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Classical Electrodynamics
Chapter 5Magnetostatics, Faraday’s Law,
Quasi-Static Fields
A First Look at Quantum Physics
2011 Classical Electrodynamics Prof. Y. F. Chen
A First Look at Quantum Physics
2011 Classical Electrodynamics Prof. Y. F. Chen
Contents§5.1 The relationship between electric field and magnetic field
§5.2 Biot and Savart law and vetor potential
§5.3 Differential equations of magnetostatics and Ampere’s law
§5.4 Vector potential and magnetic induction for a circular current loop
§5.5 Analogy between electric dipole and magnetic dipole
§5.6 Magnetic scalar potential
§5.7 Magnetic moment
§5.8 Macroscopic equations, boundary conditions on and
§5.10 Uniformly magnetized sphere
§5.11 Final remark
§5.9 Methods of solving boundary-value problems in magnetostatics
B
H
(1) The force on a charge acted by a nearby conduction wire is:
In the inertial coordinate, the charge experiences a magnetic force: BvqF
In the relative coordinate, where the observation is performed in thecoordinate of charge, it experiences a electric force: EqF
0J R
B
+ + +
- - -0net
qv
2011 Classical Electrodynamics Prof. Y. F. Chen
§5. 1 The relationship between electric field and magnetic field
(2)
(x, y, z, t) (x’, y’, z’, t’)
With the transformation of coordinates in the special relativity:
222
22
2
22
1' ,
1' ,' ,' ,
1'
cvj
cv
cv
xcvt
tzzyy
cvvtxx x
With the Gauss’s law:0
2'2 lRlrE
In the relative coordinate:
22
0
22
0
2
1
121
12
cv
vrIq
cvc
vr
jRqqEF x
2011 Classical Electrodynamics Prof. Y. F. Chen
§5. 1 The relationship between electric field and magnetic field
But when we consider the situation in the inertial coordinate:
BvqqvBvrIqF
2
0
This means that by proper transformation of the coordinate, we can justdeal with the electric force to solve the electromagnetic problems.Otherwise, the concept of the magnetic force must be introduced. Ingeneral, the force can be expressed as:
BvEqF
2011 Classical Electrodynamics Prof. Y. F. Chen
§5. 1 The relationship between electric field and magnetic field
(1) The magnetic field generated by a short segment of wire carrying current isgiven by:
''
')'(4
)(
'''
4
33
0
30
xdxx
xxxJxB
xxxxlIdBd
(2)
AxdxJxx
xdxx
xJ
xdxx
xJB
AfAfAf
')'('
14
'')'(
4
''
1)'(4
3030
30
Where the vector potential is: '')'(
430 xd
xxxJA
2011 Classical Electrodynamics Prof. Y. F. Chen
§5.2 Biot and Savart law and vetor potential
§5. 3 Differential equations of magnetostatics and Ampere’s law
(1) 0 ABAB
(2)
''
1)'(4
')'('
1')'('
14
'')'('
')'(
4
'')'(
4
320
330
3230
2
30
xdxx
xJ
xdxJxx
xdxJxx
AfAfAf
xdxx
xJxdxx
xJ
ccc
xdxx
xJB
2011 Classical Electrodynamics Prof. Y. F. Chen
)('')'('
4
as ,0'')'('
')'('
)('')'('
4'
')'('
4
'''
)(')'('
1'4
'4'
1
030
3
03030
030
2
xJxdxxxJ
Sdaxx
xJxdxx
xJ
xJxdxxxJxd
xxxJ
AfAfAf
xJxdxJxx
xxxx
2011 Classical Electrodynamics Prof. Y. F. Chen
§5. 3 Differential equations of magnetostatics and Ampere’s law
