Transcript Document 7207265

Fundamentals of Electromagnetics
for Teaching and Learning:
A Two-Week Intensive Course for Faculty in
Electrical-, Electronics-, Communication-, and
Computer- Related Engineering Departments in
Engineering Colleges in India
by
Nannapaneni Narayana Rao
Edward C. Jordan Professor Emeritus
of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign, USA
Distinguished Amrita Professor of Engineering
Amrita Vishwa Vidyapeetham, India
Program for Hyderabad Area and Andhra Pradesh Faculty
Sponsored by IEEE Hyderabad Section, IETE Hyderabad
Center, and Vasavi College of Engineering
IETE Conference Hall, Osmania University Campus
Hyderabad, Andhra Pradesh
June 3 – June 11, 2009
Workshop for Master Trainer Faculty Sponsored by
IUCEE (Indo-US Coalition for Engineering Education)
Infosys Campus, Mysore, Karnataka
June 22 – July 3, 2009
5-2
Module 5
Materials and Wave
Propagation in Material Media
5.1 Conductors and dielectrics
5.2 Magnetic materials
5.3 Wave equation and solution
5.4 Uniform waves in dielectrics and conductors
5.5 Boundary conditions
5.6. Reflection and transmission of uniform plane waves
5-3
Instructional Objectives
31. Find the charge densities on the surfaces of infinite plane
conducting slabs (with zero or nonzero net surface charge
densities) placed parallel to infinite plane sheets of charge
32. Find the displacement flux density, electric field intensity,
and the polarization vector in a dielectric material in the
presence of a specified charge distribution, for simple
cases involving symmetry
33. Find the magnetic field intensity, magnetic flux density,
and the magnetization vector in a magnetic material in the
presence of a specified current distribution, for simple
cases involving symmetry
5-4
Instructional Objectives (Continued)
34. Determine if the polarization of a specified
electric/magnetic field in an anisotropic
dielectric/magnetic material of permittivity/permeability
matrix represents a characteristic polarization
corresponding to the material
35. Write expressions for the electric and magnetic fields of a
uniform plane wave propagating away from an infinite
plane sheet of a specified sinusoidal current density, in a
material medium
36. Find the material parameters from the propagation
parameters of a sinusoidal uniform plane wave in a
material medium
37. Find the power flow, power dissipation, and the electric
and magnetic stored energies associated with electric and
magnetic fields in a material medium
5-5
Instructional Objectives (Continued)
38. Determine whether a lossy material with a given set of
material parameters is an imperfect dielectric or good
conductor for a specified frequency
39. Find the charge and current densities on a perfect
conductor surface by applying the boundary conditions
for the electric and magnetic fields on the surface
40. Find the electric and magnetic fields at points on one side
of a dielectric-dielectric interface, given the electric and
magnetic fields at points on the other side of the interface
41. Find the reflected and transmitted wave fields for a given
field of a uniform plane wave incident normally on a
plane interface between two material media
5-6
5.1 Conductors
and Dielectrics
(EEE, Secs. 4.1, 4.2; FEME, Sec. 5.1)
5-7
Materials
Materials contain charged particles that under the
application of external fields respond giving rise
to three basic phenomena known as conduction,
polarization, and magnetization. While these
phenomena occur on the atomic or “microscopic”
scale, it is sufficient for our purpose to
characterize the material based on “macroscopic”
scale observations, that is, observations averaged
over volumes large compared with atomic
dimensions.
8
5-8
Material Media can be classified as
(1) Conductors
and Semiconductors
electric property
(2) Dielectrics
(3) Magnetic materials – magnetic property
Conductors and Semiconductors
Conductors are based upon the property of
conduction, the phenomenon of drift of free
electrons in the material with an average drift
velocity proportional to the applied electric field.
5-9
electron
cloud
free electrons
+ bound
elecrons
nucleus
In semiconductors, conduction occurs not only by
electrons but also by holes – vacancies created by
detachment of electrons due to breaking of
covalent bonds with other atoms.
The conduction current density is given by
J c  E
Ohm’s Law
at a point
5-10
  conductivity (S/m)
e Ne e
 
