
Inha University 4
자기화와 자기감수율
▣자기화 : 단위 부피당 자기 쌍극자 (magnetic dipole)
▣보통 금속인 경우
◈자기화는 외부의 자기장의 크기에 비례
◈ : 자기 감수율 (magnetic susceptibility)
▣자성물질의 분류
◈상자성 : 자기감수율이 양수
◈반자성 : 자기감수율이 음수
VM
m
o
m
BM
외부
Inha University 5
Magnetic Susceptibility of Some Elements & Minerals
http://www.jmu.edu/cisr/journal/13.1/rd/igel/Igel_Table1Web.jpg
m105 SI
m : magnetic
susceptibility
: density
Inha University 6
상자성 (paramagnetism)
◈알루미늄, 나트륨, 티타늄, 텅스텐 등
◈각 원자나 분자들은 독립적으로 자기모멘트를
갖고 있으나, 모두 제멋대로 배열되어 있기
때문에 평소에는 물질이 자성을 갖지 않는다
◈외부의 자기장을 받으면, 각
자기모멘트들은 같은 방향으로
배열을 하려하나, 열운동 때문에
완전히 나란하게 배열하지 못한다
◈온도가 매우 낮으면, 물질의 자기화는 외부의
자기장에 비례하며, 온도에 반비례한다
B외부
T
BCM 외부큐리법칙:
H
HB
Inha University 7
강자성 (Ferromagnetism)
◈철, 코발트, 니켈, 또는 디스프로슘, 가돌리니움
등의 희토류 금속이나 합금처럼 외부자기장이 없더라도
자기화 되어있는 물질
◈교환상호작용때문이며, 전기적 쿨롱 상호작용 중 양자역학
적인 효과이다
◈큐리(Curie) 온도 이상에서는 상자성 상태가 된다
◈자기 구역들로 이루어져 있으며, 정렬된 정도에
따라 총 자기모멘트가 달라진다
http://en.wikipedia.org/wiki/Curie_temperaturehttp://www.phys.aoyama.ac.jp/~w3-
jun/achievements/study/oo/fig4-3_eng.gif
Inha University 8
자기이력 현상
◈ a-b-c-d-e-f-g-b-c…
◈ a : 처음에 자기화가 되어 있지 않은 상태
◈ b : 포화상태
◈ c : 외부자기장이 없더라도 잔류 자기가 남아있는 상태
◈ d : 잔류 자기를 없애기 위한 반대방향의 외부 자기장
◈ e : 반대방향의 포화상태
◈ f : 상태 c와 동일
◈ g : 상태 d와 동일
iniBo
oM BBB
솔레노이드 내부의 자기장
여기에서
: 전류 세기에 따른 솔레노이드에생성되는 내부 자기장.
BM : 솔레노이드 내부에 있는 강자성체(철심)에 의한 자기장
자석 상태
Inha University 9
반자성 (Diamagnetism)
◈구리, 비스무스 등과 같이 자석을 갖다
대면 약하게 반발한다
◈시계 반대 방향으로 돌고 있는 왼쪽 입
자에 B외부이 그림과 같이 주어지면, 렌츠
법칙에 의해 B외부의 반대방향으로 자기
선속이 증가해야 하므로, 입자의 속도가
증가한다
◈마찬가지로 시계 방향으로 돌고 있는 오
른쪽 입자는 자기 선속이 감소해야 하므
로 입자의 속도가 감소 한다
두 입자의 총 자기모멘트는 렌츠법칙에 의해 B외부와 반대방향으로 생긴다.
외부자기장이 없는 경우
외부자기장이 있는 경우
Inha University 10
6.1 Magnetization (자화)
Paramagnet:
6.1.1 Diamagnets (반자성체), Paramagnets(상자성체), Ferromagnets(강자성체)
외부 자기장이 없는 평상 상태 (내부 원자의자기쌍극자가 각기 무질서한 방향으로 배열되어 있는 상태 자기적 특성이 없음)
H
HB
외부 자기장이 상자성체에 걸리면, 원자의 자기쌍극자가 외부 자기장에 따라 나란히 배열됨.
Diamagnet:H
외부 자기장이 반자성체에 걸리면, 원자의 자기쌍극자에 의한 자기장이 외부 자기장과 서로 밀어내는 방향으로 배열됨.
