Transcript 전 자 기 학

Chapter 8. The Steady Magnetic Field
1.
Biot-Savart Law
dH 2 
I1dL1  a R12
(A/m)
4R122
cf) dE 2 
dQ1a R12
(V/m)
4o R122
 v
(Eq.(5) in Sec.5.2)
t
Dircet current only :  v is not a function of t
J  
J  0 
dH 
IdL  a R

4R 2
IdL  a R
H
4R 2
목원대학교 전자정보통신공학부 전자기학
 J  dS  0
S
Current Flow around a closed path!!
8-1
Surface current density : K
uniform current density : I  Kb
Nonuniform current density : I   KdN
IdL  KdS  Jdv
K  a R dS
S
4R 2
J  a R dv
vol 4R 2
H
H
r  a  r  z a z R12  r  r  a   z a z
a   z a z
a R12 
  z
2
dH 2 
H2  

목원대학교 전자정보통신공학부 전자기학
dL  dz a z
Idz a z  ( a   z a z )
4 (  2  z 2 ) 3 / 2
 Idz a z  ( a   z a z )

Ia
2
4 (  2  z 2 )3 / 2
z
4  2  2  z 2

Ia
dz 
4  (  2  z 2 ) 3 / 2


H2 

I
2
a
8-2
IdL  a R
dL  dza z z   tan 
4R 2
dz   sec 2  d a R  cos a   sin a z
dH 
dH 

I sec 2 da z  (cos a   sin a z )
4 (  2  z 2 )
I sec 2 d cos a
4 2 sec 2 
2
I
1
4
H   dH 
목원대학교 전자정보통신공학부 전자기학

I cos 
a d
4
(sin  2  sin 1 )a
8-3
2.
Amperes’s Circuital Law
•
The line integral of H about any closed path is exactly equal
to the direct current enclosed by that path.
 H  dL  I
2
2
0
0
 H  dL   H  d H    d H   2  I  H  
I
2
H 
I
2
  a : I encl
( a    b)
 2
I
 I 2 H 
a
2a 2
  c : H   0 (Shielding )
목원대학교 전자정보통신공학부 전자기학
  2  b2 

b    c : 2H   I  I  2
2 
 c b 
I c2  b2
H 
2 c 2  b 2
8-4
The magnetic field intensity is continuous
at all the conductor boundaries.
*Shielding
H can not vary w ith x or y. H y , H z  0. H x only exist.
H x1 L  H x 2 ( L)  K y L H x1  H x 2  K y
H x 3  H x 2  K y  H x 3  H x1
Hx 
1
1
K y ( z  0) H x   K y ( z  0)
2
2
1
H  K aN
2
A second sheet with K   K y a y at z  h.
H  K  a N (0  z  h) H  0 ( z  0, z  h)
목원대학교 전자정보통신공학부 전자기학
8-5
Solenoid
Toroid
 H  dL H  2
 K a 2 (  o  a)
목원대학교 전자정보통신공학부 전자기학
8-6
3.
Curl
•
Gauss’s law ↔ Divergence : Ampere’s law ↔ Curl
H y x 
H y (x) 





(y )
H

d
L

H


y

H

yo
yo





x
2

x
2




H x y 
H x (y ) 


(x)   H yo 
x
  H xo 

y
2

y
2




 H y H x
 


x
y

J z  lim
x , y 0
 H  dL  H
xy
x
y


xy  J z xy ( ΔI  J z xy )

H x
 H  dL  H z  H y , J  lim  H  dL  H x  H z
, J x  lim
y
y , z 0 yz
z , x 0 zx
y
y
z
z
x
목원대학교 전자정보통신공학부 전자기학
8-7
( curl H) N  lim
S N 0
 H  dL
S N
ax
 H z H y 
 H y H x 

 H x H z 




curl H  

a


a


a



x
y
z
 x
z 
x 
y 
x
 z
 y

Hx
ay

y
Hy
az

 H
z
Hz
Curl: a line integral per unit area. Circulation per unit area.
Non-time-varying conditions
 H  J
The point form of Ampere’s circuital law
 E  dL  0    E  0
목원대학교 전자정보통신공학부 전자기학
8-8
4.
Stokes’ Theorem
 H  dL
S
S
 H  dL
S
 (  H) N  (  H)  a N
 (  H)  a N S  (  H)  S
 H  dL   (  H)  dS
S
목원대학교 전자정보통신공학부 전자기학
8-9
 (  H)  dS   J  dS  H  dL  I
S
•
•
S
Stokes’ theorem relates a surface integral to a closed integral.
The divergence theorem relates a volume integral to a closed
surface integral.
 A  T

