Transcript Document

Chapter 7 Time-varying Electromagnetic Fields
Displacement Current, Maxwell’s Equations
Boundary Conditions, Potential Function
Energy Flow Density, Time-harmonic Electromagnetic Fields
Complex Vector Expressions
1. Displacement Electric Current
2. Maxwell’s Equations
3. Boundary Conditions for Time-varying Electromagnetic Fields
4. Scalar and Vector Potentials
5. Solution of Equations for Potentials
6. Energy Density and Energy Flow Density Vector
7. Uniqueness Theorem
8. Time-harmonic Electromagnetic Fields
9. Complex Maxwell’s Equations
10. Complex Potentials
11. Complex Energy Density and Energy Flow Density Vector
1. Displacement Electric Current
The displacement current is neither the conduction current nor the
convection current, which are formed by the motion of electric charges.
It is a concept given by J. C. Maxwell.
Based on the principle of electric charge conservation, we have
 J  dS  
S
For static fields
q
t
 J  dS  0
S
J  

t
 J  0
which are called the continuity equations for electric current.
For time-varying electromagnetic fields, since the charges are
changing with time, the electric current continuity principle cannot be
derived from static considerations. Nevertheless, an electric current is
always continuous. Hence an extension of earliest concepts for steady
current need to be developed.
The current in a vacuum capacitor is
neither the conduction current nor the
 convection current, but it is actually the
displacement electric current.
Gauss’ law for electrostatic fields,  D  dS  q , is still valid for timeS
varying electric fields, we obtain
D 

J


 S  t   dS  0
D 

J 
0
t 

Obviously, the dimension of D is the same as that of the current density.
t
British scientist, James Clerk Maxwell named D the density of the
t
displacement current, denoted as Jd , so that
D
Jd 
t
We obtain
  (J  J d )  0
 (J  J d )  dS  0
S
The introduction of the displacement current makes the timevarying total current continuous, and the above equations are called
the principle of total current continuity.
The density of the displacement current is the time rate of change
of the electric flux density, hence
For electrostatic fields, D  0, and the displacement current is zero.
t
In time-varying electric fields, the displacement current is larger if
the electric field is changing more rapidly.
In imperfect dielectrics, J d  J c , while in a good conductor, J d  J c .
Maxwell considered that the displacement current must also produce
magnetic fields, and it should be included in the Ampere circuital law, so
that
 H  dl   (J  J d )  dS
l
S
D
D
)  dS
 H  J 
l
S
t
t
Which are Ampere’s circuital law with the displacement current. It
i.e.
 H  dl   (J 
shows that a time-varying magnetic field is produced by the conduction
current, the convection current, and the displacement current.
The displacement current, which results from time-varying electric
field, produces a time-varying magnetic field.
The law of electromagnetic induction shows that a time-varying
magnetic field can produce a time-varying electric field.
Maxwell deduced the coexistence of a time-varying electric field and
a time-varying magnetic field, and they result in an electromagnetic
wave in space. This prediction was demonstrated in 1888 by Hertz.
2. Maxwell’s Equations
For the time-varying electromagnetic field, Maxwell summarized
the following four equations:
The integral form
The differential form
D
lH  dl   S (J  t )  dS
B
lE  dl   S t  dS
D
t
B
 E  
t
 B  dS  0
 B  0

 D  
 H  J 
S
S
D  dS  q
D
t
B
 E  
t
 H  J 
 B  0
 D  
The time-varying electric field is both divergent and curly, and the
time-varying magnetic field is solenoidal and curly. Nevertheless, the
time-varying electric field and the time-varying magnetic field cannot
be separated, and the time-varying electromagnetic field is divergent
and curly.
In a source-free region, the time-varying electromagnetic field is
solenoidal.
The electric field lines and the magnetic field lines are linked with
each other, forming closed loops, and resulting in an electromagnetic
wave in space.
The time-varying electric field and the time-varying magnetic field
are perpendicular to each other.
In order to describe more completely the behavior of time-varying
electromagnetic fields, Maxwell’s equations need to be supplemented by
the charge conservation equation and the constitutive relations:

BH
J   E  J
J  
D E
t
where J  stands for the impressed source producing the time-varying
electromagnetic field.
The four Maxwell’s equations are not independent. Equations 4 and
3 can be derived from Equation 1 and 2, respectively, and vise versa.
For static fields, we have
E D H B