For static electromagnetics, we get the Ampere’s law:
JBJ
0 0
Generally, the Ampere’s law should be modified as the Maxwell-Ampere’s law:
tEJ
xdxx
xt
J
Jxdxxtx
B
tJ
000
3
0000
030
'')'(
41
''
)'(
4
2011 Classical Electrodynamics Prof. Y. F. Chen
§5. 3 Differential equations of magnetostatics and Ampere’s law
(1)
''
4)( 0
xxldIxA
2
022
0
2
022
0
22
222
)'cos(sin2''cos
4)(
)'cos(sin2''sin
4)(
)'cos(sin2
0'sin'cos'
ˆ''cosˆ''sinˆ''
raardaIxA
raardaIxA
raar
zayaxxx
adaadaaadld
y
x
yx
aaaaaaaaaaa
rz
ry
rx
ˆsinˆcosˆ
ˆcosˆsincosˆsinsinˆ
ˆsinˆcoscosˆcossinˆ
2011 Classical Electrodynamics Prof. Y. F. Chen
§5.4 Vector potential and magnetic induction for a circular current loop
0)'cos(sin2
')'cos(4
0)'cos(sin21sincos
4
)'cos(sin2')'sin(cos
4
0)'cos(sin214
)'cos(sin2')'sin(sin
4
2
022
0
2
0
220
2
022
0
2
0
220
2
022
0
raardIaA
raarra
Iaraar
dIaA
raarra
Iaraar
dIaAr
2011 Classical Electrodynamics Prof. Y. F. Chen
§5.4 Vector potential and magnetic induction for a circular current loop
Expand the denominator of A with binomial expansion:
......sin
815010
4
sin
42cos2cos21
22cos1cos Note
......')'(cossin2!3
125
23
21
')'(cossin2!2
123
21
')'(cossin2!1
121')'cos(
4
')'cos(sin21)'cos(4
2
222
322
20
2
0
22
0
22
0
4
2
0
43
22
2
0
32
22
2
0
222
2
022
0
21
22
2
022
0
arra
ar
rIa
ddd
dar
ra
dar
ra
dar
radar
Ia
dar
raar
IaA
2011 Classical Electrodynamics Prof. Y. F. Chen
§5.4 Vector potential and magnetic induction for a circular current loop
30
302
44sin assume & let
rrm
rmrAaraIm
(2)
aarmB
rm
rmr
rB
rm
rmrr
rrB
Arr
arara
rAhAhAhqqq
ahahah
hhhAB
r
r
r
ˆsinˆcos24
4cos2
4sinsin
sin1
4sin
4sin1
sin00
ˆsinˆˆ
sin1
ˆˆˆ1
30
30
30
30
30
2
332211
321
332211
321
2011 Classical Electrodynamics Prof. Y. F. Chen
§5.4 Vector potential and magnetic induction for a circular current loop
(1) Electric dipole:
30
30
')ˆ(ˆ3
41
')'(
41
xxpnpnE
xxxxp
(2) magnetic dipole:
30
30
')ˆ(ˆ3
4
')'(
4
xxmnmnAB
xxxxmA
2011 Classical Electrodynamics Prof. Y. F. Chen
§5.5 Analogy between electric dipole and magnetic dipole
§5. 6 Magnetic scalar potential
(1)MBHJ
0 :space freein ,0 0 If
To illustrate this concept, we consider the problem as follows. For themagnetic induction at the point P with coordinate produced by an incrementof current at , the magnetic induction can be explicitly expressed as:
x
'lId
'x
dIxdR
RldI
xdxx
xxldIxdBxdd MM
44
')'(
4
3
30
where the solid angle is:
3
)(R
RAd
4
40
0IBI
MM
2011 Classical Electrodynamics Prof. Y. F. Chen
(2) As an example, find the magnetic induction at a point on the z-axis:
2322
0
2
0 2322
03
0
22
ˆ2
'ˆˆ0
4''''
'')'(
4
ˆˆ)'()'(')'(
' ,ˆˆ'
ˆ)'()'()'(
az
aaI
adaz
aaaIdzddxx
xxxJB
aaazzaIxxxJ
azxxaaazxx
azaIxJ
z
z
z
z
(i) Directly find magnetic induction:
'xx
2011 Classical Electrodynamics Prof. Y. F. Chen
§5. 6 Magnetic scalar potential
(ii) With the concept of the magnetic scalar potential:
2322
20
0z
22
220
2
0 2322
3
2
12
4
12'
'''
'''
ˆ'ˆ ,)(
az
aIz
B
azzII
azz
z
ddz
ddzAdR
aazRR
