Chapter 7Time-varying Electromagnetic Fields Displacement Current, Maxwell’s Equations Boundary...
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Transcript of Chapter 7Time-varying Electromagnetic Fields Displacement Current, Maxwell’s Equations Boundary...
Chapter 7 Time-varying Electromagnetic Fields
Displacement Current, Maxwell’s EquationsBoundary Conditions, Potential Function
Energy Flow Density, Time-harmonic Electromagnetic Fields Complex Vector Expressions
1. Displacement Electric Current2. Maxwell’s Equations3. Boundary Conditions for Time-varying Electromagnetic Fields4. Scalar and Vector Potentials5. Solution of Equations for Potentials6. Energy Density and Energy Flow Density Vector7. Uniqueness Theorem8. Time-harmonic Electromagnetic Fields9. Complex Maxwell’s Equations10. Complex Potentials11. 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.
For static fields 0d S SJ 0 J
Based on the principle of electric charge conservation, we have
t
qS
SJ d
t
J
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.
Gauss’ law for electrostatic fields, , is still valid for time-
varying electric fields, we obtain
qS
d SD
0d
SS
t
DJ
The current in a vacuum capacitor is
neither the conduction current nor the
convection current, but it is actually the
displacement electric current.
Obviously, the dimension of is the same as that of the current
density.tD
0
t
DJ
t
D
Jd
0d)( d S
SJJ 0)( d JJWe obtain
British scientist, James Clerk Maxwell named the density of the
displacement current, denoted as Jd , so thattD
The introduction of the displacement current makes the time-
varying 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, , and the displacement current is zero. 0
t
D
In time-varying electric fields, the displacement current is larger if
the electric field is changing more rapidly.
In imperfect dielectrics, , while in a good conductor, . cd JJ cd JJ
Maxwell considered that the displacement current must also produce
magnetic fields, and it should be included in the Ampere circuital law, so
thatSJJlH d)(d
d Sl
SD
JlH d)(d
Sl t t
D
JHi.e.
Which are Ampere’s circuital law with the displacement current. It
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.
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.
The law of electromagnetic induction shows that a time-varying
magnetic field can produce a time-varying electric field.
2. Maxwell’s Equations
For the time-varying electromagnetic field, Maxwell summarized
the following four equations:
SD
JlH d)(d
Sl t
SB
lE dd
Sl t
0d
S SB
qS
d SD
The integral form
t
D
JH
t
BE
0 B
D
The differential form
The time-varying electric field is both divergent and curly, and the
time-varying magnetic field is solenoidal and curly. Nevertheless, the t
ime-varying electric field and the time-varying magnetic field cannot b
e separated, and the time-varying electromagnetic field is divergent an
d curly.
In a source-free region, the time-varying electromagnetic field is sol
enoidal.
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.
t
D
JH
t
BE
0 B
D
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:
t
J ED HB JEJ
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
0
tttt
BHDE
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.
where stands for the impressed source producing the time-
varying electromagnetic field.
J
“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”.
As the founder of relativity, Albert Einstein (1879-1955), pointed
out in his book “The Evolution of Physics” that
“The field here and now depends on the field in the immediate neig
hborhood at a time just past. The equations allow us to predict what wi
ll happen a little further in space and a little later in time, if we know w
hat happens here and now”
“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”.
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 19-
th century will be judged as Maxwell’s discovery of the laws of elect
rodynamics. The American civil war will pale into provincial insigni
ficance in comparison with this important scientific event of the sam
e decade.”
In this information age, from baby monitor to remote-controlled
equipment, from radar to microwave oven, from radio broadcast to s
atellite TV, from ground mobile communication to space communi-c
ation, 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.
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.
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.
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.
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:
2t1t EE
or 0)( 12n EEe
SB
lE dd
Sl t
For linear isotropic media, the above equation can be rewritten as
2
t2
1
t1
DD
①
② en
(b) The normal components of magnetic flux intensity are
continuous at any boundary.
From the principle of magnetic flux continuity, we find
2nn1 BB
or 0)( 12n BBe
(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
SDD 1n2n
or S )( 12n DDe
where S is the surface density of the free charge at the boundary.
For linear isotropic media, we have n22 n11 HH
At the boundary between two perfect dielectrics, because of the
absence of free charges, we have
2n1n DD
(d) The boundary condition for the tangential components of th
e magnetic field intensity depends also on the property of the media.