0
t
t
t
t
Maxwell’s equations become the former equations for electrostatic
field and steady magnetic field. Furthermore, the electric field and the
magnetic field are independent each other.
As the founder of relativity, Albert Einstein (1879-1955), pointed out
in his book “The Evolution of Physics” that
“The formulation of these equations is the most important event in
physics since Newton’s time, and they are the quantitative mathematical
description of the laws of the field. Their content is much richer than we
have been able to indicate, and the simple form conceals a depth
revealed only by careful study”.
“These equations are the laws representing the structure of the field.
They do not, as in Newton’s laws, connect two widely separated events;
they do not connect the happenings here with the conditions there”.
“The field here and now depends on the field in the immediate
neighborhood at a time just past. The equations allow us to predict
what will happen a little further in space and a little later in time, if we
know what happens here and now”
Maxwell’s equations have made important impact on the
history of mankind, besides the advancement of science and
technology.
As American physicist, Richard P. Feynman, said in his book
“The Feynman Lectures on Physics”, that “From along view of the
history of mankind──seen from say, ten thousand years from now
──there can be little doubt that the most significant event of the 19th century will be judged as Maxwell’s discovery of the laws of
electrodynamics. The American civil war will pale into provincial
insignificance in comparison with this important scientific event of
the same decade.”
In this information age, from baby monitor to remote-controlled
equipment, from radar to microwave oven, from radio broadcast to
satellite TV, from ground mobile communication to space communication, from wireless local area network to blue tooth technology, and
from global positioning to navigation systems, electromagnetic waves
are used as to make these technologies possible.
The wireless information highway makes it possible that we can
reach anybody anywhere at anytime, and to be able to send text,
voice, or video signals to the recipient miles away. Electromagnetic
waves can recreate the experience of events far away, making
possible wireless virtual reality.
To appreciate the contributions Maxwell and Hertz have made to
the progress of mankind and our culture, one only needs to look at
the wide usage of electromagnetic waves.
3. Boundary Conditions for Time-varying Electromagnetic Fields
In principle, all boundary conditions satisfied by a static field can be
applied to a time-varying electromagnetic field.
(a)
The tangential components of the electric field intensity are
continuous at any boundary, i.e.
en
②
E1t  E2t
①
en  ( E2  E1 )  0
or
As long as the time rate of change of the magnetic flux density is
finite, using the same method as before we can obtain it from the
equation:
B
E

d
l


l
 S t  dS
For linear isotropic media, the above equation can be rewritten as
D1t

1

D2 t
2
(b) The normal components of magnetic flux intensity are
continuous at any boundary.
From the principle of magnetic flux continuity, we find
B1n  B2n
or
en  ( B2  B1 )  0
For linear isotropic media, we have
 1H1n   2 H 2n
(c) The boundary condition for the normal components of
electric flux density depends on the property of the media.
In general, from Gauss’ law we find
D2n  D1n   S
or
en  ( D2  D1 )   S
where S is the surface density of the free charge at the boundary.
At the boundary between two perfect dielectrics, because of the
absence of free charges, we have
D1n  D2n
For a linear isotropic dielectric, we have
 1E1n   2 E2n
(d) The boundary condition for the tangential components of the
magnetic field intensity depends also on the property of the media.
In general, in the absence of surface currents at the boundary, as
long as the time rate of change of the electric flux density is finite, we
find
H1t  H 2 t
or
en  ( H 2  H1 )  0
However, surface currents can exist on the surface of a perfect
electric conductor, and in this case the tangential components of the
magnetic field intensity are discontinuous.
Assume the boundary is formed by a perfect dielectric and a
perfect electric conductor. Time-varying electromagnetic field and
conduction current cannot exist in the perfect electric conductor, and
they are able to be only on the surface of the perfect electric conductor.
E ≠ 0  J = E  
  
H≠0  E≠0
E(t), B (t), J (t) = 0
 H≠0
J≠0
The tangential component of the electric field intensity and the normal
component of magnetic field intensity are continuous at any boundaries,
and they cannot exist on the surface of a perfect electric conductor.
Consequently, the time-varying electric field must be perpendicular
to the surface of the perfect electric conductor, while the time-varying
magnetic field is tangential to the surface.
en
et
 ,
E
H2t
H

