Chapter 7.
Time-Varying Fields and
Maxwell’s Equations
Electrostatic & Time Varying Fields
•
Electrostatic fields
  E  0, D  
B  0,   H = J
D  E
H
•
1

B
In the electrostatic model, electric field and magnetic fields are not related
each other.
Faraday’s law
•
A major advance in EM theory was made by M. Faraday in 1831
who discovered experimentally that a current was induced in a
conducting loop when the magnetic flux linking the loop changed.
d
electromotive force (emf): V  
dt
V 
Faraday’s law
•
Fundamental postulate for electromagnetic induction is
V 
•
d
B
 V   Edl   Edl   
ds
C
C
S
dt
t
B
E  
t
The electric field intensity in a region of time-varying magnetic flux
density is therefore non conservative and cannot be expressed as
the negative gradient of a scalar potential.
• The negative sign is an assertion that the induced emf will cause a
current to flow in the closed loop in such a direction as to oppose the
change in the linking magnetic flux  Lentz’s law
V 
d
dt
E  
B
t
7-2.3. A moving conductor in a static magnetic field
•
Charge separation by magnetic force
Fm  qu  B
•
To an observer moving with the conductor, there is no apparent
motion and the magnetic force can be interpreted as an inducted
electric field acting along the conductor and producing a voltage.
V21    u  B dl
2
1
•
Around a circuit, motional emf or flux cutting emf
V21    u  B dl
C
A moving conductor in a static magnetic field
•
Example 7-2
(a) Open voltage V0 ?
(b) Electric power in R
(c) Mechanical power required to move the sliding bar
(a) V0  V1  V2    u  B dl    a x u  a z B0  a y dl   uB0 h
1
C
2
uB0h 
V uB h
(b) I  0  0  Pe  I 2R 
R
R
R
2
W 
(c ) PM  FM u, FM  mechanical force to counteract the magnetic force Fm
1
Fmag  I  dl  B  a x IB0 h
2
I
uB0 h
 FM  a x
R
 Pe  PM
u  B0 h 
R
N
 FM  Fmag
2
 PM 
u 2  B0 h 
R
2
W 
A moving conductor in a static magnetic field
•
Example 7-3. Faraday disk generator
V0    u  B dl =   a r  B0a z  a r dr  
C
3
0
 B0b 2
  B0  rdr  
V
b
2
4
F  q E  u  B
Magnetic force & electric force
When a charge q0 moves parallel to the current on a wire,
the magnetic force on q0 is equivalent to the electric force on q0.
At the rest frame on wire
FB  q0 v  B
I
At the moving frame on charge
FE  q0 E'
L  L0 1  v / c     0 / 1  v / c 
2
2
E'
I
 To observer moving with q0 under E and B fields, there is no apparent motion.
But, the force on q0 can be interpreted as caused by an electric field, E’.
7-2.4. A moving circuit in a time-varying magnetic field
 To observer moving with q0 under E and B fields, there is no apparent motion.
But, the force on q0 can be interpreted as caused by an electric field, E’.
E  E  u  B
Now, consider a conducting circuit with contour C and surface S
moves with a velocity u under static E and B fields.
u
Changing in magnetic flux due to the circuit movement produces an emf, V:
d

 VB
dt
On the other hand, the moving circuit experiences an emf, V’, due to E’:

C
E 'dl  VE '
 Is it true that
VB  VE '
VB  VE ' ??
E  E  u  B
From the Faraday's law of

C
B
d s,
S t
E dl   
by replacing E with E = E  - u  B,
u

C
E  dl   
S
B
 d s    u  B  dl
C
t
time variation
at rest
Note that
VE '   Edl
C
V 
motional emf
VB  
Therefore, we need to prove that
d
d

dt
dt
d
dt

S

B ds 
S
Bd s

S
B
 d s    u  B  dl
C
t
VB  VE ' ??
•
Time-rate of change of magnetic flux through the contour
d d

dt dt
1 
B  t  t d s 2   B  t d s1 


S
S
S1
t 0 t 
 2
B  t 
B  t  t   B  t  
t  H.O.T.,
t
B  t 
1 
d
d

B

s
d s  lim
Bd s 2   Bd s1  H.O.T.