In general, in the absence of surface currents at the boundary, as lo
ng as the time rate of change of the electric flux density is finite, we find
t21t HH
0)( 12n HHeor
However, surface currents can exist on the surface of a perfect elect
ric conductor, and in this case the tangential components of the magn
etic field intensity are discontinuous.
For a linear isotropic dielectric, we have 2n2 n11 EE
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.
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.
E(t), B (t), J (t) = 0
E ≠ 0 J = E
H ≠ 0 E ≠ 0
J ≠ 0 H ≠ 0
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.
Due to , we find 01n D
SD 2n or SDen
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,
, we find01t H
SH J2t SJHe nor
E
H ,
en
et ①②
H1t
H2t JS
Example. The components of the time-varying electromagnetic fiel
d in a rectangular metal waveguide of interior cross-section a b are
) sin(π
cos0 zktxa
HH zzz
) cos(π
sin0 zktxa
HH zxx
) cos(π
sin0 zktxa
EE zyy
Find the density of the dis
placement current in the
waveguide and the charge
s and the currents on the i
nterior walls of the waveg
uide. The inside is vacuu
m.
a
z
y
x
b
x
z y
x
y
z
g
b
a
Magnetic field lines Electric field lines
Solution: (a) We find the displacement current as
t
D
Jd ) sin(π
sin 0 zktxa
E zyy
e
(b) At the interior wall y = 0 , we
have yyyS E ) ( Ee
zxxzzxyS HH eeHHeJ )(
At the interior wall y = b, we have
yyyS E ) ( Ee
zxxzzxyS HH eeHHeJ )(
At the lateral wall x = 0 , , we have0xH
) sin() sin( 00 zktHzktH zzyzzzxS eeeJ
) sin()) sin(( 00 zktHzktH zzyzzzxS eeeJ
At the lateral wall x = a , , we find 0xH
At the lateral walls x = 0 and x = a, due to , and . 0yE 0S
z
y
x
The currents on the interior walls
4. Scalar and Vector Potentials
JH
H
2
2
t
Suppose the medium is linear, homogeneous, and isotropic, fro
m Maxwell’s equation we find
tt
JE
E 2
2
Using , and considering and ,then the above equations become
0 BAAA 2 D
JH
H
2
22
t
1
2
22
tt
JEE
JH
H
2
22
t
1
2
22
tt
JEE
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.
AB
where A is called the vector potential. Substituting the above equation
into givest
BE
)( AE
t
Due to , hence B can be expressed in terms of the curl of
a vector field A, as given by
0 B
The above equation can be rewritten as 0
t
AE
The vector is irrotational. Thus it can be expressed in terms
of the gradient of a scalar , so that
t
AE
t
AE
where is called the scalar potential, and we have
t
AE
The vector potential A and the scalar potential are functions of t
ime and space.
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 so
urces, from the definition of the potentials and Maxwell’s equations w
e obtain
t
tt
A
AJA
2
2
JA
AA
tt 2
22 )(
)(2 At
Using , the above equations becomeAAA 2
The curl of the vector field A is given as , but the divergence m
ust be specified.
BA
t
Φ
A
Then the above two equations become
JA
A 2
22
t
2
22
t
ΦΦ
Lorentz condition
After the divergence of the vector potential A is given by the Lorent
z condition, the equations are simplified. The original equations are two
coupled equations, while new equations are decoupled.
In principle, the divergence can be taken arbitrarily, but to simplify
the application of the equations, we can see that if let
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.
The original Equations are two vector equations with complicated
structure, and in tree-dimensional space, six coordinate components
need to be solved.
New potential equations are a vector equation and a scalar
equation, respectively.
JH
H
2
22
t
1
2
22
tt
JEE
Consequently, the solution of Maxwell’s equations is related to that
of the equations for the potential functions, and the solution is
simplified.
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 。
JA
A 2
22
t
2
22
t
ΦΦ
Here we find the solution by using an analogous method based
on the results of static fields.
5. Solution of Equations for Potentials
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 .
