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Chapter 10. Time-Varying Fields and
Maxwell’s Equations
1.
Faraday’s Law
•
•
•
A time-varying magnetic field produces an electromotive force
(emf) which may establish a current in a suitable closed circuit.
An electromotive force is merely a voltage that arises from
conductors moving in a magnetic field or from changing
magnetic fields.
Faraday’s Law: emf   d V
dt
•
1.
2.
3.
The minus sign is an indication that the emf is in such a direction
as to produce a current whose flux, if added to the original flux,
would reduce the magnitude of the emf.(Lenz’s Law)
A time-changing flux linking a stationary closed path
Relative motion between a steady flux and a closed path
A combination of the two.
목원대학교 정보통신공학전공
9-1
• If the closed path is taken by an N-turn filamentary conductor
emf   N
• Define:
emf   E  dL
d
dt
emf   E  dL  
d
B  dS
dt S
• The fingers of our right hand indicate the direction of the
closed path, and our thumb indicates the direction of dS.
• A flux density B in the direction of dS and increasing with time
thus produces an average values of E which is opposite to the
positive direction about the closed path
• Stationary Path
B
 dS and
S t
emf   E  dL   
  E  dS   B  dS
t
 E  dL     E  dS (Stokes' Theorem)
S
  E   B
t
Electrosta tics :  E  dL  0 and   E  0
목원대학교 정보통신공학전공
9-2
Example) Within th e cylindrica l region   b, B  Bo e kt a z
Choosing the circular path   a, a  b, in the z  0 plane
1
emf  2aE  kBo e kt a 2  E   kBo e kt a 
2
 ( E )
1
  Ez  kBo e kt  1
 kBo e kt  2  E
 
2
1
E   kBo e kt a 
2
A filamentar y conductor of resistance R would have a current flowing in the negative a 
and this current establish a flux withi n the circular loop in the negative a z direction.
[Time - constant flux and a moving closed path]
  Byd
d
dy
emf  
  B d   Bvd
dt
dt
 E  dL  EL   Bvd
* Lenz' s law
목원대학교 정보통신공학전공
9-3
F  Qv  B
F
 vB
Q
The force per unit charge : the motional electric field intensity E m
Em  v  B
If the moving conductor were lifted off the rails, this electric field intensity would force
electrons to one end of the bar (the far end) until the static field due to these charges just
balanced the field induced by the motion of the bar. The resultant tangential electric field
intensity would then be zero along the length of the bar.
emf   E m  dL   v  B   dL   vBdx   Bvd
0
d
In the case of a conductor
moving in a uniform constant magnetic field
emf   E  dL   E m  dL   v  B   dL
If the magnetic flux density is changing with time ,
B
 dS   v  B   dL
S t
transfor mer emf motional emf
emf   E  dL   
emf  -
d
dt
목원대학교 정보통신공학전공
9-4
2.
Displacement Current
B
t
Steady magnetic field   H  J
E  
 v
t
 v

D
0    J    G Thus   G 
 (  D)   
t
t
t
    H  0    J but the eq. of continuity :   J  
H  J G
G
D
t
H  J 
D
t
D
: displaceme nt current density
t
Conduction current density, the motion of charge (zero net charge density) : J  E
Convection current density, the motion of volume charge density : J   v v
  H  J  Jd Jd 
In a nonconduct ing medium,   H 
목원대학교 정보통신공학전공
D
(J  0)
t
9-5
D
 dS
S t
I d   J d  dS  
S
The time varying version of Ampere' s cirsuital law :
Applying Stokes' theorem :
D




H

d
S

J

d
S

S
S
S  dS
t
D
 dS
S t
 H  dL  I  I d  I  
Circuit : a filamentar y loop and a parallel - plate capacitor
A magnetic field varying sinusoidal ly with ti me  emf  Vo cos t
Negligible resistance and inductance : I  C
H
k
k
dV
S
 CVo sin t   Vo sin t
dt
d
 dL  I k
 V cos t 
Within the capacitor : D  E    o

d


D
S
Id 
S   Vo sin t
t
d
Displaceme nt current is associated with time - varying
electric fields and therefore exists in all imperfect
conductors carrying a time - varying conduction current.
목원대학교 정보통신공학전공
9-6
3.
Maxwell’s Equations in Point Form
B
t
(If a changing magnetic field is present, E may have circulatio n.)
D
H  J 
t
  D   v (Charge density is a source of electric flux density)
E  
  B  0 (no magnetic charges)
D  E   o E  P
B  H   o (H  M )
J  E conduction current density
J   v v convection current density
For linear materials, P   e  o E M   m H
Lorentz force equation ( the force per unit volum e)
f   v (E  v  B)
목원대학교 정보통신공학전공
9-7
4.
Maxwell’s Equations in Integral Form
B
t
D
H  J 
t
  D  v
B
 dS (Faraday' s law)
S t
D
H

d
L

I


S t  dS (Ampere' s circuital law)
E  
 E  dL   
B  0
 D  dS  
 B  dS  0
S
vol
S
 v dv (Gauss' s law for the electric field)
(Gauss' s law for the magnetic field)
[Boundary conditions ]
Et1  Et 2
H t1  H t 2 (K  0) DN 1  DN 2   S
BN 1  BN 2
In a perfect conductor, E  0 H  0 J  0
If region 2 is a perfect conductor,
E t1  0 H t1  K D N 1   S
목원대학교 정보통신공학전공
BN 1  0
9-8
5.
The Retarded Potentials
 v dv 2

 V   v (static), E  V (static)
vol 4R

Jdv
Vector magnetic potential : A  
(dc),  2 A   J (dc), B    A (dc)
vol 4R
Scalar Electric potential : V  
In time - varying fields, B    A (   B  0)
B
E  
 E  V  N (   V  0)
t
B

A
A
E  0  N N  
  N     A    
N t
t
t
t
E  V -
A
t
D
H  J 
t
 V  2 A 
  A  -  A  J    
 2 

t
t 






  D   v      V -   A    v  2V    A    v
t
t



V
Define   A   
t
v
2A
 2V
2
2
 A   J   2
and  V     2
t

t
9-9
2
목원대학교 정보통신공학전공

V  2 A 

    A  J     


t

t


1