vol
(    A)dv   Tdv
vol
Divergence Theorem :  (  A)  dS   Tdv  0
S
vol
( The applicatio n of Stokes' theorem to a closed surface  0)
  A  0
 H  J   J  0
목원대학교 전자정보통신공학부 전자기학
8-10
5.
The Magnetic Flux and Magnetic Flux Density
•
•
Define: The Magnetic Flux Density
The permeability
 o  4  10 7 H/m
Magnetic Flux    B  dS Wb
S
 B  dS  0
(cf,    D  dS  Q)
S
B  0
S
•
B   o H (free space only)
Maxwell’s equations(static electric field & steady magnetic field)
D  
E  0
H  J
B  0
D   oE
B  o H
E  V
 D  dS  Q    dv
 E  dL  0
 H  dL  I  J  dS
 B  dS  0
S
vol
v
S
S
목원대학교 전자정보통신공학부 전자기학
8-11
H 
I
2
B  o H 
( a    b)
o I
a
2
   B  dS  
S
6.
d
0

b
a
o I
 Id b
a  ddza  o ln
2
2
a
The Scalar and Vector Magnetic Potentials
Define H  Vm
  H  J     Vm 
The curl of the gradient of any scalr is identicall y zero.
If H is to be defined as the gradient of a scalar megnetic potential, then current density
must be zero throughou t the region in which t he scalar megnetic potential is so defined.
H  Vm
목원대학교 전자정보통신공학부 전자기학
(J  0)
8-12
•
The scalar magnetic potential also satisfies Laplace’s equation.
  B  o  H  0
 o   (  Vm )  0
 2Vm  0 (J  0)
J  0 in the region a    b
I
H
a
2
I
1 Vm
  V m   
2
 
Vm
I
I

Vm  


2
2
At P,  
 9 17
4
,
4
,
4
,, or -
7 15 23
,,,
4
4
4
The electric potential V is single valued once a zero reference is assigned.
Vm is not a single - valued function of position.
목원대학교 전자정보통신공학부 전자기학
8-13
•
The reason for this multivaluedness
b
 E  dL  0, therefor e V   E  dL (independe nt of path)
  H  0 (wherever J  0) but  H  dL  I
  E  0 and
ab
a
b
Vm,ab   H  dL (specified path)
a
•
Vector Magnetic Potential(A), useful in studying radiation from
antennas, from apertures, and radiation leakage from
transmission lines, waveguides, and microwave ovens
B  0
H
1
o
B   A
 A H  J 
목원대학교 전자정보통신공학부 전자기학
1
o
 A
A
 o IdL
4R
dA 
 o IdL
4R
8-14
Vector magnetic potential field about a differential filament
IdL  KdS  Jdv
A
 o IdL
4R
 o KdS
S
4R
A
dL  dza z
dA 
 o IdL
 o Idza z

4R
4  2  z 2
 o Idz
dA z 
dH 
4   z
2
1
o
  dA 
2
 o Jdv
vol 4R
A
, dA  0, dA  0
1  dAz

 o  

Idz

a  
4  2  z 2

목원대학교 전자정보통신공학부 전자기학


3/ 2
a
8-15
7.
Derivation of the Steady-Magnetic-Field Laws
o Jdv
IdL  a R
(3)
B


H
(32)
B



A
(46)

A

(51)
o
2

vol
?
4R
4R
Eq. (51) is correct and agrees with the three definition s (3), (32), and (46).
H
Prove Ampere’s circuital
law in point form
    A  (  A)   2 A
2 A  2 Axa x  2 Aya y  2 Aza z (The Laplacian of a vector)
 v dv

  2V   v
vol 4 R
o
o
 J dv
Ax   o x   2 Ax   o J x ,  2 Ay   o J y ,  2 Az   o J z
vol 4R
V 
  A  0 (p. 264 - 265)
H 
1
o
목원대학교 전자정보통신공학부 전자기학
 A 
1
o
[(  A)   2 A] 
 2 A   o J
1
o
 o J   J
8-16
Between conductors , J  0
 2 A   o J  0
Cartesian coordinate s :  2 A   2 Ax a x   2 Ay a y   2 Az a z
Cylindrica l coordinate s :  2 A   2 A a    2 A a    2 Az a z
However,  2 A   2 Az
z
1   Az

Therefore,  Az 
   
 1  2 Az  2 Az
  2

0
2
2
z
  
1   Az 

0
Az is a function only of  :
    
2
Az  C1 ln   C 2
Assume a zero reference at   b, Az  C1 ln
 A  
b
Az
C
C
a    1 a   B H   1 a


o 
2
 H  dL  I   
0
Az 

C1
o 
a  da  
2C1
o
 C1  
o I
2
o I b
I
ln
and H  
2

2
목원대학교 전자정보통신공학부 전자기학
8-17