(상자성체)
(반자성체)
Ferromagnet:
(강자성체)외부 자기장이 없는 평상 상태에서도 내부 원자의 자기쌍극자가 모두 한 방향으로 나란하게 배열되어 있는 상태 자기적 특성을 보임)
Inha University 11
I I
I
I
Fb3
Fb1
Fa4
Fa2
6.1 Magnetization (자화)
For a current flowing rectangular loop placed at an
angle with respect a magnetic field B in the z-direction,
6.1.2 Torques and Forces on Magnetic Dipoles
0ˆ cosˆ cos42 xIaBxIaBFF aa
- Net forces acting on the sloping sides of the loop :
Current flowing
infinitesimal rectangles
Magnetic field
in z-direction
When the current flowing
infinitesimal rectangle is
tilted at an angle from
the z axis,- A Torque acting on the loop :
0ˆ ˆ 31 yIbByIbBFF bb
- Net forces acting on the horizontal sides of the loop :
BmxBIabxIbBa
xIbBa
xIbBa
yFza
yFza
Fr bb
ˆ sinˆ sin
ˆ sin2
ˆ sin2
ˆˆ sin2
ˆˆ sin2
13
where nIAnIabm ˆ ˆ
: the magnetic dipole moment of the current loop
( : a unit vector along the normal direction of the surface area A = ab)n
Bm
: Torque acting on the magnetic dipole moment
in a uniform magnetic field
(6.1)
Inha University 12
6.1 Magnetization (자화)
For an electric dipole in an electric field
6.1.2 Torques and Forces on Magnetic Dipoles
From Chapter 4
No net torque
nIAnIabm ˆ ˆ
This torque
accounts for
“paramagnetism”.
Bm
- continued (1)
The torque acting on an electric dipole moment
in a uniform field E :
Ep
The magnetic dipole moment
The electric dipole moment
dqp
For a magnetic dipole in a magnetic field
The torque acting on a magnetic dipole moment
in a uniform field B :
+
-
-
ⓔn
nⓔ
ⓔ
n The paramagnetic materials
require odd number of electrons
For an infinitesimal loop, with magnetic dipole m, in a magnetic
field B, the net force on the loop is BmF
For a electric dipole p, in an electric field E, the net
force on the dipole is EpF
See next
page
: an analogy to that in the magnetic
case (See next page)
Inha University 13
R
m
6.1 Magnetization (자화)
6.1.2 Torques and Forces on Magnetic Dipoles - continued (2)
- the net force on a current loop is
0 BldIBldIF
For a magnetic dipole in a uniform magnetic field
- the net force on a circular current loop is
BldIF
For a magnetic dipole in a nonuniform magnetic field
F
Fhor
Fdown
F
Fhor
Fdown
- the net downward force on the circular current loop is
R
Bm
R
Bm
R
BIR
BRIF
cos cos
cos 2
2
nIRnIAm ˆ ˆ 2
For an infinitesimal loop, with magnetic dipole m, in a
magnetic field B, the net force on the loop is
BmF
(6.2)
(6.3)
Inha University 14
6.1 Magnetization (자화)
6.1.2 Torques and Forces on Magnetic Dipoles - continued (3)
Electric dipole
+
p
-
N
m
S
Magnetic dipole
m
I
Magnetic dipole
(Gilbert model)
If we break them,
+
-
N
S
Is this correct?
Inha University 15
6.1 Magnetization (자화)
Consider a circular motion of the electron around the nucleus
(without considering the electron spin yet)
6.1.3 Effect of a Magnetic Field on Atomic Orbits
v
RT
2- Period of the circular motion :
- The orbital dipole moment :
R
ev
T
eI
2
- The current caused by the electron motion :
zevRnRInIAm ˆ 2
1ˆ ˆ 2
For a magnetic field B perpendicular to the plane of the orbit,
Bm
(Tilting the entire orbit with this torque is hard, but change of the electron’s speed is relatively easy.)
+
-R
- The torque acting the orbital dipole moment in a magnetic field B :
- The electrical attraction force between the electron and proton = the centripetal force :
R
vm
R
ee
2
2
2
04
1
R
vmBve
R
ee
2
2
2
04
1
(6.4)
(6.5)
(6.6)
Under the situation of the magnetic field B applied, vv
spin angular
momentumorbital angular
momentum
Inha University 16
6.1 Magnetization (자화)
Consider a circular motion of the electron around the nucleus
(without considering the electron spin yet)
6.1.3 Effect of a Magnetic Field on Atomic Orbits
Eq. (6.5) Eq. (6.6) :
When
The electron speeds up when B is turned on.
