PHY 7 11 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103
PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 (Make up lecture 2/17/2014 at 9 AM)
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Transcript of PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 (Make up lecture 2/17/2014 at 9 AM)
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PHY 712 Spring 2014 -- Lecture 13 102/17/2014
PHY 712 Electrodynamics10-10:50 AM MWF Olin 107(Make up lecture 2/17/2014 at 9 AM)
Plan for Lecture 13:Continue reading Chapter 5
1. Hyperfine interaction 2. Macroscopic magnetization density M3. H field and its relation to B4. Magnetic boundary values
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PHY 712 Spring 2014 -- Lecture 13 202/17/2014
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PHY 712 Spring 2014 -- Lecture 13 3
Interactions between magnetic dipolesSources of magnetic dipoles and other sources of magnetism in an atom:
• Intrinsic magnetic moment of a nucleus• Intrinsic magnetic moment of an electron• Magnetic field due to electron current
Interaction energy between a magnetic dipole m and a magnetic field B:
02/17/2014
Ne
intE m B
In this case: (0)Nint e NE
eJB B
( )eJ r
Nr e
303
ˆ ˆ3 ( ) 8( ) ( )4 3r
N N
NNμ
r μ r μB r μ r
Hyperfine interaction energy:
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PHY 712 Spring 2014 -- Lecture 13 402/17/2014
Hyperfine interaction energy: -- continued
(0)Nint e NE
eJB B
Evaluation of the magnetic field at the nucleus due to the electron current density:The vector potential associated with an electron in a bound state of an atom as described by a quantum mechanical wavefunction can be written:( )
lnlm r
2
302 2
( )ˆ( )
4 | | sinl
e
nlml
e
e m d rm r
J
rz rA rr r
We want to evaluate the magnetic fieldin the vicinity of the nucleus
B A( 0).r
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PHY 712 Spring 2014 -- Lecture 13 502/17/2014
Hyperfine interaction energy: -- continued 2
302 20
0
( )ˆ(0)
4 | | sinl
e e
nlml
e
e m d rm r
J J r
r
rz rB Ar r
2
302 2
0
2
303 2 2
( )ˆ( ) ( )( )4 | | sin
( )ˆ( )( )4 sin
l
l
nlml
e
nlml
e
e m d rm r
e m d rm r r
o 3
r
o
rr r z rB rr r
rr z rB 0
2 22 2
3 30 03 2 2 3
ˆ ˆˆ ˆ ˆ ˆ( ) ( cos ) cos sin cos cos sin sin ).
( ) ( )ˆ sin ˆ( )4 sin 4
l lnlm nlml l
e e
e r em d r m d rm r r m r
2
o
r z r z 1 θ x y
r rzB 0 z
03
1ˆ =4 'l
e
e mm r
z
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PHY 712 Spring 2014 -- Lecture 13 602/17/2014
Hyperfine interaction energy: -- continued
(0)Nint HF e NE H
eJB B
Putting all of the terms together:
30HF 3 3
ˆ ˆ3( )( ) 8 ( ) .4 3 e
eHr m r
N e N e NN e
μ r μ r μ μ L μμ μ r
In this expression the brackets indicate evaluating the expectation value relative to the electronic state.
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PHY 712 Spring 2014 -- Lecture 13 702/17/2014
Macroscopic dipolar effects --Magnetic dipole moment
rJrm 21 3rd
Note that the intrinsic spin of elementary particles is associated with a magnetic dipole moment, but we often do not have a detailed knowledge of J(r).
30
4 rrmrA
Vector potential for magnetic dipole moment
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PHY 712 Spring 2014 -- Lecture 13 802/17/2014
Macroscopic magnetization
i
ii rrmrM 3
Vector potential due to “free” current Jfree(r) and macroscopic magnetization M(r). Note: the designation Jfree(r) implies that this current does not also contribute to the magnetization density.
3
30
''
' '
4 rrrrrM
rrrJ
rA ''rd free
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PHY 712 Spring 2014 -- Lecture 13 902/17/2014
''
'4
''
''
'1'
''
: thatNote
''
' '
4
30
3
330
rrrMrJ
rA
rrrM
rrrM
rrrM
rrrrrM
rrrrrM
rrrJ
rA
''rd
''
''
''rd
free
free
Vector potential contributions from macroscopic magnetization -- continued
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PHY 712 Spring 2014 -- Lecture 13 1002/17/2014
rJrMrB
rMrJ
rMrJrr
rArArBA
rrrMrJ
rA
free
free
free
free
''rd
''rd
00
0
330
2
30
''4'4
:0 that case for the that Note
''
'4
Vector potential contributions from macroscopic magnetization -- continued
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PHY 712 Spring 2014 -- Lecture 13 1102/17/2014
rJrH
rMrBrH
rJrMrB
free
free
)( )(
:densityflux magnetic theDefine
00
00
Magnetic field contributions
rJrH
rMrBrHrHrB
rJrMrB
free
free
)( )(
field magnetic the)( Definedensityflux magnetic the that Note
00
00