h N h e   e N e e
conductors
semiconductors
  Mobility
Nh,e = Density of holes ( h) or electrons ( e)
The effect of conduction is taken into account
explicitly by using J = Jc on the right side of
Maxwell’s curl equation for H.
5-11
Ohm’s Law
V  El
Jc = E =
A
V
l
A
I  Jc A 
V
l
l
VI
A
V  IR Ohm’s Law
l
Resistance
R=
A
l

E, Jc
I
V
5-12
D4.1
I
0.1
Jc   –4  10 3 A m 2
A 10
(a) For cu,
  5.8  10 7 S m
103
E

 17.24 V m
7
 5.8  10
(b)    h  e  Ne e
Jc
 1700  3600 104  2.5 1013 106
1.602 1019
 2.1227 S m
103
E

 471.1 V m
 2.1227
Jc
5-13
l
(c) From R 
1
A
l
1
10 6



Sm
–6
RA   10

Jc
10 3
E
 6
 3.14 mV m
 10 
5-14
Conductor in a static electric field
E
E
5-15
Plane conducting slab in a
uniform electric field
–
–
–
–
– rS = e0E0
E0az
z=d
z=d
z=0
z=d
z=0
rS0
z=0
+
+
+
+
+ rS = e0E0
–
–
–
–
– rS = –e0E0
+
–
+
+
E =0
–
–
+
–
+
–
+
+
+
+ rS = e0E0
+
rS = e0E0
rS = –e0E0
+
–
+
–
+
–
+
–
+
–
E = – rS0 az
e0
–rS0
rS 0
–
a z  E0 a z  0
e0
rS 0  e 0E0
5-16
P4.3
(a) rS 0  rS1  rS 2
r S1
Ei  0
z
rS 2
rS1
rS 2
Ei  –
az 
az  0
2e 0
2e 0
 r S1  r S 2
1
 rS 0
2
5-17
r S11
(b)
Ei1  0
r S12
r S 21
Ei 2  0
r S 22
rS11  rS12  rS1
rS 21  rS 22  rS 2
(1)
(2)
Write two more equations and solve for the four
unknowns.
5-18
1
 r S11  r S12  r S 21  r S 22   0 (3)
2e 0
1
 r S11  r S12  r S 21  r S 22   0 (4)
2e 0
Solving the four equations, we obtain
1
r S11  r S 22   r S1  r S 2 
2
1
r S12   r S 21   r S1  r S 2 
2
5-19
Dielectrics
are based upon the property of polarization, which is the
phenomenon of the creation of electric dipoles within the
material.
Electronic polarization: (bound electrons are displaced to
form a dipole)
Q
E
+

+
d
Q

Dipole moment
p = Qd
5-20
Orientational polarization: (Already existing dipoles are
acted upon by a torque)
QE
Torque  QEd sin q
+
d
q
q

E
Direction into the paper.
 T  Qd × E
 p×E
QE
Ionic polarization: (separation of positive and negative ions
in molecules)
5-21
The phenomenon of polarization results in a polarization
charge in the material which produces a secondary E.
Applied
Field, E a
+
Total Field
+
E  Ea  Es
Dielectric
Secondary Field, E s
Polarization
11
5-22
Plane dielectric slab in a
uniform electric field
rS0
z=d







Ea
e  e 0
z=0
+
+
+
+
rS0
z=d
z=0
+

+

+
+

+

+

+

+

+

+

+

+

+

+

+

+
r pS  r pS 0
+
+

+
ES

r pS   r pS 0

+


+

+


+
+

+

t

+
+

E+

+
+
5-23
Polarization Current (in the time-varying case)
5-24
5-25
5-26
5-27
The effect of polarization needs to be taken into account by
adding the contributions from the polarization charges and
the polarization current to the right sides of Maxwell’s
equations.
For free space,
D  e 0E
 D = r   e 0E  r
D

