电磁场与电磁波第13讲磁化强度磁场强度和相对导磁率

2 b2 z
0 Ib2
2 3/ 2

(T)
6
EXAMPLE 6-7 Find the magnetic flux density at a distant point of a small circular loop of radius b that carries current I (a magnetic dipole).
ˆ J ms M an J m M
磁 偶 极 子
m IS
电磁对偶性
12
2. Magnetization and Equivalent Current Densities
dF Idl B
an
F
d B S a b F c F a d S b B
4
Main topic
Steady Magnetic Fields
1. The Magnetic Dipole 2. Magnetization and Equivalent Current Densities 3. Magnetic Field Intensity and Relative Permeability
2 ˆ J m M (A/m ) Jms M an
A/m
17
3. Magnetic Field Intensity and Relative Permeability
Fundamental equation of magnetostatics in free space
' Fdv ' F ds '
S'
0 ˆ M an ' ' M ' 0 A V ' R dv 4π S ' R ds 4π
'
Where an’ is the unit outward normal vector from ds’ and S’ is the surface bounding the volume V’ . Similar
Bn
F c F a
B
F
F
d S

b F
c Bt
F
F
磁偶极子受磁场力而转动
13
B0
B'
14
To analyze the macroscopic effect of magnetic dipoles we define a magnetization vector, M,
M lim
mk
10
y
Magnetic dipole:
0 Ib2 ˆ A a sin 2 4R
Electric dipole:
0 Ib2 ˆ ˆ B (aR 2cos asin ) 3 4R
E p 4 0 R
3
ˆ P aR qd cos V 2 4π 0 R 4π 0 R 2
0 M aR ' ˆ dA dv 2 4πR
15
Integrating over the volume V’ of the dielectric, we obtain the vector magnetic potential due to the magnetized dielectric
b
’

dl’ a’ ar ’
P ''
y
P'
8
R12 R 2 b 2 2bR cos R 2 b 2 2bR sin sin ' 1 1 b 2 2b (1 2 sin sin ' )1/ 2 R1 R R R
z

R
P( R, , / 2)
Postulates of Magnetostatics in Free Space
Differential form
Integral form
B 0
B 0 J
S B ds 0 B dl 0 I
C
3
3. Vector Magnetic Potential
M 1 ' ' ' 1 ( ) M M ( ) R R R
'
we can rewrite as
0 M ' 0 M ' A dv ( )dv ' V ' R V ' 4π 4π R
16
Be aR ' 0 ˆ ' 1 A ' dA M 2 dv = M ( )dv ' V 4π V ' R 4π V ' R
Where R is the distance from the elemental volume dv’ to a fixed field point. Recalling the vector identity,
P( R, , / 2)

R R1
4R
Magnetic dipole:
I
o
b ’ 0 Ib2 ˆ A a sin 2 P ' dl’ 4R x 2 0 Ib ˆ ˆ B A (aR 2 cos a sin ) 3 4R

P ''
a’ ar ’
a’’
z

R
P( R, , / 2)
R1 o
ar’’
I
'' ˆ d l bd ' a '' '' '' x ˆ ˆ ˆ dx d l ax bd ' a '' ax bd 'sin ' '' '' ˆ ˆ ˆ dy d l a y bd ' a '' a y bd 'cos ' '' ˆ ˆ ˆ ˆ d l dx '' ax dy '' a y ( ax sin ' a y cos ')bd '
I
o az
a''
a''r
b
a'
y
x
P ' a'
r
ˆ' ˆ ˆ ˆ ˆ dl ' R bd ' a zaz ba 'r a 'r bzd ' az b 2 d '
0 I B 4
a ˆ
c
z
b2 z
b 2 d '
2 3/ 2

ˆ az

2
0
1 b ˆ (1 sin sin ' )( ax )b sin ' d ' R R
2 0 Ib b ˆ (ax ) (1 sin sin ' ) sin ' d ' 0 4 R R z /2 0 Ib b ˆ a (1 sin sin ' ) sin ' d ' / 2 2 R R 0 Ib 2 ˆ a sin 2
ˆ (a R 2cos aˆsin )
ˆ ˆ ˆ m az I b2 az IS az m (A m2 )
Where is defined as the magnetic dipole moment, which is a vector whose magnitude is the product of the current in and the area of the loop and whose direction is the direction of the thumb as the fingers of the right hand follow the direction of the current.
R1 o
I
b
2 2 2
b P ' dl’ R b , 2 0 x R 1 1 2b 1 b ' 1/ 2 (1 sin sin ) (1 sin sin ' ) R1 R R R R
’

a’ ar ’
P ''
y
9
0 I A 4
' I dl 0 C' R1 4
ˆ 0 I b 2 0 m aR 0 m ˆ ˆ B (aR 2 cos a sin ) ˆ A a sin 3 2 2 4 R 4 R 4 R
11
模型 电 偶 极 子
极化，磁化
产生的电场与磁场
p qd
p P ˆ ps P an
B 0
B 0 J
In the magnetized medium, the magnetic field can be considered as that produced by the conducting current I and the magnetizing current I in vacuum. In this way,
Field and Wave Electromagnetic
电磁场与电磁波
第13讲
作业情况
1班：人 合计：人 情况:
2
Review
1. Introduction
Lorentz’s force equation
F qE qu B
2. Fundamental Postulates of Magnetostatics in free space
R b
z

R
P( R, , / 2)
Spherical coordinates
' 0 I dl A C 4 ' R1
R1
I
b
o
’

dl'
' dl ? R1 ?
P
y
x
P'
7
' ˆ d l bd ' a ' ' ' ˆ ˆ ˆ dx d l ax bd ' a ' ax bd 'sin ' ' ' ˆ ˆ ˆ dy d l a y bd ' a ' a y bd 'cos ' ' ˆ ˆ ˆ ˆ d l dx ' ax dy ' a y (ax sin ' a y cos ')bd '
k 1
nv
v 0
v
(A/m)
as where mk be the magnetic dipole moment of an atom，n is the number of atoms per unit volume and the numerator represents the vector sum of the magnetic dipole moments contained in a very small volume v. The vector M, is the volume density of magnetic dipole moment. The magnetic dipole moment dm of an elemental volume dv’ is dm=Mdv’ , will produce a vector magnetic potential
0 A 4

V'
0 J ' d v ;A R 4

S'
Js ' ds R
The effect of the magnetization vector is equivalent to both a volume current density and a surface current density.
B 0 B＝ A
A 0 J
2
Ax 0 J x ; Ay 0 J y ; Az 0 J z ;
2 2 2
0 J ' A V ' R dv (Wb/m) 4 4. The Biot-Savart Law and Applications ' aR 0 I ' R 0 I ˆ B C' d l R2 4 C' d l R3 C' dB (T) 4
5
1. The Magnetic Dipole
Example 6-6 cylindrical coordinate
0 I B 4 ' R C' d l R3 (T)
- a'r
z
P
az
ˆ' dl ' a bd ' ˆz ba'r ˆ R op op ' za