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.
rt
r
vr
r0 0
) (1) (2
2
22
2
1
vwhere
In the open space excluding the origin, the scalar potential
function satisfies the following equation
The above equation is the homogeneous wave equation for the
function ( r), and the general solution is
v
rtf
v
rtfr 21
We will know that the second term is contrary to the physical
situation, and it should be excluded. Therefore, we find the scalar
electric potential as
rvr
tftΦ
1
),(r
The electric potential produced by the static elemental charge
at the origin isVq d
r
V
π4
d )(
r
Comparing the above two equations, we
know
Vvr
t
v
rtf d
π4
1
Hence, we find the electric potential produced by the time-
varying elemental charge at the origin as
Vrvr
tt d
π4
),( d
r
where r is the distance to the field point from the charge dV .
From the above result,
the electric potential produced
by the volume charge in V can be obtained as
Vv
t
tV
dπ4
1),(
rr
rr,r
r
r' r
z
y
x
(r, t)
V '
dV'
vt
rrr ,
r' - r
O
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.
x
xx J
t
AA
2
22
yy
y Jt
AA
2
22
zz
z Jt
AA
2
22
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
Vv
t
tV
d
,
π4),(
rr
rrrJ
rA
Vv
t
tV
dπ4
1),(
rr
rr,r
r
V
vt
tV
d
,
π4),(
rr
rrrJ
rA
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 moment .
vt
rr
It means that the field produced by the source at r needs a
certain time to reach r, and this time difference is . v
rr
vt
rr In other words, the field at t does not depend on the source at
the same moment, but on the source at an earlier time .
The quantity is the distance between the source
point and the field point, and v stands for the propagation
velocity of the electromagnetic wave.
rr
From we can see that the propagation velocity of electroma
gnetic wave is related to the properties of the medium. In vacuum, 1
v
m/s103m/s2997924581 8
00
v
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.
Radiation is associated with a time-varying electromagnetic
field while static field must be tied to a source, and the static field
is called the bound field.
The transition from rear-field to far-field depends not only on
the distance but also the time rate of change of the source.
At a point close to a time-varying charge or current, the field
varies almost in synchronism with the source. The field in this
region is called the near field, which is quasi-static in nature.
At a point very far away from the source, the delay in the action
of the field with respect to the source will become highly noticeable.
The field in this region is referred to as the far field, and it is called
radiation field.
A transmission antenna needs to be excited by a high frequency
current in order to radiate efficiently, while the 50Hz power line cur
rent has little radiation effect.
The change with respect to time in he 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.
Since the time factor implies that the evolution of the fiel
d precedes that of the source, it violates causality, and must
be abandoned.
v
rtf2
v
rt
v
rtThe time factor can be rewritten as
v
rt
v
rt
)(
For a point charge placed in open free space this reflective
wave cannot exist.
Hence, the function can be considered as a wave traveling
toward the origin as a reflected wave from a distant location.
v
rtf2
For surface or line charge and current, consider the following
substitutions:
All of the above equations are valid only for linear,
homogeneous, and isotropic media.
lSV lS ddd lJJ ddd ISV S
S
S
Sv
t
t d
,
π4
1),(
rr
rrr
r
S
S
Sv
t
t d
,
π4),(
rr
rrrJ
rA
l
l
lv
t
t d
,
π4
1),(
rr
rrr
r
l
lv
tI
t d
,
π4),(
rr
rrr
rA
We obtain
6. Energy Density and Energy Flow Density Vector
),( 2
1),( 2
e tEtw rr ),( 2
1),( 2
m tHtw rr ),( ),( 2 tEtpl rr
The density of energy of a time-varying electromagnetic field is
given by ),( ),(
2
1),( 22 tHtEtw rrr
For a linear isotropic media
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. 。
All the formulas for the density of electrostatic energy , the
density of steady magnetic energy , and the density of power
dissipation pl for a steady current field are valid for time-varying
electromagnetic fields.
mwew
In order to measure the flow direction and the magnitude of the
energy, we define a vector, whose direction stands for the flow
direction of the energy, and its magnitude is the amount of energy
flowing through a unit cross-sectional area per unit time. This vector
is called the energy flow density vector, denoted as S, with the unit
W/m2.
In the western textbooks, the energy flow density vector is called
the Poynting vector, and in Russian textbooks it is called the Woum
ov vector.
The energy flow density vector is denoted as S, with the unit W/m2.