JS
H1t
②
①
Due to D1n  0 ，we find
D2n   S
or
en  D   S
Since there exists surface current JS on the surface of perfect
electric conductor,considering the direction of the surface current
density and the integral path complying with the right hand rule,
H1t  0 , we find
H 2t  J S
or
en  H  J S
Example. The components of the time-varying electromagnetic field
in a rectangular metal waveguide of interior cross-section a  b are
y
π 
E y  E y 0 sin  x  cos( t  k z z )
a 
π 
H x  H x 0 sin  x  cos( t  k z z )
a 
π 
H z  H z 0 cos x  sin(  t  k z z )
a 
b
a
z
x
z





x
y


a



g
x
z
b
y
Electric field lines
Magnetic field lines
Find the density of the
displacement current in
the waveguide and the
charges and the currents
on the interior walls of the
waveguide. The inside is
vacuum.
Solution: (a) We find the displacement current as
D
π 
 e y E y 0  sin  x  sin(  t  k z z )
Jd 
t
a 
(b) At the interior wall y = 0，we have
 S  e y  ( E y )   E y
J S  e y  ( H x  H z )  e z H x  e x H z
At the interior wall y = b, we have
 S  e y  ( E y )   E y
J S  e y  ( H x  H z )  e z H x  e x H z
At the lateral wall x = 0 , H x  0 , we have
J S  e x  e z H z 0 sin(  t  k z z )  e y H z 0 sin(  t  k z z )
At the lateral wall x = a , H x  0 , we find
J S  e x  e z ( H z 0 sin(  t  k z z ))  e y H z 0 sin(  t  k z z )
At the lateral walls x = 0 and x = a, due to E y  0 , and  S  0 .
y
The currents
on the interior walls
z
x
4. Scalar and Vector Potentials
Suppose the medium is linear, homogeneous, and isotropic, from
Maxwell’s equation we find
2H
 H   
 J
2
t
2E
J
    E    2  
t
t
Using     A    A   2 A , and considering   B  0 and   D   ，
then the above equations become
2E
J 1
 E   2  
 
t
t 
2
2H
 H  
   J
t 2
2
2E
J 1
 E   2  
 
t
t 
2H
2
 H  
   J
2
t
2
The relationship between the field intensities and the sources is
quite complicated. To simplify the process, it will be helpful to solve
the time-varying electromagnetic fields by introducing two auxiliary
functions: the scalar and the vector potentials.
Due to   B  0, hence B can be expressed in terms of the curl of a
vector field A, as given by
B   A
where A is called the vector potential. Substituting the above equation
into   E   B gives
t
 E  

(  A)
t
The above equation can be rewritten as    E  A   0
t 

A 
The vector  E 
is irrotational. Thus it can be expressed in terms of
t 

the gradient of a scalar , so that
A
E
 
t
where  is called the scalar potential, and we have
A
 
t
The vector potential A and the scalar potential  are functions of
time and space.
E
If they are independent of time, then the results are the same as
that of the static fields. Therefore, the vector potential A is also called
the vector magnetic potential and the scalar potential  is also called
the scalar electric potential. 。
In order to derive the relationship between the potentials and the
sources, from the definition of the potentials and Maxwell’s equations
we obtain
 2 A
 

    A   J    2  
t 
 t

 A 
         

 t 
Using     A    A   2 A , the above equations become
 2 A
 
   J
 A  (  A)    2  
t 
 t
2
 2 


(  A)  
t

The curl of the vector field A is given as   A  B , but the divergence
must be specified.
In principle, the divergence can be taken arbitrarily, but to simplify
the application of the equations, we can see that if let
Φ
  A   
t
Then the above two equations become
2 A
 A   2    J
t
2
Lorentz condition
 2Φ

 Φ   2  
t

2
After the divergence of the vector potential A is given by the Lorentz
condition, the equations are simplified. The original equations are two
coupled equations, while new equations are decoupled.
The vector potential A only depends on the current J, while the scalar
potential  is related to the charge density  only.
If the current and the charge are known, then the vector potential A
and the scalar potential  can be determined. After A and  are found,
the electric and the magnetic fields can be obtained.
Consequently, the solution of Maxwell’s equations is related to that
of the equations for the potential functions, and the solution is simplified.
The original Equations are two vector equations with complicated
structure, and in tree-dimensional space, six coordinate components need
to be solved.
2E
J 1
 E  


 
t 2
t 
2
2H
 H  
   J
t 2
2
New potential equations are a vector equation and a scalar
equation, respectively.
2 A
 2Φ