×H J
  × H  J   e 0E 
t
t
5-28
For dielectrics,
 e 0E  r  r p  r   P

 × H  J  J p   e 0E 
t
P 
J
 e 0E 
t t
Where P is the polarization vector, or the dipole
moment per unit volume. Rearranging,
 e 0E  P    D  r

D
 × H  J  e 0E + P   J 
t
t
where now, D  e 0E  P
5-29
Thus, to take into account the effect of polarization, we
define the displacement flux density vector, D, as
D  e0E  P
= e 0 E  e 0 e E
= e 0 1  e  E
= e0 er E
= e E C m2
e = permittivity, F m
e r = relative permittivity
e r and e vary with the material, implicitly taking into account
the effect of polarization.
5-30
As an example, consider
 rS 0







z=d
e
+
+
+
+
rS 0
z
+
Then, inside the material,
+
+
z=0
rS 0
 rS 0
E
az 
 az 
2e
2e
rS 0

az
e
D  e E  rS 0 a z
5-31
D4.3
1 C m2







z=d
z
e  4e 0
+
+
+
+
+
+
1 C m2
For 0 < z < d,
6
2
D

r
a

10
a
C
m
(a)
S0 z
z
+
z=0
5-32
1
6
(b) E 

10
az 

e 4e 0
D
36
6


10
az
9
4  10
 9000 az V m
(c) P  D  e E
0
= 106 az  0.25  106 az
6
 0.75  10 az C m
2
5-33
Isotropic Dielectrics:
D is parallel to E for all E.
y
Dx  e Ex
D
Dy  e Ey
Dz  e Ez
D  eE
E
x
Anisotropic Dielectrics:
D is not parallel to E in general. Only for certain directions
(or polarizations) of E is D parallel to E. These are known
as characteristic polarizations.
5-34
Dx  e xx Ex  e xy Ey  e xz Ez
Dy  e yx Ex  e yy Ey  e yz Ez
Dz  e zx Ex  e zy Ey  e zz Ez
y
E
D
x
 Dx  e xx
 D   e
 y   yx
 Dz  e zx

e xy e xz   Ex 
 
e yy e yz   Ey 
e zy e zz   Ez 
5-35
D4.4
8 2 0 
e    2 5 0 
 0 0 9 
(a) E  E0az
 Dx 
8 2 0   0 
 0 
 
2 5 0  0   e  0 
D

e
0
0
 y

 
 