S
S
S
S1
0
t


dt
t
t  2
•

Bd s  lim
In going from C1 to C2, the circuit covers a region bounded by S1, S2, and S3

V
B d  0   Bd s 2   Bd s1   Bd s3
S2
S1
S3
d s3  dl  ut   Bd s 2   Bd s1  t   u  B dl
S2

•
u
d
dt

S
S1
C
emf = V    Edl
B  t 
d s    u  B dl
S
C
t
Bd s  
Therefore, the emf induced in the moving circuit C is equivalent
to the emf induced by the change in magnetic flux
VE '  VB

V 
d
dt
: same form as not in motion.
C
A moving circuit in a time-varying magnetic field
•
Example 7.3
– Determine the open-circuit voltage of the Faraday disk generator
   Bds  B0 
b
0
S

t
0
b2
rdrd  B0 t 
2
 B0b 2
d
VB  

dt
2
Compare!
VE ' 

C
E  dl   
S
B
 d s    u  B  dl
C
t
VE '    u  B dl =   a r  B0a z  a r dr  
C
3
0
 B0b 2
  B0  rdr  
b
2
4
VB  VE '
VE ' 

C
E  dl   
S
B
 d s    u  B  dl
C
t
VB  
d
d

dt
dt

S
Bd s
(Example 7-4) Find the induced emf in the rotating loop under B(t )  a y Bo sin t
(a) when the loop is at rest with an angle .
   Bds   a y Bo sin t  a n hw   B0 hw sin t cos 
Va  
d
  B0 S cos t cos  ,  S  hw : the area of the loop 
dt
(b) When the loop rotates with an angular velocity ω about the x-axis
1
3
w
w

Va    u  B dl   [(an  )  (a y B0 sin t )]  ax dx   [( an  )  (a y B0 sin t )]  ax dx
C
2
4
2
2
w
 2(  B0 sin t sin t )h  B0 S sin t sin  ( = t)
2
VE '  Va  Va  B0 S (cos 2 t  sin 2 t )   B0 S cos 2t
Compare!
  t   B  t  a n  t  S   B0 S sin t cos   B0 S sin t cos t 
VB  
d
d 1

   B0 S sin  2t     B0 S cos 2t
dt
dt  2

1
B0 S sin  2t 
2
VB  VE '
7-3. How Maxwell fixed Ampere’s law?
•
We now have the following collection of two curl eqns. and two divergence eqns.
B
,   H =J
t
D   ,
B =0
E  
•
Charge conservation law  the equation of continuity

J  
t
•
The set of four equations is now consistent with the equation of continuity?
Taking the divergence of   H =J ,
   H   J  0     H   0 from the vector null identity
J does not vanish in a time-varying situation
and this equation is, in general, not true.
Maxwell’s equations
• A term ∂ρv /∂t must be added to the equation.

D 

   H   0  J 
  J 
  D  
t
t 

D
H  J 
t
E
  B   0 J   0 0
t
• The additional term ∂D/∂t means that a time-varying
electric field will give rise to a magnetic field,
even in the absence of a free current flow (J=0).
• ∂D/∂t is called displacement current (density).
Maxwell’s equations
•
Maxwell’s equations
B
E  
t
D
  H =J 
t
D  
B =0
Continuity equation

J  
t
Lorentz’s force
F  q( E    B)
•
These four equations, together with the equation of continuity and
Lorentz’s force equation form the foundation of electromagnetic
theory. These equations can be used to explain and predict
all macroscopic electromagnetic phenomena.
•
The four Maxwell’s equations are not all independent
– The two divergence equations can be derived from the two curl
equations by making use of the equation of continuity
Maxwell’s equations
• Integral form & differential form of Maxwell’s equations
Differential form
B
E  
t
D
H = J 
t
D  
B =0
Integral form

C
B
d
 dS  
S t
dt
Faraday’s law
D
 ds
S t
Ampere’s law
Edl   

C
Significance
H dl  I  

Dds  Q
Gauss’s law

B  ds  0
No isolated
magnetic charge
C
S
Maxwell’s equations
•
Example 7-5
– Verify that the displacement current = conduction current in the wire.
iC : conduction current in the connecting wire
dC
 C1V0 cos t
dt
A
C1 
d
iC  C1
E
C
d
: uniform between the plates
C  V0 sin t
V0
 D   E   sin t
d
D
 V0

iD  
ds     cos t A  C1V0 cos t  iC
A t
 d

Maxwell’s equations
•
Example 7-5
– Determine the magnetic field intensity at a distance r from the wire
Two typical open surfaces with rim C may be chosen:
(a) a planar disk surface S1 ,  b  a curved surface S 2
For the surface S1, D = 0

C
H dl  2 rH    J ds  iC  C1V0 cos t  H  
S1
C1V0
 cos t
2 r
Since the surface S 2 passes through the dielectric medium,
no conducting current flows through S 2  displacement current
D
ds iC
A t
2 rH   iD  
 H 
C1V0
 cos t
2 r
7-5. Electromagnetic Boundary Conditions