2
2
 A   2    J
 Φ   2  
t
t

In three-dimensional space, only four coordinate components need to
be found. In particular, in rectangular coordinate system the vector
equation can be resolved into three scalar equations。
5. Solution of Equations for Potentials
Here we find the solution by using an analogous method based on
the results of static fields.
The scalar potential caused by a point charge is obtained first, then
use superposition principle to obtain the solution of the scalar potential
due to a distribution of time-varying charge.
If the source is a time-varying point charge placed at the origin, the
distribution of the field should be a function of the variable r only, and
independent of the angles  and  .
In the open space excluding the origin, the scalar potential function
satisfies the following equation
where v 
1

 2 ( r ) 1  2 ( r )
 2
0
2
2
r
v
t
0r 
The above equation is the homogeneous wave equation for the
function ( r), and the general solution is
 r
 r
 v
 v
We will know that the second term is contrary to the physical
 r  f1  t    f 2  t  
situation, and it should be excluded. Therefore, we find the scalar
electric potential as
 r
f1  t  
v
Φ (r , t )  
r
The electric potential produced by the static elemental charge
q   dV at the origin is
 dV
 (r ) 
4 π r
Comparing the above two equations, we know
 r
f1  t   
 v
 r
 v  dV
4π 
 t  
Hence, we find the electric potential produced by the timevarying elemental charge at the origin as
 r
v
d (r , t )  
dV
4π r
 t  
where r is the distance to the field point from the charge dV .

  r , t 
z

r' - r
dV'
r  r 

v 
 (r, t)
V'
r'
O
x
From the above result,
the electric potential produced
by the volume charge in V 
can be obtained as
r
y
 (r , t ) 
1
4π

r  r 

  r , t 
v 

dV 
V 
r  r
To find the vector potential function A, the above equation can
be expanded in rectangular coordinate system, with all coordinate
components satisfying the same inhomogeneous wave equation, i.e.
2

Ax
 2 Ax  
  J x
2
t
2

Ay
2
 Ay  
  J y
2
t
 2 Az
2
 Az  
  J z
2
t
Apparently, for each component we can find a solution similar
to that of scalar potential equation. Incorporating the three
components gives the solution of the vector potential A as

r  r 



Jr ,t 
v 


A(r , t ) 
dV 


V

4π
r r
 (r , t ) 
1
4π

r  r 



  r ,t 
v 

dV 
V 

r r

r  r 



Jr ,t 
v 


A(r , t ) 
dV 


V
4π
r  r
Both equations show that the solution of the scalar or the vector
potential at the moment t is related to the source distribution at the

r  r 
moment  t 
.


v


In other words, the field at t does not depend on the source at the

r  r 
same moment, but on the source at an earlier time
t  .


v 

It means that the field produced by the source at r needs a
r  r
certain time to reach r, and this time difference is
.
v
The quantity r  r  is the distance between the source point
and the field point, and v stands for the propagation velocity of the
electromagnetic wave.
From v 
1

we can see that the propagation velocity of
electromagnetic wave is related to the properties of the medium. In
vacuum,
v
1
 0 0
 299792458m/s  3 108 m/s
which is the propagation velocity of light in vacuum, also called the
speed of light, usually denoted as c.
It is worth noting that the field at a point away from the source
may still be present at a moment after the source ceases to exist.
Energy released by a source travels away from the source and
continuous to propagate even after the source is taken away. This
phenomenon is a consequence of electromagnetic radiation.
The change with respect to time in the scalar electric potential 
and the vector magnetic potential A is always lagging behind the
sources. Hence the functions  and A are called the retarded
potentials.
r
Since the time factor  t   implies that the evolution of the field
v

precedes that of the source, it violates causality, and f 2  t  r  must be
 v
abandoned.
r
The time factor  t   can be rewritten as

v
r
( r )
t t
v
v
Hence, the function f 2  t  r 
can be considered as a wave traveling
 v
toward the origin as a reflected wave from a distant location.
For a point charge placed in open free space this reflective wave
cannot exist.
For surface or line charge and current, consider the following
substitutions:
JdV  J S dS  Idl
 dV   S dS  l dl
We obtain