 Dz 
0 0 9   E0 
9e0 
D  9e0 E0az  9e0E
e eff  9e 0 , e reff  9
5-36

(b) E  E0 a x  2a y

 Dx 
8 2 0  E0 
 4 E0 
 
 2 5 0  2 E   e  8E 
D

e
0
0
0
0
 y





 Dz 
0 0 9   0 
 0 
D  4e 0 E0  ax  2a y   4e 0 E
e eff  4e 0 , e reff  4
5-37

(c) E  E0 2ax  a y

 Dx 
8 2 0   2 E0 
18E0 
 
2 5 0  E   e  9E 
D

e
0
0
 y

 0 
 0
 Dz 
0 0 4  0 
 0 
D  9e 0 E0  2ax  a y   9e 0 E
e eff  9e 0 , e reff  9
5-38
Review Questions
5.1. Distinguish between bound electrons and free electrons
in an atom.
5.2. Briefly describe the phenomenon of conduction.
5.3. State Ohm’ law valid at a point, defining conductivity.
How is conduction current taken into account in
Maxwell’s equations?
5.4. Discuss the formation of surface charge at the boundaries
of a conductor placed in a static electric field.
5.5. Briefly describe the phenomenon of polarization in a
dielectric material. What are the different kinds of
polarization?
5.6. What is an electric dipole? How is its strength defined?
5-39
Review Questions (Continued)
5.7. Discuss the effect of polarization in a dielectric material,
involving polarization charge and polarization current.
5.8. What is the polarization vector? How is it related to the
electric field intensity?
5.9. Discuss how the effect of polarization in a dielectric
material is taken into account in Maxwell’s equations.
5.10. Discuss the revised definition of the displacement flux
density and the permittivity concept.
5.11. What is an anisotropic dielectric material? When can an
effective permittivity be defined for an anisotropic
dielectric material?
5-40
Problem S5.1. Finding the electric field due to a point
charge in the presence of a conductor
5-41
Problem S5.1. Finding the electric field due to a point
charge in the presence of a conductor (Continued)
5-42
Problem S5.2. Finding D, E, and P for a line charge
surrounded by a cylindrical shell of dielectric material
5-43
Problem S5.3. Expressing an electric field in terms of the
characteristic polarizations of an anisotropic dielectric
5.2 Magnetic Materials
(EEE, Sec. 4.3; FEME, Sec. 5.2)
5-45
Magnetic Materials
are based upon the property of magnetization, which is the
phenomenon of creation of magnetic dipoles within the
material.
Diamagnetism:
A net dipole moment is induced by changing the angular
velocities of the electronic orbits.
I
e
+
A
I
Dipole moment
m = IA an
5-46
Paramagnetism:
Already existing dipoles are acted upon by a torque.
I dl × B
I
B
I
I dl × B
Other: Ferromagnetism, antiferromagnetism,
ferrimagnetism
5-47
The phenomenon of magnetization results in a magnetization
current in the material which produces a secondary B.
Applied
Field, Ba
+
+
Total Field
B  Ba  Bs
Magnetic Material
Secondary Field, Bs
Magnetization
5-48
 J S 0a y
z=d
Ba
m  m 0
z=0
J S 0a y
J mS   J mS 0a y
z=d
Bt
Bs
z=0
JmS  JmS 0a y
Magnetization Current
5-49
5-50
5-51
5-52
The effect of magnetization needs to be taken into account
by adding the contributions from the magnetization current
to the right sides of Maxwell’s equations.
For nonmagnetic materials,
H
B
0
D
B
D
×H=J+
 ×
J+
t
0
t
5-53
For magnetic materials,
D
×
 J + Jm 
0
t
B
D
 J + Jm 
t
D
 J + ×M 
t
where M is the magnetization vector, or the magnetic dipole
moment per unit volume.
5-54
Rearranging,
B

D
 ×  M   J +
t
 0

D
×H =J+
t
where now
H=
B
0
M
5-55
Thus, to take into account the effect of magnetization, we
define the magnetic field intensity vector, H, as
H
B
0
M
m B


 0 1  m  0
B
B

 01  m 


B
 0 r
B

  permeability, H m
r  relative permeability
A m
r and  vary with the material, implicitly taking into account
the effect of magnetization.
As an example, consider
5-56
 JS 0a y
z
y
Then inside the material,
B=

2
J S 0a y
J S 0a y × a z 
  J S 0ax
B
H   J S 0ax

×x

J

2
a  ×  az 
S0 y
5-57
D4.6
 0.1 ay
z= d
z
  100  0
y
z=0
0.1 a y
For 0 < z < d,
(a) H  0.1 a y × az  0.1 ax A m
×x
5-58
(b) B =  H = 100  0 0.1 ax 
 10  0ax Wb m
2
 4  106 ax Wb m2
(c) M 
B
0
H
= 10 ax  0.1 ax
 9.9 ax A m
5-59
Materials and Constitutive Relations
Summarizing,
J c   E Conductors
D  eE
B
H

Dielectrics
Magnetic materials
E and B are the fundamental field vectors.
D and H are mixed vectors taking into account the
dielectric and magnetic properties of the material
implicity through e and , respectively.