+
-R
- The magnetic dipole moment change becomes
vvvvR
mvv
R
mBve ee 22
(6.7)
(6.8)
The change in m is opposite to the direction of the field B.
- continued (1)
,smallvvv vvR
mBve e 2
em
eRBv
2
zevRm ˆ
2
1
Bm
RezR
m
eRBezRvem
ee
4ˆ
22
1ˆ
2
1 2
The increment of m is antiparallel to the field B.
The mechanism is responsible for diamagetism.
This is observed mainly in atoms with even numbers of electrons.
Inha University 17
6.1 Magnetization (자화)
6.1.4 Magnetization
(6.9)
: magnetization (자화)
Paramagnetism: The dipole associated with the spins of unpaired electrons
experience a torque tending to line them up parallel to the field.
H
HB
Diamagnetism:
(analogues to the polarization P in electrostatics)
(상자성)
(반자성)H
The orbital speed of the electrons is altered in such a way as to
change the orbital dipole moment in a direction opposite to the field.
+
-
R
spin angular
momentum
orbital angular
momentum
M magnetic dipole moment per unit volume
State of magnetic polarization:
Inha University 18
Chapter 6. Magnetic Fields in Matter
6.1 Magnetization
6.2 The Field of a Magnetized Object
6.3 The Auxiliary Field H
6.4 Linear and Nonlinear Media
• 각 원자들의 전자 궤도 운동에 의한 전류 : Bound currents
- Volume current :
- Surface current :
MJb
nMKbˆ
- 이들 전류에 의한 벡터 포텐셜 :
''
4'v
'
4
00 das
rKd
s
rJrA bb
• 균일하게 자화된 구에 의한 자기장RM
MB
03
2
MRm 3
3
4 mrrm
rrBdip
ˆ ˆ3
1
4 3
0
(구 내부)
(구 외부)
Inha University 19
6.2 The Field of a Magnetized Object
For a single magnetic dipole m :
6.2.1 Bound Currents
the vector potential at point p is 2
0ˆ
4 r
rmrAdip
m
I
r
p
(6.10)
For a piece of magnetized material :
the vector potential at point p is
(with the magnetic dipole moments per unit volume M )
'vˆ'
4 2
0 ds
srMrA
s
p
'vd 'rM
(6.11)
Since
Appendix 1
, ˆ1
'2s
s
s
''1
4'v ''
1
4
'v '
''v ''1
4
'v 1
''4
00
0
0
adrMs
drMs
ds
rMdrM
s
ds
rMrA
Vector product rule
From Chapter 1,
fAAfAf
AfAffA
Appendix 2
SadrM
sd
s
rM''
1'v
''
v
From Appendix 2,
(6.12)MJb
nMKb
ˆ
Vector potential of a volume current Vector potential of a surface current ndaad ˆ
from Chapter 5 (5.85)
Inha University 20
6.2 The Field of a Magnetized Object
6.2.1 Bound Currents
For a piece of magnetized material :
the volume current :
s
p
'vd 'rM
(6.13)
''
4'v
'
4
00 das
rKd
s
rJrA bb
For the electric field of polarized object
MJb
nMKbˆ
- continued
the surface current : (6.14)
where is the normal unit vector of the surface element. n
Thus, the vector potential can be written as
(6.15)
where nPbˆ
: surface charge density
Pb
: volume charge density
(4.11)
(4.12)
v'
0S
0
v'4
1
4
1d
rda'
rrV bb
(4.13)
: Bound
current
A
p
r
Inha University 21
6.2 The Field of a Magnetized Object
Let us choose the z axis along the direction of M.
For a rotating spherical shell of uniform surface charge ,
the corresponding surface current density becomes
(6.16)
[Example 6.1]
(Solution)
0 MJb
ˆ sinˆ MnMKb
ˆ sin RvK
n
n
z
v
sin v
RM
From the comparison of above two equations, we can get
From [Example 5.11], the magnetic field of the spinning spherical shell is
MRB 003
2
3
2
MB
03
2
Since the volume current density ,0 MJb
the field of a uniformly magnetized sphere is
identical to the field of a spinning spherical shell.
inside the sphere.