r  r 

 S  r , t 
v
1

 dS 
 (r , t ) 
4π  S 
r  r

r  r 

J S  r , t 
v


 dS 
A(r , t ) 
4π  S 
r  r

r  r 



l  r , t 
v 
1

 (r , t ) 
dl 
4π l 
r  r

r  r 



I r ,t 
v 


A(r , t ) 
dl 
4π l 
r  r
All of the above equations are valid only for linear, homogeneous,
and isotropic media.
6.
Energy Density and Energy Flow Density Vector
All the formulas for the density of electrostatic energywe , the
density of steady magnetic energywm
, and the density of power
dissipation pl for a steady current field are valid for time-varying
electromagnetic fields.
For a linear isotropic media
1
we (r , t )   E 2 (r , t )
2
wm (r , t ) 
1
 H 2 (r , t )
2
pl (r , t )   E 2 (r , t )
The density of energy of a time-varying electromagnetic field is
given by
1
w (r , t )   E 2 (r , t )   H 2 (r , t )
2
Since a time-varying electromagnetic field changes with respect to
space and time, the density of its energy is also a function of space and
time, giving rise to a flow of energy. 。
Suppose there is a linear isotropic medium in the region V
without any impressed source ( J   0 ,   0) , and the parameters of
the medium are  ,  , . Then in this region the electromagnetic
field satisfies the following Maxwell’s equations:
E
 H  E 
t
H
  E  
t
  ( H )  0
, , 
E, H
V
  ( E )  0
Using   ( E  H )  H    E  E    H , we have
  H2
  (E  H )   
t  2
   E2 
 
  E2
 t  2 
Integrating both sides of the equation over the volume V, we have

V
  ( E  H )dV  

1
2
2
2
(

E


H
)
d
V


E
 V dV
t  V 2
Considering

V
  ( E  H )dV   ( E  H )  dS , we have
S
 1
2
2
(

E


H
)dV
S

V
V
t 2
Based on the definition of energy density, the above equation can
be rewritten as

  wdV   ( E  H )  dS   pl dV
S
V
t V
which is called the energy theorem for time-varying electromagnetic
( E  H )  dS    E 2 dV  
field, and any time-varying electromagnetic field satisfying Maxwell’s
equation must obey the energy theorem. The vector ( E  H ) represents
the power passing through unit
S
cross-sectional area and that is
, , 
just the energy flow density
vector S, given by
E, H
S  EH
S  EH
This equation states that S  E and S  H. S, E, H are perpendicular
to each other in space. They comply with the right hand rule.
The instantaneous value of the energy
flow density is
E
S (r , t )  E (r , t ) H (r , t )
S
H
The instantaneous value of the energy
flow density is equal to the product of the
instantaneous electric field intensity and
the instantaneous magnetic field intensity.
The energy flow density is maximum only if the two field intensities
are maximum. If the electric field intensity or the magnetic field
intensity is zero at a certain moment, the energy flow density will be
zero at that moment.
7. Uniqueness Theorem
In a region V bound by a closed surface S, if the initial values of the
electric field intensity E and the magnetic field intensity H are given at
the time t = 0, as long as the tangential component Et of the electric field
intensity or the tangential component Ht of the magnetic field intensity
is given at the boundary for t > 0, then the electromagnetic field will be
uniquely determined by Maxwell’s equations at any moment t (t > 0) in
the region V.
S
E(r, 0)
&
H(r, 0 )
E( r, t), H(r, t )
V
E t (r, t)
or
H t (r, t)
Based on the energy theorem and using the method of contrary, this
theorem can be proved.
8. Time-harmonic Electromagnetic Fields
A special kind of time-varying electromagnetic field that has
sinusoidal time dependence, with its general mathematical form
given by
E (r , t )  Em (r ) sin(  t  e (r ))
where Em(r) is a function of space only, referred to as the amplitude
of the sinusoidal function.  is the angular frequency and e(r) is the
initial phase.
The sinusoidal electromagnetic field is also called a time-harmonic
electromagnetic field.
From Fourier analyses, we know that any periodic or aperiodic
time function can be expressed in terms of the sum of sinusoidal
functions if it satisfies certain conditions. Hence, there is justification
for us to concentrate on sinusoidal electromagnetic fields.
A sinusoidal electromagnetic field is produced by charge and
current that have sinusoidal time dependence. The time dependence
of the field is the same as that of the source in a linear medium. Thus
the field and the source of a sinusoidal electromagnetic field have the
same frequency.
A complex vector can be used to represent these sinusoidal
quantities with the same frequency. Namely, only the amplitude and
the space phase
while the time dependent
 e (rare
) considered,
part  t is omitted.
The electric field intensity can be expressed in terms of a
complex vector E m (r ) as
E (r )  E (r ) e j e ( r )
m
m
The original instantaneous vector can be expressed in terms of
the complex vector as
E (r , t )  Im[ E (r ) e j t ]
m
In practice, the measured values are usually effective values
of the sinusoidal quantity, and it is denoted as E (r ).
Then
E (r )  E (r )e j e ( r )
E m (r )
2
where
E (r ) 
Hence
E m (r )  2 E (r )
The complex vector is a function of space only, and it is
independent of time.
In addition, this complex vector representation and the
operational method that makes it possible are only applicable to
sinusoidal functions of the same frequency.
9.
Complex Maxwell’s Equations
The field and the source of a sinusoidal electromagnetic field
have the same frequency in a linear medium. Hence Maxwell’s
equations can be expressed in terms of complex vectors.
Consider the derivative of a sinusoidal function with respect to
time t such that
E ( r , t )
 Im( j E m (r )e j t )  Im( j 2 E (r )e j t )
t
Hence,   H   E  