5-60
Review Questions
5.12. Briefly describe the phenomenon of magnetization in a
material. What are the different kinds magnetization?
5.13. What is a magnetic dipole? How is its strength defined?
5.14. Discuss the effect of magnetization in a magnetic
material, involving magnetization current.
5.15. What is the magnetization vector? How is it related to
the magnetic flux density?
5.16. Discuss how the effect of magnetization in a magnetic
material is taken into account in Maxwell’s equations.
5.17. Discuss the revised definition of the magnetic field
intensity and the permeability concept.
5.18. Summarize the constitutive relations for a material
medium.
5-61
Problem S5.4. Finding H, B, and M for a wire of current
surrounded by a cylindrical shell of magnetic material
5.3 Wave Equation and Solution
(EEE, Sec. 4.4; FEME, Sec. 5.3)
5-63
Maxwell’s equations for a material medium
B
H
×E= 
 
t
t
D
D
×H=J+
 Jc +
t
t
E
Ee
t
 D=r
 D=0
5-64
Waves in Material Media
E  Ex  z, t  ax , H = H y  z, t  ay
H y  z , t 
Ex  z , t 
 
z
t
H y  z , t 
Ex  z , t 
  Ex  z , t   e
z
t
Ex
  j H y
z
H y
  Ex  je Ex     je  Ex
z
5-65
Combining, we get
 Ex
 j   je  Ex
2
z
2
Define
    j 
j   je 
Then
2 Ex
2


Ex
2
z
Wave equation
5-66
Solution:
Ex  z   Ae z  Be z
jt

Ex  z , t   Re  Ex  z  e 
 z
z
jt

 Re  Ae  Be  e 


j
q


z

j

z
j

t
j
q
 z j  z jt 

 Re Ae e e e  Be e e e


 Ae z cos t   z  q  
z
 Be
cos t   z  q


5-67
Ae z cos t   z  q  

attenuation    wave
B e z cos t   z  q  

attenuation    wave
 = attenuation constant, Np/m
 = phase constant, rad/m
 = propagation constant, m1
5-68
f  z, t   e
 z
cos t   z 
f
1
0


t 
4
t 0
2

1
t 
2
z
5-69
g  z, t   e z cos t   z 
g
t 
2
t 
4
1
t 0
z
2


0

1
5-70
Ex
  j H y
z
1 Ex
Hy  
j z
1 
 Ae z  Be  z 

j z
1
  Ae z  Be  z 

j
where  
 intrinsic impedance of the medium.
  je
5-71
Summarizing,
    j 
j   je 
j
  e 
  je
j
Conversely,
1


j

  Re


e  Im


1
5-72
2

 e 



 1 

  1
2 

 e 



vp 
 e 
2



12

 

 1 
  1
e 





2
12
2

2 
 

 1 
  1
e 

 e 


1 2
Characteristics of Wave Propagation
The quantity  characterizes attenuation, which is a function
of frequency. The quantity vp characterizes phase velocity,
which is a function of frequency, giving rise to dispersion. The
 in phase difference between the
complex nature of results
electric and magnetic fields.
5-73
 ,e , 
 ,e , 
x
y
JS
z 0
For J   JS 0 cos t ax , z  0
E  z, t  
 JS 0
e
2
JS 0
H  z, t   
e
2
z
z
cos t  z    for z >< 0
cos t  z  for z >< 0
z
5-74
Three-dimensional depiction of wave propagation
5-75
E5.1
For dry earth,   105 S/m, e  5e0 , and   0 .
Let us compute  ,  , vp ,  , and  for f  100 kHz.
Solution:
 

j   je 



j je 1 

j
e


 j e 1  j

2 f e
5
2


10
 5 1  j 0.36
 j
3 108
5-76
 j 0.004683 1.0628 19.8
 j 0.004683 1.0309  9.9
 j 0.004683 1.0155  j0.1772 
 0.00083  j 0.004756
  0.00083 Np/m
  0.004756 rad/m
5-77
5

2


10
8
vp  
 1.32110 m/s
 0.004756
2
  2 
 1321.05 m
 0.004756
j

  je

j
je
1
1   je
5-78


e
1  j  je
 120
5
1
1  j 0.36
 168.6
1
1
1.0309  9.9
 163.559.9
 161.1  j 28.1 
5-79
Power Flow and Energy Storage