The magnetic field outside the spherical shell is the same as that of a pure dipole. MRm 3
3
4
Rv
Appendix 3
Appendix 4
Inha University 22
Comparison between Magnetization & Polarization
6.2 The Field of a Magnetized Object
6.2.1 Bound Currents
4.2 The Field of a Polarized Object
•For a uniformly polarized sphere•For a uniformly mangnetized sphere
n
0 MJb
ˆ sinˆ MnMKb
ˆ sin RvK
RM
MRB 003
2
3
2
MB
03
2
Inside the sphere.
Outside the sphere.
MRm 3
3
4
,3
4 3PRp
cosˆ PnPb
: the total dipole moment of the sphere.
0 Pb
: identical to that of a perfect dipole at the origin.
RrPzzP
dz
dVE
for
3
1ˆ
3 00
Since r cos = z , the field inside the sphere is uniform :
where
Rr for
The potential outside the sphere becomes
ˆ sinˆ cos2 4 2
0 rr
mArBdip
mrrmr
rBdip
ˆ ˆ3
1
4 3
0
ˆ3
sinˆ
3
cos23
0
3
3
0
3
r
PRr
r
PRE
Rrr
rpV
for
ˆ
4
12
0
Inha University 23
6.2 The Field of a Magnetized Object
6.2.2 Physical Interpretation of Bound Currents
All the internal
currents cancel out.
taMMm v
The magnetic dipole moment is
( : a unit vector directing outward)n
Thus, the surface current is
in terms of the magnetization M.
aIm in terms of the circulating current I.
M tI
Mt
IKb
nMKbˆ
There is no current on the top or bottom surface of the slab.
Inha University 24
6.2 The Field of a Magnetized Object
6.2.2 Physical Interpretation of Bound Currents
taMMm v
When the magnetization is nonuniform along z-direction,
aIm
The corresponding volume current density is
M tI
y
zx
xb
M
dzdy
IJ
nMKbˆ
In general,
dzdyy
M dzyMdyyMI z
zzx
When the magnetization is nonuniform along y-direction,
z
M
dzdy
IJ
yx
xb
dydzz
M dyzMdzzMI
y
yyx
Thus, the total current density is z
M
y
MJ
yz
xb
MJb
For a steady-current, the conservation law should be satisfied 0 MJb
- continued
Inha University 25
6.2 The Field of a Magnetized Object
6.2.2 The Magnetic Field Inside Matter
Microscopic magnetic field
(The averaged field over regions
containing many atoms)
The magnetization M is “smoothed out”.
Macroscopic magnetic field
(fields for a specific point or atom)
Inha University 26
Next Class
Chapter 6. Magnetic Fields in Matter
6.1 Magnetization
6.2 The Field of a Magnetized Object
6.3 The Auxiliary Field H
6.4 Linear and Nonlinear Media
Inha University 27
[Appendix 1] Problems 1.13 & Its Solution
[Problem 1.13] Let s be the separation vector from a fixed point (x’, y’, z’) to the point
(x, y, z), and let s be its length. Show that
(a)
(b)
(c) What is the general formula for ?
zzzyyyxxxs ˆ 'ˆ 'ˆ '
ss
22
(a)
szzzyyyxxx
zzzyyxxz
yzzyyxxy
xzzyyxxx
s
2ˆ '2ˆ '2ˆ '2
ˆ '''ˆ '''ˆ '''2222222222
2/ˆ/1 sss
ns 222
''' zzyyxxs
(b)
23
23-222
23-222
23-22223-222
21-222
21-22221-222
ˆ1ˆ 'ˆ 'ˆ ''''
ˆ '2'''2
1
ˆ '2'''2
1ˆ '2'''
2
1
ˆ '''
ˆ '''ˆ '''1
s
ss
szzzyyyxxxzzyyxx
zzzzzyyxx
yyyzzyyxxxxxzzyyxx
zzzyyxxz
yzzyyxxy
xzzyyxxxs
Inha University 28
[Appendix 1] Problems 1.13 & Its Solution
[Problem 1.13]
(c) What is the general formula for ?