C
B
 dS  0 when h  0
S t
Edl   
 E1t  E2t
D 

H

dl

J

C
S  t ds ;
 D 
S  t ds  0 when h  0
 an 2   H1  H 2   J s
 D  dS  
s
V
 H1t  H 2t  J sn 
 d  a n  D1  D2    s when h  0
2
 D1n  D2 n   s
 B  dS  0
s
 B1n  B2 n  0
 1H1n  2 H 2 n 
Electromagnetic Boundary Conditions
Both static and time-varying electromagnetic fields satisfy the same boundary conditions:
E1t  E2t
 The tangential component of an E field is continuous across an interface.
H1t  H 2t  J sn
 The tangential component of an H field is discontinuous across an interface
where a surface current exists.
B1n  B2 n
 The normal component of an B field is continuous across an interface.
D1n  D2 n   s
 The normal component of an D field is discontinuous across an interface
where a surface charge exists.
Boundary conditions at an interface between two lossless linear media
Between two lossless media () with  = 0, and S = 0, Js = 0
D1t 1

E1t  E2t 
D2t  2

B
D


E
,
H


 


B1t 1
H1t  H 2t 

B2t 2
D1n  D2 n  1 E1n   2 E2 n
B1n  B2 n  1 H1n  2 H 2 n
Boundary conditions at an interface between dielectric and perfect conductor
In a perfect conductor (  infinite, for example, supercondiuctors),
 E2  0  D2 , H 2  0  B2
Medium 1(dielectric)
Medium2
(perfect metal)
E1t  0
E2t  0
H1t  J s
H 2t  0
D1n   s
D2n  0
B1n  0
B2n  0
Boundary conditions
•
Table
7-4. Potential functions
B   A
T
•
Vector potential
•
Electric field for the time-varying case.
E  
   B  0

A 







A
E



0
t
t 

A
E
 V
t
 E  V 
A
t
 V/m 
Due to charge distribution 
Due to time-varying current J
Wave equation for vector potential A
A 

From  H  B    A, D   E    V 
,
t 

D 

H  J 

t 


A 
    A   J    V 

t 
t 
2A
 V 
2
   A    A   J    
   2 or
t 
t

2A
V 

 A   2    J     A  

t
t 

2A
2
 A   2    J
t
2
0 (Lorenz condition, or gauge)
(# Show that the Lorentz condition is consistent
with the equation of continuity. Prob. P.7-12)
Non-homogeneous wave equation for vector potential A
traveling wave with a velocity of
1

Wave equations for scalar potential V
ρv
A
and E  ,
From E  V 
ε
t
A  ρv

 V 


t  ε

ρ

V
 2V    A    v   A   με
ε
t
t
ρv
 2V
 V  με 2  
ε
t
2
for scalar potential V
V




A



0


t


 The Lorentz condition uncouples the wave equations for A and for V.
 The wave equations reduce to Poisson’s equations in static cases.
Gauge freedom
•
Electric & magnetic field
•
Gauge transformation
E  V 
A
, B =  A
t
If A  A   , B remains unchanged.
A




   V 
t
t
t

 A

 t

 Thus, if V is further changed to V  V 
, E also remains same.
t
 E  V 
 Gauge invariance
E & B fields are unchanged if we take any function (x,t) on simultaneously A and V via:
A  A  
V V 

t
V
 A  
0
t
 2
    2  0
t
2
 The Lorentz condition can be converted to a wave equation.
7-6. Solution of wave equations
•
The mathematical form of waves
f (x, t =0)
t=0
up
t = t0
x = upt0
x
2 f 1 2 f
f  x, t   f  x  u p t   2  2 2  0 : wave equation
x
u p t
Wave equation
•
Simple wave
– http://navercast.naver.com/science/physics/1376
  x  t 
y  x , t   A cos 

  A cos  kx  t 
T



2
2

 2 f , k 
, up 

T
k

Solution of wave equations from potentials
ρv
 2V
 V  με 2  
ε
t
2
: Nonhomogeneous wave equation
for scalar electric potential
R
d '
  t 
First consider a point charge at time t, (t)v’, located at a origin.
Except at the origin, V(R) satisfies the following homogeneous equation ( = 0):
1  2 V 
R
 for spherical symmetry V  R,  ,    V  R 
2 
R  R 
1   2 V 
 2V
 2
R
   2  0
t
R R  R 
1
Introducing a new varible, V  R, t   U  R, t 
R

 2U
 2U
R
1
 2   2  0  U  R, t   U  t   or U  R  u p t  , u p 
 up 
R
t




R
 Thus, we can write in a form of V  R, t   V  t  
 up 


Since  2V 
Solution of wave equations
The potential at R for a point charge   t   is,
R
  t   
V  R  
4 R

R
V  R, t   V  t 
 up

d '
  t 



 t  R / u p  
  V t  R / u p 
4 R



R
Now consider a charge distribution over a volume V’ .
V  R, t  
  R, t  R / u p 
1
4 

