Transcript 6.1 Magnetization

Chapter 6 Magnetostatic 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
6.1 Magnetization
6.1.1 Diamagnets, Paramagnets, Ferromagnets
6.1.2 Torques and Forces on Magnetic Dipoles
6.1.3 Effect of a Magnetic Field on Atomic Orbits
6.1.4 Magnetization
6.1.1 Diamagnets, Paramagnets, Ferromagnets
Magnetized: a net alignment of magnetic dipoles
by an applied magnetic field
E.S.
  
M.S.
m
m
E
mm
B
m in B direction m in ( B) direction
paramagnets
diamagnets
Ferromagnets retain a substantial magnetization indefinitely
after the external field has been removed.
6.1.2 Torques and Forces on Magnetic Dipoles

B
B
?
6.1.2 (2)

B
uniform B

N  S1F sin iˆ

F
 S1 (S2 IB) sin  iˆ
 mBsin  iˆ
s1
m  S1S2 I
N  m B
s2

F
F2  S2 I x  Bz
magnetic dipole moment
•This torque lines the magnetic dipole up
parallel to B : paramagnetism.
• In a uniform field, the net force is zero.
F  I  (dl  B)  I (  dl )  B  0
s1
F  S2 IB
6.1.2 (3)
In a nonuniform field,
F  2 RIB cos zˆ
zˆ

m  R 2 Izˆ
F  ( m  B )
B : external field
net force is not zero.
(m  B)  m  (  B)  B  (  m)  (m ) B  ( B )m
B  (  m)  0
( B  )m  0
 F  m  (  B)
0 J ext ( at m 0)
 F  ( m  ) B
force on m itself
 (m  ) B
6.1.3 Effect of a Magnetic Field on Atomic Orbits
a classical model
kˆ
( in 6.1.2

r
e spin 
s2
s  const , I  const )
s1
2 r 2
time  T 

v
wc
-e motion gives a steady-like current
I  e  ev
T 2 r
no B
e

r
m   r 2 I   1 evrkˆ
2
1 e2  m v 2
e r
4 0 r 2
6.1.3
with B
1
e
2
2
v'
 ev ' B  me
2
4 0 r
r
( assume r = constant)

r
v  erB
2me
me 2 2
ev ' B 
(v '  v )
r
me

( v ' v )( v ' v )
r
v  v
B↑ Δv > 0 v↑
2 2
e
1
ˆ
m   e(v)rk   r B
2
4me
antiparallel to B
[ This is just a model.
diamagnetism
Quantum is needed, indeed.]
6.1.4 Magnetization
e
e

spin
→ paramagnetism
 orbiting → diamagnetism
magnetization
M  magnetic dipole moment per unit volume
6.2 The Field of a Magnetized Object
6.2.1 Bound Currents
6.2.2 Physical Interpretation of Bound Currents
6.2.1 Bound Currents
0 m  Rˆ
A
4 R 2
for a single dipole m
m  Md '
0 M  Rˆ
A( r ) 
d '

2
4
R
0
1

 ( M   ' )d '
4
R
R


ˆ
1
(
r
'

r
)
r

r
'
R
( '   ' 1   ' 1  
 2  2)
2
r r'
r ' r
R
R
R
R
1
1
1
 ' ( M )  ( ' M )  M  ( ' )
R
R
R
6.2.1(2)
1
0 1
A(r ) 
[  ( ' M )d '    ' ( M )d ' ]
4 R
R
0 1
0 1
'

(

'

M
)
d

'

(
M

da
)


4 R
4 R
Jb
Kb
0
0
A(r ) 
d ' 
da '


volume
surface
4
R
4
R
J b ( r ')   ' M ( r ')
Kb ( r ')  M ( r ')  nˆ
bound volume current
bound surface current
6.2.1(3)
Ex.1.
zˆ
B?
uniform M
Jb   M  0 Kb  M  nˆ  M sin  ˆ


Ex
.11
of
chapter
5
a
rotating
spherical
shell


 K   v   R sin  ˆ



 B  2 0 zR zˆ

(inside)
3


inside
B  2 0 M
(uniform)
3
outside m  4  R3M
3
6.2.2 Physical Interpretation of Bound Currents
for uniform M
m  Mat
 Ia
I  Mt
 K bt
Kb  M
Kb  M  nˆ
6.2.2
for non-uniform M
nonuniform in yˆ
I x  [ M z ( y  dy )  M z ( y )]dz
M z

dydz
y
Ix
M z
Jx 

dydz
y
M y
for non-uniform in zˆ J x  
z
M z M y
 ( Jb ) x 

y
z
Jb   M
  Jb  0
6.3 The Auxiliary Field H
6.3.1 Ampere’s Law in Magnetized Materials
6.3.2 A Deceptive Parallel
in E.S.
b   f
 E 
0
b    P
 D   f
D  0E  P
6.3.1 Ampere’s Law in Magnetized Materials
  B  0 J  0 ( J b  J f )  0 (  M  J f )
  ( 1 B  M )  J f
0
 H  J f
 Ampere ' s loop H  dI  I f , enc
H  1 BM
0
Ampere’s Law
Auxiliary field
6.3.2 A Deceptive Parallel
 B  0
but   H    M
Ex.2. Copper ( diamagnet )
H(r < R) = ?
rR
Ampere’s
loop
H (2 r )  I f ,enc
H
r
uniform
distributed
rR
H(r > R) = ?
I rˆ
2 R 2
 r2
I
 R2
H (2 r )  I
I
H
ˆ
2 r
0 I
B  0 H 
ˆ
2 r
M 0
6.4 linear and Nonlinear Media
6.4.1 Magnetic Susceptibility and Permeability
6.4.2 Ferromagnetism
6.4.1 Magnetic Susceptibility and Permeability
in E.S.,
P   0 e E
Polarization
electric susceptibility

(  E  )
0
in M.S.,
M  1 m B
0
magnetization
(linear dielectric)
D  E
   0 (1  e )
(linear medium)
magnetic susceptibility
M  m H
  B  0 J
6.4.1(2)
H  1 BM
0
  B  0 ( J f  J b )
 0 ( J f    M )
 B  0 ( H  M )
 0 (1   m ) H
  ( 1 B  M )  J f
0
H
BH
  0 (1   m )
permeability
Permeability of free space
6.4.1(3)
kˆ
Ex.3.
xm
sol.
B?
 H  dl  Ienc
I
N turns per unit
length
H  NIkˆ
B  0 (1   m ) NIkˆ
M
Kb  M  nˆ   m NI (kˆ  nˆ )   m NIˆ
6.4.2 Ferromagnetism
Paramagnetic
no B
M 0
remove B
applied B
Ferromagnet
no B
with B
B

M 0
6.4.2(2)
6.4.2(3)
Iron is ferromagnetic,
but if T>Tcurie ( 770。 for iron )
curie point
it become a paramagnetic.
This is a phase transition.