zzzyyyxxxs ˆ 'ˆ 'ˆ '
(c) x
snss
x
nn
1
ns 222''' zzyyxxs
snsss
ns
zzzyyyxxxs
nszz
sy
y
sx
x
snss
nn
nnn
ˆ1
ˆ 'ˆ 'ˆ '1
ˆ ˆ ˆ
11
11
s
xxxxzzyyxxzzyyxx
xx
s ''2'''
2
1'''
21-22221222
Back
- continued
For the case of , ' ns ''
1
x
snss
x
nn
s
xxxxzzyyxxzzyyxx
xx
s ''2'''
2
1'''
''
21-22221222
snsss
nszzzyyyxxxs
nszz
sy
y
sx
x
snss nnnnn ˆ
1 ˆ 'ˆ 'ˆ '
1 ˆ
'ˆ
'ˆ
'' 1111
Inha University 29
[Appendix 2] Problem 1.61(b) & Its Solution
[Problem 1.61] Although the gradient, divergence, and curl theorems are the fundamental
integral theorems of vector calculus, it is possible to derive a number of corollaries
from them. Show that
(b)
The divergence theorem :
S
add
vv vv
c
v
(b) S
adcdc
vv vv
[Hint: Replace v by in the divergence theorem.]
S
add
vvvv
Vector product rule #4 BAABBA
( : a constant vector)c
0 c vvvv
cccc
adcadccadadc
vvvv
S
adcdc
vv vv
S
add
vv v v
BAC
ACBCBA
From Chapter 1,
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Inha University 30
s
5.4 Magnetic Vector Potential
[Example 5.11]
(Solution)
Eq. (5.66b)
''
4
0 das
rKrA
(5.66b)
where v
K
'cos222 RrrRs
'''sin' 2 ddRda
The velocity of a point r’ in a rotating rigid body is 'v r
zyxRω
RRR
zyx
r
ˆ'sin'sinsinˆ'cossin'cos'sincosˆ'sin'sincos-
'cos'sin'sin'cos'sin
cos0sin
ˆˆˆ
'v
[Appendix 3]
Inha University 31
s
5.4 Magnetic Vector Potential
[Example 5.11] (Solution)
Since
'.cosu
- Continued (1)
We let
. when 3
2
when 3
2
2
2
Rr r
R
R,r R
r
'''sin
'cos2
'
4'
'
4
2
22
00
ddR
RrrR
rda
s
rKrA
and
yRr ˆ'cossin'v
,
Inha University 32
s
5.4 Magnetic Vector Potential
[Example 5.11] (Solution)
Since
- Continued (2)
The field inside this spherical shell is
yrr ˆ sin
when 33
sin
3
2
2
sin
when 33
sin
3
2
2
sin
3
4
0
3
4
0
2
3
0
00
2
3
0
Rrrωr
R
r
rR
r
RR
RrrωRrR
R
rR
rA
For the original coordinates with in the z-direction,
x
y
(r,,)z
ˆ sinrr
when ˆ sin
3
when ˆ sin3
2
4
0
0
Rrr
θR
RrrR
rA
ˆ v
v1
ˆ vv
sin
11
ˆ v
vinsin
1v
θ
r
rrrr
rrr
rsr
zRrR
rR
rrr
rrR
sr
rArr
rAsr
AB
ˆ 3
2ˆ sinˆ cos3
2
ˆ sin3
1ˆ sin
3 in
sin
1
ˆ 1
ˆ insin
1
00
00
r
z
RB 03
2
: uniform
'4
0 dar
KA
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5.4 Magnetic Vector Potential
[Example 5.11] (Solution)
The vector potential outside the spherical shell is
- Continued (3)
The field outside this spherical shell is
when ˆ sin
3 2
4
0 Rrr
θRrA
ˆsinˆ
cos
ˆsinˆ
sin
sin
ˆˆsin
2
3
3
1
3 in
1
1
in1
33
4
0
2
4
0
2
4
0
rr
r
θR
r
Rr
rrr
r
Rs
r
rArr
rAsr
AB
R
z
d
R dR sin
(5.70b) ˆsinˆcos 23
3
4
0 rθr
RB
Inha University 34
5.4 Magnetic Vector Potential
[Example 5.11] (Solution) - Continued (4)
R
z
d
R dR sinThe total charge on the shaded ring is dRRdq sin2
The time for one revolution is
2dt
The current in the ring is
dR
dRR
dt
dqI sin
2
sin2 2
The cross sectional area of the ring is 2sin Rda
The magnetic moment of the ring is
dRRdRdaIdm sinsin sin 3422
The total magnetic moment of the shell is
44
0
4
0
4
0
34
3
4
4
3
12
1
4
3
12
1cos
4
33cos
12
1
sin33sin4
1 sin
RRR
dRdRdmm
4
3
4Rm (5.70c)
Eq. (5.70c) → Eq. (5.70b) :
ˆsinrθcosr
mB 2
4
3
0
(5.70d)
sin33sinsin3
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