 E×H   H  ×E  E  ×H 
A Vector Identity
D
B
  E× H   E J  E
H
t
t
For J = J 0  J c  J 0   E,
 1 2  1
2
E J0   E   e E     H   
t  2
 t  2

2
P  E× H
Poynting Vector
 E× H 
5-80
Poynting’s Theorem for Material Medium
1 eE 2  dv   1 H 2  dv  P  dS
  E  J 0  dv   E 2 dv    



 t 2
 t  2
V
V
V
Power dissipation
density
Source
power density
Electric stored
energy density
V
Magnetic stored
energy density
S
5-81
Interpretation of Poynting’s Theorem
Poynting’s Theorem for the material medium says that the power
delivered to the volume V by the current source J0 is accounted for
by the power dissipated in the volume due to the conduction
current in the medium, plus the time rates of increase of the
energies stored in the electric and magnetic fields, plus another
term, which we must interpret as the power carried by the
electromagnetic field out of the volume V, for conservation of
energy to be satisfied.
5-82
In the case of the infinite plane sheet of current, work is
done by an external agent (source) for the current to flow, and
 E J0  represents the power density (per unit volume)
associated with this work.


P  E × H  Power flow density W m2 associated with the
electromagnetic field
pd   E 2  Power dissipation density due to conduction current
flow in the material


1 2
we  e E  Energy density J m3 stored in the electric field
2
wm 


1
 H 2  Energy density J m3 stored in the magnetic field
2
5-83
Review Questions
5.19. Summarize Maxwell’s equations for a material medium.
5.20. What is the propagation constant in a material medium?
Discuss the significance of its real and imaginary parts.
5.21. What is the intrinsic impedance of a material medium?
Discuss the significance of its complex nature.
5.22. Discuss the consequence of the frequency dependence
of the phase velocity in a material medium.
5.23. Discuss the solution for the electromagnetic field due to
to an infinite plane current sheet of sinusoidally timevarying current density embedded in a material medium.
5.24. How would you obtain the electromagnetic field due
to a current sheet of nonsinusoidally time-varying
current density embedded in a material medium?
5-84
Review Questions (Continued)
5.25. State and discuss Poynting’s theorem for a material
medium.
5.26. What are the power dissipation density, the electric
stored energy density, and the magnetic stored energy
density associated with an electromagnetic field in a
material medium?
5-85
Problem S5.5. Finding the magnetic field and material
parameters from a specified electric field
5-86
Problem S5.6. Finding E and H for a current sheet of
nonsinusoidal current density in a material medium
5-87
Problem S5.7. For showing that the time average powers
delivered to, and dissipated in, a medium are the same
5.4 Uniform Plane Waves
in Dielectrics and Conductors
(EEE, Sec. 4.5; FEME, Sec. 5.4)
5-89
Special Cases:
Case 1. Perfect dielectric   0
  j je  j e
  0
no attenuation
   e

j


, purely real
je
e
Behavior same as in free space except that e0  e
and 0  .
5-90
For materials with nonzero conductivity, there are two special
cases, depending on the relationship between  and e . The
quantity /e is known as the loss tangent.
From
D
 × HJ
t
E
Ee
,
t
it can be seen that /e is the ratio of the amplitudes of the
conduction current density and the displacement current
density in the material.
5-91
Case 2. Imperfect Dielectric   0 but 
e 
j   je 
 

2

 j e
e
 
 