The relationship between this vector and the field intensities
will be derived.
t
E
EH
t
H
E
0) ( H
0) ( E
Using , we haveHEEHHE )(
222
2
2
)( E
E
t
H
t
HE
Integrating both sides of the equation over the volume V, we have
V V V
VEVHEt
V
222 d d) (2
1d)( HE
, ,
E, HV
Suppose there is a linear isotropic medium in the region V
without any impressed source , and the parameters of
the medium are . Then in this region the electromagnetic
field satisfies the following Maxwell’s equations:
)0 0( ,J , ,
Considering , we have V S
V
d)(d)( SHEHE
V VS
VHEt
VE
222
d) (
2
1d d)( SHE
Based on the definition of energy density, the above equation
can be rewritten as
V lSV
VpVwt
dd )(d SHE
which is called the energy theorem for time-varying electromagnetic
field, and any time-varying electromagnetic field satisfying Maxwell’s
equation must obey the energy theorem. The vector ( ) represents th
e power passing through unit c
ross-sectional area and that is j
ust the energy flow density vec
tor S, given by
HE
, ,
E, H
HES
S
HES
S
E
H
The instantaneous value of the energy
flow density is
),(),(),( tHtEtS rrr
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.
This equation states that and . S, E, H are perpendicular t
o each other in space. They comply with the right hand rule.
ES HS
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 t
he time t = 0, as long as the tangential component Et of the electric field i
ntensity or the tangential component Ht of the magnetic field intensity is
given at the boundary for t > 0, then the electromagnetic field will be un
iquely determined by Maxwell’s equations at any moment t (t > 0) in the
region V.
Based on the energy theorem and using the method of contrary,
this theorem can be proved.
V
SE t (r, t) or H t (r, t)
E(r, 0)&
H(r, 0 )
E( r, t), H(r, t )
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))( sin()(),( em rrErE tt
where Em(r) is a function of space only, referred to as the amplitude o
f the sinusoidal function. is the angular frequency and e(r) is the i
nitial phase.
From Fourier analyses, we know that any periodic or aperiodic
time function can be expressed in terms of the sum of sinusoidal fun
ctions if it satisfies certain conditions. Hence, there is justification fo
r us to concentrate on sinusoidal electromagnetic fields.
The sinusoidal electromagnetic field is also called a time-
harmonic electromagnetic field.
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.
The electric field intensity can be expressed in terms of a
complex vector as)(m rE)(j
mmee )()( rrErE
The original instantaneous vector can be expressed in terms of
the complex vector as
]e )(Im[),( jm
tt rErE
A complex vector can be used to represent these sinusoidal
quantities with the same frequency. Namely, only the amplitude
and the space phase are considered, while the time
dependent part is omitted.
)(e r
t
In practice, the measured values are usually effective values
of the sinusoidal quantity, and it is denoted as .)(rE
2
)()( m rE
rE where
)(2)(m rErE Hence
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.
)(j ee)()( rrErE Then
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.
)e)( jIm(),( j
mt
t
t rErE
)e)(2jIm( j t rE
Consider the derivative of a sinusoidal function with respect to
time t such that
ttt DJH jjj e2 jIme2Im)e2Im(
tt DJH jj e2 j2Im)e2(Im or
Hence, can be expressed ast
E
EH
In the same way, we find
BE j
0 B
D
j J
ED HB
JEJ
and
The above equations are called Maxwell’s equations in the
frequency domain, with all field variables being complex.
Since the above equation holds for any moment t, the
imaginary symbol can be eliminated.
DJH 2 j22 DJH j
tt DJH jj e2 j2Im)e2(Im
Example. The instantaneous value of the electric field
intensity of a time-variable electromagnetic field in vacuum is
) sin() π10sin(2),( zktxt zy erE
Find complex magnetic field intensity.
Solution: From the instantaneous value we have the complex
vector of the electric field intensity as zk
yzx je) π10sin()( erE
Since the electric field has only the y-component, and it is
independent of the variable y, so that , then0
y
E y
x
E
z
E yz
yx
eeE zk
zzk
zxzz xxk jj e ) π10cos( π10e) π10sin(j ee
Due to HBE 0jj EH 0
j
zkz
zx
zxxk j
0 0
e) π10cos(
π10j) π10sin(
eeHWe find
10. Complex Potentials
The equations for the potential functions can be expressed in
terms of the complex vector as
Consider the time delay factor , for a sinusoidal functio
n it leads to a phase delay of .