E
can be expressed as
t


 
  Im( 2 H e jt )  Im 2 Je jt  Im j 2 D e jt
or


 e jt )  Im
Im   ( 2 H

2 J  j

 
 e jt
2D


 e jt )  Im
Im   ( 2 H

2 J  j
 
 e jt
2D
Since the above equation holds for any moment t, the
imaginary symbol can be eliminated.
  J  j D

 H
  2 H  2 J  j 2 D
In the same way, we find
  E   j B
  B  0
  D  
and
  J   j 
D   E
B   H
J   E  J 
The above equations are called Maxwell’s equations in the
frequency domain, with all field variables being complex.
Example. The instantaneous value of the electric field intensity
of a time-variable electromagnetic field in vacuum is
E (r , t )  e y 2 sin( 10π x) sin(  t  k z z )
Find complex magnetic field intensity.
Solution: From the instantaneous value we have the complex
vector of the electric field intensity as
E (r )  e y sin( 10π x)e jk z z
Due to   E   jB   j0 H
H 
j

 E
0
Since the electric field has only the y-component, and it is
E y
independent of the variable y, so that
 0 , then
  E  e x
We find
E y
z
 ez
E y
x
y
 e x jk z sin( 10π x)e  jk z z  e z 10π cos(10π x) e  jk z z

  jk z z
kz
10π
H   e x
sin( 10π x)  e z j
cos(10π x)e




0
0


10. Complex Potentials
The equations for the potential functions can be expressed in
terms of the complex vector as
2 A
 A   2    J
t

 2   2     
2
 2Φ  

   2  A
   J
2 A
Φ



t 2

2

Consider the time delay factor   r  r  , for a sinusoidal function

v


it leads to a phase delay of    r  r  .

v 

k    
Let
Then

r  r
v

v
 k r  r 


We obtain
1
 (r , t ) 
4π

r  r 

  r , t 
v 

dV 
V 
r  r

r  r 



Jr ,t 
v 


A(r , t ) 
dV 


V
4π
r  r
1
 (r ) 
4π 

V
 (r )e  jk r - r
r  r

dV 
 (r )e  jk r - r 
J
A (r ) 
dV 


V
4π
r  r

The complex Lorentz condition is
Φ
 (r )   j   (r )
 A
t
The complex electric and magnetic fields can be expressed in
terms of the complex potentials as
  A   
B   A
A
E
 
t

B    A

  A




E   j A     j A 
j  
11.
Complex Energy Density and Energy Flow Density Vector
The instantaneous energy densities of electric and magnetic
origins, respectively, are
1
we (r , t )   E 2 (r , t )
2
wm (r , t ) 
1
 H 2 (r , t )
2
The corresponding maximum values
1
wem (r )   Em2 (r )
2
wmm (r ) 
1
 H m2 (r )
2
Or in complex vector as
1
1
wem (r )   E m  E m*
wmm (r )   H m  H m*
2
2
where E m* and H m* are the conjugates of the complex vectors E m
and H , respectively.
m
The effective value of a sinusoidal quantity is the root-meansquare of the instantaneous value. Hence, the average over a period
value of the energy densities of a sinusoidal electromagnetic field is
w av
i.e.
1

T

T
0
w(r , t ) dt 
 1
2  T

T
0
  1
E 2 (r , t )dt   
 2 T
1
1
wav   E 2 (r )   H 2 (r )
2
2
Which can be rewritten as
1
1
wav   E  E *   H  H *
2
2
Or in terms of the maximum value as
1
1
wav   E m  E m*   H m  H m*
4
4
or
wav 
1
( wem  wmm )
2