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PHY 712 Spring 2014 -- Lecture 13 1202/17/2014
0
)(
)(
:ticsmagnetosta of equations ofSummary
0
0
rBrMrHrB
rJrHrJrB
free
total
For the case that 0 :
( ) 00
free
J r
H rB r
n̂1
2
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PHY 712 Spring 2014 -- Lecture 13 1302/17/2014
00)(
: that case For the
rBrH
rJ free
n̂1
2nBnBnHnH
ˆˆˆˆ
:boundaryAt
21
21
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PHY 712 Spring 2014 -- Lecture 13 1402/17/2014
Example magnetostatic boundary value problem
ararM
0ˆ
)( 0zrMM0
rMr
rMrHrBrMrHrB
rrHrH
)(
)(0)(
)()( 0)(
2
0
0
H
H
Φ
Φ
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PHY 712 Spring 2014 -- Lecture 13 1502/17/2014
Example magnetostatic boundary value problem -- continued
ararM
0ˆ
)( 0zrMM0
'''
41
'1''
''' '
41
''''
41)(
)(
3
3
3
2
rrrM
rrrM
rrrM
rrrMr
rMr
rd
rd
rdΦ
Φ
H
H
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PHY 712 Spring 2014 -- Lecture 13 1602/17/2014
ararM
0ˆ
)( 0zrMM0
Example magnetostatic boundary value problem -- continued
20
0
2 20
0
330
0 3
For this example:
1( ) 4 ' '4
For : ( )2 6 3
For : ( )3 3
a
H
H
H
MΦ r dr
z r
M za rr a Φ Mz
M a zar a Φ Mz r r
r
r
r
'''
41)( 3
rrrMr
rdΦH
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PHY 712 Spring 2014 -- Lecture 13 1702/17/2014
ararM
0ˆ
)( 0zrMM0
Example magnetostatic boundary value problem -- continued
53
30
3
30
00
3ˆ3
)( )( 3
)( :For
ˆ3
)( )( 3
)( :For
rz
raMΦ
rzaMΦar
MΦzMΦar
HH
HH
rzrrHr
zrrHr
53
30
0
53
30
00
0
0
3ˆ3
)(
3ˆ3
)( :For
3ˆ2 )(
3ˆ
)( :For
)( )( )(
rz
raM
rz
raMar
MMar
rzrB
rzrH
zrBzrH
rMrHrB
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PHY 712 Spring 2014 -- Lecture 13 1802/17/2014
3
30
53
30
00
ˆˆ3
ˆ)ˆ(
3ˆ3
)( :For
ˆˆ3
ˆ)ˆ( 3
ˆ )( :For
aaMa
rz
raMar
MaMar
rzrrH
rzrH
rzrrHzrH
Check boundary values:
31ˆˆ 3
ˆ)ˆ(
3ˆ3
)( :For
ˆˆ 3
2 ˆ)ˆ( 3
ˆ2 )( :For
5
2
3
30
0
53
30
0
00
00
aa
aaMμa
rz
raMar
MaMar
rzrrB
rzrB
rzrrBzrB
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PHY 712 Spring 2014 -- Lecture 13 1902/17/2014
Variation; magnetic sphere plus external field B0
arar
0)( 0MrMM0 B0
00
0
00
00
000
000
23
material, ic"paramagnet" isotropican For 32
311
32
:For :ionsuperpositBy
BM
rHrBBrHrB
MBrH
MBrB
ar
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PHY 712 Spring 2014 -- Lecture 13 2002/17/2014
0
)(
)(
:ticsmagnetosta of equations ofSummary
0
0
rBrMrHrB
rJrHrJrB
free
total
00)(
: that case For the
rBrH
rJ free
n̂1
2nBnBnHnH
ˆˆˆˆ
:boundaryAt
21
21
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PHY 712 Spring 2014 -- Lecture 13 2102/17/2014
Magnetism in materials
)hysteresis(with materials agneticantiferrom tic,ferromagneFor
material cdiamagneti material icparamagnet
:magnetismlinear with materialsFor )(
0
0
0
HB
HB
rMrHrB
f
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PHY 712 Spring 2014 -- Lecture 13 2202/17/2014
Example: permalloy, mumetal /0 ~ 104
0 0
a b
B0Spherical shell a < r < b :
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PHY 712 Spring 2014 -- Lecture 13 2302/17/2014
Example: permalloy, mumetal /0 ~ 104 -- continued
0 0
ab
B0
)(
00)(
:case For this
rHrBrBrH
continuousˆcontinuousˆ
:boundariesat Continuity
nBnH
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PHY 712 Spring 2014 -- Lecture 13 2402/17/2014
Example: permalloy, mumetal /0 ~ 104 -- continued
0 0
ab
B0
coscos)( For
cos)( For
cos)( 0For
0)( 0
)()( :Let
10
0
1
2
lll
lH
lll
lllH
ll
llH
H
H
Pr
rBbr
Pr
rbra
Prar
r
r
r
rrB
rrH
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PHY 712 Spring 2014 -- Lecture 13 2502/17/2014
Example: permalloy, mumetal /0 ~ 104 -- continued
0 0
ab
B0
21
0
021
1
31
0
031
10
21
11
31
10
1
22 At
2 At
:)contribute terms1(only conditionsboundary Applying
bα
μBb
bγ bβ
bB
bbr
aγaβaδ
aar
l
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PHY 712 Spring 2014 -- Lecture 13 2602/17/2014
Example: permalloy, mumetal /0 ~ 104 -- continued
0 0
ab
B0
0
03
0
0
02
03
00
01
/12/9
/1
1//22/1/2/9
:clearsdust When the
Bba
Bba
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PHY 712 Spring 2014 -- Lecture 13 2702/17/2014
Energy associated with magnetic fields
rDrE
rHrB
BmBm
BmFBm
21 :analogy toIn
21 :shown that becan It
-- energies cMacroscopi
:bygiven is field externalan in dipole magnetic a aligning with associatedenergy that implies This
:is field externalan in dipole magnetic aon force the
-- proof without used previously We:Note
3
3
rdW
rdW
U
E
B