1

j


e 
2e 
Behavior essentially like in a perfect dielectric except
for attenuation.
5-92
Case 3. Good Conductor 
 
e 
j   je 
  f  1  j 
     f 
j
 f


1  j 
  je


2 f 

45
Behavior much different from that in a dielectric.
5-93
Characteristics of wave propagation in a good conductor
     f  
f
Skin effect: Concentration of fields near the skin of
the conductor.
Skin depth, : Distance in which fields are attenuated
by the factor e1, that is,  = 1.
1
1
 

 f 
Ex: For copper,   5.80 107 S m,   0.066
f m.
5-94
Even at a low frequency of 1 MHz,  = 0.066 mm. This
explains the phenomenon of shielding by good conductors,
such as copper, aluminum, etc.
2
2




 f 
4
f 
4 f 
 conductor
4 f e


 dielectric

1 f e

2e

1
5-95
  conductor
dielectric . Furthermore, the lower the frequency,
the smaller is the ratio. Coupled with the fact that  
this property makes low frequencies more suitable for
communication with underwater objects.
f,
Ex: For sea water,   4 S m, e  80e0 ,   0 . The value
of the ratio of the two wavelengths for f = 25 kHz is 1/134.

2 f 

45
5-96
For copper,   3.69 107 f . Even at a frequency of
1012 Hz, this is equal to 0.369 , a very low value. In fact, if
we note that
 
2 f 

e



e
dielectric . Hence, for the same
we see that  conductor
electric field, the magnetic field inside a good conductor is
much larger than that inside a dielectric.
5-97
Case 4. Perfect Conductor    
Idealization of good conductor in the limit
that   .
  ,   0
No waves can penetrate into a perfect conductor.
No time-varying fields inside a perfect conductor.
5-98
Review Questions
5.27. What is loss tangent? Discuss its significance.
5.28. What is the condition for a material to be a perfect
dielectric? How do the characteristics of wave
propagation in a perfect dielectric medium differ from
those of wave propagation in free space?
5.29. What is the condition for a material to be an imperfect
dielectric? What is the significant feature of wave
propagation in an imperfect dielectric as compared to
that in a perfect dielectric?
5.30. What is the condition for a material to be a good
conductor? Give two examples of materials that behave
as good conductors for frequencies up to several
gigahertz.
5.31. What is skin effect? Discuss skin depth, giving some
numerical values.
5-99
Review Questions (Continued)
5.32. Why are low-frequency waves more suitable than highfrequency waves for communication with underwater
objects?
5.33. What is the consequence of the low intrinsic impedance
of a good conductor as compared to that of a dielectric
medium having the same ε and μ.
5.34. Why can there be no time-varying fields inside a perfect
conductor?
5-100
Problem S5.8. Plotting field variations for an infinite plane
sheet current source in a perfect dielectric medium
5-101
Problem S5.9. Calculating parameters for good conductor
materials to satisfy specified conditions
5.5 Boundary Conditions
(EEE, Sec. 4.6; FEME, Sec. 5.5)
5-103
Why boundary conditions?
Medium
1
Inc.
wave
Ref.
wave
Medium
2
Trans.
wave
5-104
Maxwell’s equations in integral form must be satisfied
regardless of where the contours, surfaces, and volumes are.
Example:
C3
C1
Medium 1
C2
Medium 2
5-105
Boundary Conditions
Jn1
JS
Ht 2
Ht 1
an
Jn2
Medium 1, z > 0
e
Bn1
Dn1
rs
Et1
Bn2 Dn2
Et2
z 0
z
Medium 2, z < 0
 e
x
y

5-106
Example of derivation of boundary conditions
d
C E d l   dt S B d S
Medium 1
an
Lim
ad 0
bc 0

abcda
as
a
b
d
c
E dl  
Lim
ad 0
bc 0
Medium 2
d
area B d S

dt abcd
5-107
Eab  ab   Ecd  cd   0
Eab  Edc  0
aab
a s × an
 E1  E2   0
 E1  E2   0
as an ×  E1  E2   0
an ×  E1  E2   0
or,
Et1  Et 2  0
5-108
Summary of boundary conditions
an ×  E1  E2   0
or
Et1  Et 2  0
an ×  H1  H2   JS or
Ht1  Ht 2  JS
 D1  D2   rS
or
Dn1  Dn 2  rS
 B1  B2   0
or
Bn1  Bn 2  0
an
an
5-109
Perfect Conductor Surface
(No time-varying fields inside a perfect conductor. Also
no static electric field; may be a static magnetic field.)
Assuming both E and H to be zero inside, on the surface,
an ×E = 0
or
Et  0
an × H = JS or
Ht  JS
an D  rS
or
Dn  rS
an B  0
or
Bn  0
5-110
an
E