A  R, t  
4
V

V
R
J  R, t  R / u p 
R
d 
d 
V
 Wb/m 
d
R
V’
  R,t 
 The potentials at a distance R from the source at time t depend on the values of  and J
at an earlier time (t- R/u)  Retarded in time
 Time-varying charges and currents generate retarded scalar potential, retarded vector potential.
Source free wave equations
Maxwell’s equations in source-free non-conducting media (ε, μ, σ=0).
  E  
•
H
E
,   H =
, E  0, H =0
t
t
Homogeneous wave equation for E & H.
  E 2E  2E  H=
E
t

 2E
 E     H   2
t
t
1  2E
2
 E 2 2  0
u p t
1 2 H
In an entirely similar way,  H  2 2  0
up t
2
Review –The use of Phasors
Consider a RLC circuit
L
di
1
 Ri   idt  V  t , i  t   I 0 cos t   
dt
C
Phasor method (exponential representation)



V  t   V0 cos t  Re  V0 e j 0 e jt   Re Vs e jt



i  t   Re  I 0 e j e jt   Re I s e jt
Vs  V0 e j 0
I s  I 0e
j


 (Scalar) phasors that contain amplitude and phase information
but are independent of time t.
If we use phasors in the RLC circuit,

 R 

 I

di
 Re j I s e jt ,  idt  Re  s e jt 
dt
 j

1 

j L 
I  Vs
C   s


 i (t )  Re Vs


/ R 

1   it

j L 
  e
C


 


Time-harmonic Maxwell’s & wave equations
•
Vector phasors.
E  x, y, z , t   Re E  x, y, z  e jt 
H  x, y, z , t   Re  H  x, y, z  e jt 
•
Time-harmonic (cos t) Maxwell’s equations in terms of vector phasors
D  
B
E  
t
B =0
D
  H =J 
t

 E 

  E   j H
H  0
  H  J  j E
Time-harmonic Maxwell’s & wave equations
Time-harmonic wave equations (nonhomogeneous Helmholtz’s equations)
2
 ( R, t )

V ( R, t )
2
 V ( R, t )  

2

t
 ( R)
2
2
  V ( R)    j  V ( R)  

 2

k  wave number =  

up

 V ( R)  k V ( R)  

2
2
2


A

2
2
2
 A( R)  k A( R)    J    A   2    J 
t


Time-harmonic retarded potential
•
Phasor form of retarded scalar potential
V  R 
•
4 
V
R
d 
V
cos t  kx 
 e j t  kx   e jt e  kx
Phasor form of retarded vector potential.

A  R 
4
•
  R  e jkR
1

J  R  e jkR
V
When kR  2
R
R

d 
 Wb/m 
 1
k 2 R2
 1  jkR 
 ...  1
e
2
V  R  , A  R   static expressions(Eq. 3-39 & Eq. 5-22)
 jkR
Time-harmonic retarded potential
•
Example: Find the magnetic field intensity H and the value of β when = 90

E  z , t   a y 5cos 109 t   z
ax
1 
H  z  
j0 x
0
  10 (1/s)
9
E  z   a y 5e  j  z
H z  
1
j0
  V/m 
E

ay

y
5e  j  z
  j z   

 j z 
e
e
5
5



a
 a x  H x ( z )a x
 x
 
j0 
z
  0

1
 2
1  

Ez 
H 
ay Hx   ay  2
j
j  z

   0
3
    0  3 0 0 
 10  rad/m 
c
1
az

z
0
  j z
 5e


5e  j  z  a x  0.0398  e  j10 z
0

5e  j  z  a x  0.0398  cos 109 t  10 z 
H  z , t   a x
0
H  z   a x
The EM Waves in lossy media
•
If a medium is conducting (σ≠0), a current J=σE will flow
  H =J 
D
t

 
  H    j  E  j   
 E  j c E,
j 

 c  complex permittivity     j      j
•
•
•
•


When an external time-varying electric field is applied to material bodies,
small displacements of bound charges result, giving rise to a volume density
of polarization. This polarization vector will vary with the same frequency as
that of the applied field.
As the frequency increases, the inertia of the charged particles tends to
prevent the particle displacements from keeping in phase with the field
changes, leading to a frictional damping mechanism that causes power
loss.
This phenomenon of out of phase polarization can be characterized by a
complex electric susceptibility and hence a complex permittivity.
Loss tangent, δc
The EM Waves in lossy media
•
Loss tangent, δc

 
  H  j c E  j   
E
j 

 c     j 
tan  c 
•
  

,  c : loss angle
  
Good conductor if

E
 
  H =J  
 j   
E
t
j 

J
c

>>1


<<1

•
Good insulator if
•
Moist ground : loss tangent ~ 1.8104@1kHz, 1.810-3 @10GHz
E
The electromagnetic spectrum
•
Spectrum of electromagnetic waves