v
rr
v
rr
Letv
k
rrrr
kv
Then
JA
A 2
22
t
2
22
t
ΦΦ JAA 22
22
We obtain
The complex Lorentz condition is
The complex electric and magnetic fields can be expressed in
terms of the complex potentials as
Vv
t
tV
dπ4
1),(
rr
rr,r
r
Vv
t
tV
d
,
π4),(
rr
rrrJ
rA
VV
k
de)(
π4)(
j
rr
rJrA
r-r
VV
k
de)(
π4
1)(
j
rr
rr
r-r
t
Φ
A )( j)( rrA
t
AE
AB AB
j j j
AAAE
11. Complex Energy Density and Energy Flow Density Vector
The instantaneous energy densities of electric and magnetic
origins, respectively, are
),( 2
1),( 2
e tEtw rr ),( 2
1),( 2
m tHtw rr
The corresponding maximum values
)( 2
1)( 2
mem rr Ew )( 2
1)( 2
mmm rr Hw
*EEr mmem 2
1)( w *HHr mmmm
2
1)( w
Or in complex vector as
where and are the conjugates of the complex vectors
and , respectively.
*Em *Hm
mE
mH
The effective value of a sinusoidal quantity is the root-mean-
square of the instantaneous value. Hence, the average over a period
value of the energy densities of a sinusoidal electromagnetic field is
ttwT
wT
d ),(1
0 av r
ttH
TttE
T
TTd ),(
1
2d),(
1
2
0
2
0
2 rr
)( 2
1)(
2
1 22av rr HEw i.e.
Or in terms of the maximum value as
** HHEE mmmmav 4
1
4
1 w
)(2
1mmemav www or
*HHEE 2
1
2
1 *av w
Which can be rewritten as
)(2
1mmemav www
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 expr
essed in terms of the complex vector, and the maximum value is
*EErr mm2mm )()( Epl
*mm
2av
2
1 )( )( EEEErr * Ep l
The time average is
The time average of the power dissipation per unit volume is also
the half of the maximum value.
),(),(),( ttt rHrErS ) sin() sin()()( hemm ttrHrE
The instantaneous value of the energy flow density vector S is
The time average is
ttT
Td ),(
1)(
0 av rSrS )cos()()(2
1h emm rHrE
Where Sc a complex energy flow density vector, and and
are the effective values.
)(rE )(rH *
)()(2
1)( mmc rHrErS *
It can be written in terms of the maximum value as
)()()( *c rHrErS Let
Then the real part and the imaginary part of the complex energy
flow density vector Sc, are
)cos()()(2
1)Re( hemmc rHrES
)sin()()(2
1)Im( hemmc rHrES
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( avc rSS
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.
If the electric field and the magnetic field are in phase,
π2he n
If they are oppositely
phased,
π)12(he n
If they have a phase difference that is an odd multiple of , 2
π
2
π)12(he n
)cos()()(2
1)Re( hemmc rHrES )sin()()(
2
1)Im( hemmc rHrES
In other cases, the real and the imaginary parts are both non-zero.
Then the real part is the positive maximum, and the imaginary part is
zero.
Then the real part is the negative maximum, and the imaginary part
is still zero.
Then the real part is zero, while the imaginary part is the positive or
the negative maximum.
Electric field intensity Magnetic field intensity
t
t
t
t
Energy flow density
E
H
S
V
V
V
VS
d) ( j
d d)(
*
**
*
EEHH
EESHE
The energy theorem can also be expressed by complex vectors as
Vww
V
V
V lS
d )()(2 j
d )( d)(
eavmav
c
rr
rPSrS
i.e.
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 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) .
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.
V
SE(r, 0)
&H(r, 0 )
E( r, t), H(r, t ) E( r), H(r)Et (r, t) orHt (r, t)
Et (r) orHt (r)
) sin() π10sin(2),( zktxt zy erE
Find the average of the energy flow density vector.
Solution: we have found the electric and magnetic field
intensities as
Example. The instantaneous value of the electric field
intensity of a time-variable electromagnetic field in vacuum is
From Sav = Re (Sc) we find
*c HES ) π20sin(
2
π10j) π10(sin
0
2
0
xxk
xz
z ee
Then
) π10(sin
2
0 av x
kzz
eS
zkz
zx
zxxk
- j
0 0
e) π10cos(
π10j) π10sin(
)(
eerH
zky
zx je) π10sin()( erE