RAd
M
M
a
z
2011 Classical Electrodynamics Prof. Y. F. Chen
§5. 6 Magnetic scalar potential
(iii) Since this problem has the property of symmetry, we can expand themagnetic scalar potential with the help of the Legendre polynomial. Besides,the magnetic scalar potential at any point can be obtained with the knowingthe magnetic scalar potential on the z-axis:
(a) For r < a: l
ll
lM PrA )(cos
l
llMl zAzPzr )( 1)1( , 0
Expand M(z) with the binomial expansion, and note that the constant termof the M(z) can be dropped without loss:
......!3
125
23
21
!21
23
21
!11
211
2
12
)(
642
21
2
az
az
az
aIz
az
aIzzM
2011 Classical Electrodynamics Prof. Y. F. Chen
§5. 6 Magnetic scalar potential
......)(cos165
)(cos83)(cos
2)(cos
2) ,(
...... ,325 ,
163 ,
4 ,
2
......325
163
42
77
7
55
5
33
3
1
7151321
7
7
5
5
3
3
Par
ParP
arP
arIr
aIA
aIA
aIA
aIA
aIz
aIz
aIz
aIz
M
2011 Classical Electrodynamics Prof. Y. F. Chen
§5. 6 Magnetic scalar potential
(b) For r > a: l
llM zBAz 10
1)(
l
llMl zAzPzr )( 1)1( , 0
Expand M(z) with the binomial expansion, and note that the constant termof the M(z) can be dropped without loss:
...... ,325 ,
163 ,
4 ,
2
......325
163
42
......!3
125
23
21
!21
23
21
!11
211
2
12
)(
65
43
210
6
6
4
4
2
2
642
21
2
aIBaIBaIBIA
zIa
zIa
zIaI
za
za
zaI
zaIzM
2011 Classical Electrodynamics Prof. Y. F. Chen
§5. 6 Magnetic scalar potential
The figure below shows the simulation of the magnetic scalar potentialviewed from radial axis with the parameters of I = 0.1 and a = 1:
......)(cos165
)(cos83)(cos
21
2) ,(
56
6
34
4
12
2
Pr
a
PraP
raIrM
2011 Classical Electrodynamics Prof. Y. F. Chen
§5. 6 Magnetic scalar potential
§5.7 Magnetic moment
For a localized current density, we can use Taylor expansion:
......'1'
13
xxx
xxx
0'')'( 3 ldIxdxJ no magnetic monopole
......'')'(4
)(
......'')'(4
'')'(
4)(
33
0
33
030
xdxxJxxxA
xdxxxJx
xdxx
xJxA
ii
2011 Classical Electrodynamics Prof. Y. F. Chen
O( )J x
xx
P
In the text book, it is pointed out that:
'''')'( 33 xdxJxxdxxJxj
jiji
0''' 3 xdJxJx ijji
')'('21
'''21'')'(
3
33
xdxJxx
xdJxJxxxdxxJxj
ijjiji
It is customary to define the magnetic moment: ')'('21 3xdxJxm
Consequently, the magnetic dipole vector potential is: 30
4)(
xxmxA
And the magnetic induction outside the localized source is: 30 )ˆ(ˆ3
4)(
xmnmnxB
2011 Classical Electrodynamics Prof. Y. F. Chen
§5.7 Magnetic moment
B
H
(1) ABB
0
(2) Magnetization: i
ii mNM
(3) With the bulk magnetization and a macrosopic current density:
SxdxxxM
fAAfAf
xdxx
xMxx
xJ
xdxx
xxxMxx
xJxA
xxVxxxM
xxVxJxA
as ,0'')'('
'')('
''
1')'(')'(
4
''
')'(')'(
4)(
'')'(
4')'(
4)(
3
30
33
0
300
2011 Classical Electrodynamics Prof. Y. F. Chen
§ 5.8 Macroscopic equations, boundary conditions on and
interpret:
''
)'(')'(4
30 xdxx
xMxJ
MJxM
)(
JMBH
JJB M
0
0
If the material is linear:
cdiamagneti :
tic paramagne: :
0
0
HB
(4) Boundary conditions: (the same discussion as for the electrostatics)
densitycurrent surface a is where, :