E
 
an
JS
H
 
H
JS
5-111
Dielectric-Dielectric Interface
rS  0, JS  0
an ×  E1  E2   0
or
Et1  Et 2
an ×  H1  H2   0
or
Ht1  Et 2
an
 D1  D2   0
or
Dn1  Dn 2
an
 B1  B2   0
or
Bn1  Bn 2
5-112
an
Mediume0
Dn1
En1
Dn2
En2
Bn1
Hn1
Bn2
Hn2
Et1
Et2
Mediume0
an
Medium0
Ht1
Ht2
Medium0
5-113
D4.11 At a point on a perfect conductor surface,


(a) D  D0 ax  2a y  2a z and pointing away from
the surface. Find rS . D0 is positive.
D
an 
D
rS  an
D
D
D=
D=
D
D
2
 D  D0 ax  2a y  2az  3D0
5-114


(b) D  D0 0.6 ax  0.8 a y and pointing toward the
surface. D0 is positive.
D
an  
D
rS  an
D
D
D=
D=
D
D
2
  D   D0 0.6 ax  0.8 a y
  D0
5-115
E5.2
z
E1  E0 az for r < a.
r>a
ee0
(0, 0, a)
(a) At  0, 0, a  ,
an  az
E1 is entirely normal.
 D2  D1  2 e0 E1
E2 
D2
e0
 2E1  2 E0 az
a a 

0,
,


2 2

(0, a, 0)
r<a
ee0
y
5-116
(b) At  0, a, 0  ,
an  a y
E1 is entirely tangential.
E2  E1  E0 az
a a 

,
(c) At  0,
,
2 2

1
an 
a y  az 

2
an ×  E2  E1   0 
 Solve.
an  D2  D1   0 
5-117
Review Questions
5.35. What is a boundary condition? How do boundary
conditions arise and how are they derived?
5.36. Summarize the boundary conditions for the general case
of a boundary between two arbitrary media, indicating
correspondingly the Maxwell’s equations in integral
form from which they are derived.
5.37. Discuss the boundary conditions on the surface of a
perfect conductor.
5.38. Discuss the boundary conditions at the interface
between two perfect dielectric media.
5-118
Problem S5.10. For application of boundary conditions
on a perfect conductor surface
5-119
Problem S5.11. Applying boundary conditions at a dielectric
interface in the presence of a point charge
5-120
Problem S5.11. Applying boundary conditions at a dielectric
interface in the presence of a point charge (Continued)
5-121
Problem S5.12. For application of boundary conditions on
a the surface of a magnetic material
5-122
5.6 Reflection and Transmission
of Uniform Plane Waves
(EEE, Sec. 4.7; FEME, Sec. 5.6)
5-123
5-124
5-125
5-126
5-127
5-128
5-129
Review Questions
5.35. Discuss the determination of the reflected and
transmitted wave fields of a uniform plane wave
incident normally onto a plane boundary between two
material media.
5.36. Define reflection and transmission coefficients for a
uniform plane wave incident normally onto a plane
boundary between two material media.
5.37. Discuss the reflection and transmission coefficients for
the special case of two perfect dielectric media.
5.38. What is the consequence of a wave incident on a perfect
conductor?
5-130
Problem S5.13. Normal incidence of uniform plane waves
on interface between free space and water
5-131
Problem S5.14. Eliminating reflection of uniform plane
waves from a dielectric slab between two media
5-132
Problem S5.14. Eliminating reflection of uniform plane
waves from a dielectric slab between two media (Continued)
The End