T
0

H 2 (r , t ) dt 

1
wav  ( wem  wmm )
2
which states that the average value of the energy density of a sinusoidal
electromagnetic field is equal to half of the sum of the maximum values
of the electric energy density and the magnetic energy density.
Similarly, the power dissipation per unit volume can also be
expressed in terms of the complex vector, and the maximum value is
plm (r )  Em2 (r )  E m  E m*
The time average is
1 
*


p l av (r )   E (r )   E  E   E m  E m*
2
2
The time average of the power dissipation per unit volume is also
the half of the maximum value.
The instantaneous value of the energy flow density vector S is
S (r , t )  E (r , t )  H (r , t )  Em (r )  H m (r )sin(  t  e ) sin(  t  h )
The time average is
1
S av (r ) 
T
Let

T
0
S ( r , t ) dt 
1
E m (r )  H m (r )cos( e  h )
2
S c (r )  E (r )  H * (r )
Where Sc a complex energy flow density vector, and
E (r )
are the effective values.
It can be written in terms of the maximum value as
Sc (r ) 
1 
E m (r )  H m* (r )
2
and
H * (r )
Then the real part and the imaginary part of the complex energy
flow density vector Sc, are
Re( S c ) 
1
E m (r )  H m (r )cos( e  h )
2
1
Im( S c )  E m (r )  H m (r )sin(  e  h )
2
we can see that the real part of the complex energy flow density
vector is just the time average of the energy flow density vector, i.e.
Re( Sc )  Sav (r )
The real part and the imaginary part of the complex energy flow
density vector depend not only on the magnitudes of the electric and
the magnetic fields but also on the phases.
Re( S c ) 
1
E m (r )  H m (r )cos( e  h )
2
Im( S c ) 
1
E m (r )  H m (r )sin(  e  h )
2
If the electric field and the magnetic field are in phase,
e  h  2nπ
Then the real part is the positive maximum, and the imaginary
part is zero.
If they are oppositely phased, e  h  (2n  1)π
Then the real part is the negative maximum, and the imaginary
part is still zero.
If they have a phase difference that is an odd multiple of
 e   h  (2n  1)
π
2
π
,
2
Then the real part is zero, while the imaginary part is the positive
or the negative maximum.
In other cases, the real and the imaginary parts are both non-zero.
t
E
t
S
t
H
t
Electric field intensity
Magnetic field intensity
Energy flow density
The energy theorem can also be expressed by complex vectors as
  ( E  H * )  dS    E  E * dV 
S
V
j  (  H  H *   E  E * )dV
V
i.e.
  Sc (r )  dS  
S
Pl (r ) dV 
V
j  2wmav (r )  weav (r ) dV
V
which is called the complex energy theorem.
The real part of the flux of complex energy flow density vector
flowing into the closed surface S is equal to the power dissipation
within the closed surface. It shows clearly that the real part of Sc
stands for the energy flowing along a certain direction.
For a sinusoidal electromagnetic field, the initial condition is no
longer required when the uniqueness theorem is applied. A sinusoidal
electromagnetic field is determined uniquely by the tangential
components of the electric or magnetic fields at the boundary.
S
E(r, 0)
&
H(r, 0 )
E(E(
r, t),
r), H(r,
H(r)t )
V
(r, t)
EEt t(r)
or
or
(r, t)
HHt t(r)
For convenience in notation, hereafter the effective values of the
complex vectors for the sinusoidal electromagnetic field will be
represented by E(r), H(r) or E, H, with the top symbol ‘ · ’ omitted,
and the instantaneous values will be expressed in terms of E(r, t), H(r, t)
or E(t), H(t) .
Example. The instantaneous value of the electric field intensity
of a time-variable electromagnetic field in vacuum is
E (r , t )  e y 2 sin( 10π x) sin(  t  k z z )
Find the average of the energy flow density vector.
Solution: we have found the electric and magnetic field
intensities as
E (r )  e y sin( 10π x)e jk z z


k
10π
H (r )  -e x z sin( 10π x)  e z j
cos(10π x)e  jk z z
 0
  0

Then
Sc  E  H  e z
*
kz
 0
sin 2 (10 π x)  e x j
10 π
sin( 20π x)
2  0
From Sav = Re (Sc) we find
Sav  e z
kz
 0
sin 2 (10π x)