: 0
12
21
KKHHJH
BBB
tt
nn
2011 Classical Electrodynamics Prof. Y. F. Chen
B
H
§ 5.8 Macroscopic equations, boundary conditions on and
(1) Generally applicable method of the vector potential:
equation Poisson :
0 :gauge Coulomb choose
11
if
0
2
2
JA
A
AAA
JABH
HB
ABB
(2)
equation Laplace: 0
0 if
0 entialscalar pot magnetic 0
2
2
M
M
M
BHB
HHJ
2011 Classical Electrodynamics Prof. Y. F. Chen
§ 5.9 Methods of solving boundary-value problems in magnetostatics
(3) Hard ferromagnetic 0 given, JM
equation Poisson :
defineand ,0
0
20
MM
M
M
MMHB
HJ
(i) In free space and no surface contribution:
''
)'('41'
')'(
41)( 33 xd
xxxMxd
xxxx M
M
(ii) With surface contribution:
'
')'(ˆ
41'
')'('
41)( 3 da
xxxMnxd
xxxMxM
2011 Classical Electrodynamics Prof. Y. F. Chen
§ 5.9 Methods of solving boundary-value problems in magnetostatics
Moreover:
......(*)'
')'(
41
)(
')'('
141
0 ,0'ˆ')'('
')'('
')'('
1'41'
')'('
41
'')('
''
)'('41)(
3
3
3
33
3
xdxxxM
MfMffM
xdxMxx
SdanxxxMxd
xxxM
xdxMxx
xdxxxM
MfMffM
xdxxxMxM
2011 Classical Electrodynamics Prof. Y. F. Chen
§ 5.9 Methods of solving boundary-value problems in magnetostatics
Note that the expression (*) is generally applicable even for the limit ofdiscontinuous contributions magnetic surface charge density:
'')'('
41'
')'('
41
')'('
1'41
')'('
141
'')'(
41)(
33
3
3
3
xdxxxMxd
xxxM
xdxMxx
xdxMxx
xdxxxMxM
Surface charge contribution
2011 Classical Electrodynamics Prof. Y. F. Chen
§ 5.9 Methods of solving boundary-value problems in magnetostatics
Consider a sphere of radius a, with a uniform permanent magnetization ofmagnitude M0 and parallel to the z-axis:
M
0 2and symmetry,-
) ,()' ,'(12
4)'(cos
)'(cos'
1 :remember
'''sin''cos
4'
''cos
41)(
'cosˆand ,0
ˆ
2
00,
*
01
2020
0
0
mde
YYl
P
Prr
xx
ddxx
aMdaxx
Mx
MMnM
zMM
mim
lm
l
lmlml
lll
l
M
M
2011 Classical Electrodynamics Prof. Y. F. Chen
§5.10 Uniformly magnetized sphere
cos3
)(
1 12
2)()( :ityorthogonalBy
'''sin'cos)'(cos)(cos4
)(
)(cos4
12) ,(
1
20
'
1
1-'
2
0 001
20
0
l
l
M
llll
ll
ll
l
M
ll
rraMx
ll
dxxPxP
ddPPrraMx
PlY
(1) Inside the sphere: cos3
) ,( , 0 rMrrrar M
(2) Outside the sphere: cos3
) ,( , 2
30
raMrarrr M
2011 Classical Electrodynamics Prof. Y. F. Chen
§5.10 Uniformly magnetized sphere
2011 Classical Electrodynamics Prof. Y. F. Chen
The magnetic scalar potential and the lines of and are shown below. The linesof are continuously closed paths, but those of terminate on the surfacebecause there is an effective surface charge density.
B
H
B
H
§5.10 Uniformly magnetized sphere
Find the magnetic dipole moment of a uniformly charged spherical shell of radiusa rotating with angular frequency about the z-axis:
zaaM
aVmM
adam
adaadm
adaqTT
qtqdI
aA
ˆ
34sin
sin22
sin
sin222
2sin
4
0
34
2
2
We can use this model to explain some magnetized behavior of the atomic system.
z
e
2011 Classical Electrodynamics Prof. Y. F. Chen
§5.11 Final remark
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