# Chapter 2 Maxwell's Equations and Plane EM Waves

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16 Νοε 2013 (πριν από 4 χρόνια και 5 μήνες)

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Chapter 2 Maxwell’s Equations and Plane EM Waves

2
-
1
Dielectric and Conductor

Displacement vector
:

E
E
E
P
E
D
r
e

0
0
0
1

Polarization vector
:

v
P
P
v
n
k
k
v

1
0
lim



'
2
0
'
4
1
v
R
dv
R
a
P
V

,

2
2
2
2
'
'
'
z
z
y
y
x
x
R

'
'
'
'
z
z
y
y
x
x

2
1
'
R
a
R
R



'
0
'
1
'
4
1
v
dv
R
P
V





'
'
0
'
'
'
'
4
1
v
v
dv
R
P
dv
R
P





'
'
0
'
'
'
'
4
1
v
s
n
dv
R
P
S
d
R
a
P



Surface charge density
:

ρ
ps
=

n
a
P
.

Volume charge density
:
ρ
p
=

P

Total charge
:
Q
=
0
'
'
'
'




v
s
n
dv
P
S
d
a
P

P
E
E
p
0
0
1

Define

P
E
D
0

Q
S
d
D
D
s



Note:

Generally,

E
D

or

z
y
x
z
y
x
E
E
E
D
D
D
33
32
31
23
22
21
13
12
11

.

For

biaxial dielectric,

33
22
11
0
0
0
0
0
0

Eg. For an anisotropic medium characterized by

z
y
x
z
y
x
E
E
E
D
D
D
9
0
0
0
5
2
0
2
8
0

,
find the value of the effective relative permittivity for (a)
0
ˆ
E
z
E

, (b)
)
ˆ
2
ˆ
(
0
y
x
E
E

, (c)
)
ˆ
ˆ
2
(
0
y
x
E
E

.

(Sol.) (a)
0
0
0
0
0
0
1
0
0
9
9
0
0
1
0
0
9
0
0
0
5
2
0
2
8
E
E
E
D
D
D
z
y
x

,

r

=
9

(b)
0
0
0
0
0
0
0
2
1
4
0
8
4
0
2
1
9
0
0
0
5
2
0
2
8
E
E
E
D
D
D
z
y
x

,
r

=
4

(c)

0
0
0
0
0
0
0
1
2
9
0
9
18
0
1
2
9
0
0
0
5
2
0
2
8
E
E
E
D
D
D
z
y
x

,
r

=
9

Hall Effect
:

Current density:
v
Nq
J
y
J

0
ˆ

If the m
aterial is a conductor or an
n
-
type semiconductor the cha
rge

carrier are
e
lect
r
ons
:

q

< 0

Hall field
:

0
0
0
0
ˆ
)
ˆ
(
)
ˆ
(
B
v
x
B
z
v
y
B
v
E
h

Hall voltage
:

d
B
v
dx
E
V
d
h
h
0
0
0

Hall coefficient
:
0
1

Nq
B
J
E
C
z
y
x
h

If the material is a
p
-
type semiconductor, the ch
a
rge

carries are
holes
:
q

> 0

Hall field
:

0
0
ˆ
B
v
x
E
h

Hall voltage
:

d
B
v
V
h
0
0

Hall coefficient:

0

h
C

2
-
2
Boundary Conditions

of
Electromagnetic

Fields

Boundary
c
onditions for
electric f
ields:

Eg. Show that
E
t
=
0

o
n the conductor plane
.

(Proof)

T
he
E
-
field inside

a conductor

is zero
,

0

W
E
l
d
E
t
0

t
E

0
0

s
n
s
n
E
S
S
E
S
d
E



Eg. Show that

E
1t
=

E
2t

and

s
n
D
D
a

2
1
2

on the interface between two
dielectric
.

(Proof)

0
2
1

W
E
W
E
l
d
E
t
t
abcda
,
E
1
t
=
E
2
t



s
s
n
n
n
S
S
D
D
a
S
a
D
a
D
S
d
D

2
1
2
1
2
2
1

s
n
D
D
a

2
1
2

or
D
1n
-
D
2n
=
ρ
s

If

ρ
s
=0
, then
D
1n
=
D
2n

o
r
ε
1
E
1n
=
ε
2
E
2n

Eg.

Two dielectric media are separated by a charge free boundary. The electric
field intensity in media 1 at the point
P
1

ha
s a magnitude
E
1

and makes an angle
with the normal. Determine the magnitude and direction of the electric field
intensity at point
P
2

in medium 2.

[

]

(Sol.)

1
1
2
2
s i n
s i n

E
E

,
1
1
1
2
2
2
cos
cos

E
E

1
2
1
2
tan
tan

2
2
2
2
2
2
2
2
2
2
2
cos
sin

E
E
E
E
E
n
t

2
/
1
2
1
1
2
1
2
1
1
cos
sin

E
E
2
/
1
2
1
2
1
1
2
1
cos
sin

E

Eg. Assume that
z
=0 plane separates two lossless dielectric regions with

ε
r1
=2 and
ε
r2
=3
. If

1
E

in region 1 is

z
z
x
y
y
x

5
3
2
, find

2
E

and

2
D

at
z
=0 in
region 2.

(Sol.)

5
3
2
1

z
x
y
y
x
E
,

x
y
y
x
z
E
z
E
t
t
3
2
0
0
2
1

,

0
3
0
2
0
0
2
1
2
1

z
E
z
E
z
D
z
D
n
n
n
n

3
10
5
3
2
0
2

z
z
z
E
n
,

3
10
3
2
0
2

z
x
y
y
x
z
E

0
2
3
3
10
3
2
0

z
x
y
y
x
z
D

Eg.

A lucite sheet

(
ε
r
=3.2)

is introduced perpendicularly in a unifo
rm electric
field

0
0
E
x
E

in free space. Determine

i
i
D
E
,

and

i
P

inside the lucite.

[

]

(Sol.)

0
0
0
E
x
D
x
D
x
D
i
i

2
.
3
1
1
0
0
E
x
D
D
E
i
r
i
i

)
/
(
6875
.
0
2
.
3
1
1
0
0
0
0
0
m
C
E
x
E
x
E
D
P
i
i
i

E
g.

Dielectric lenses can

be used to collimate electromagnetic fields.
T
he left
surface of the lens

is that of a circular cylinder, and right surface is a plane. If

1
E

at point
P
(
r
0
,
45°
,
z
)

in region 1 is
3
5

a
a
r
, what must be the dielectric
con
stant of the lens in order that

3
E

in region 3 is parallel to the
x
-
axis?

(Sol.)

Assume

2
2
2
E
a
E
a
E
r
r

,

3
2
2
1

E
E
E
E
t
t

For

3
E
axis
x

//

2
E
axis
x

//
3
2
2
2

r
r
E
E
E

2
1
D
a
D
a
n
n
2
2
1
1
r
r
E
E

,
3
5
3
5
2
2
0
0

r
r

Eg
.

A positiv
e point charge
Q

is at the center of a spherical dielectric shell

of an
R
i

R
o
. The dielectric constant of the shell is

ε
r
.
Determine

D
V
E
,
,
, and

P

as functions of the radial distance
R
. [

]

(Sol.)

E
E
D
P
r
1
0
0

R
>
R
o
:

2
0
4
ˆ
R
Q
a
E
R


,
R
Q
V
0
4


2
4
ˆ
R
Q
a
D
R

and
0

P

R
i
<
R
<
R
o
:

2
2
0
4
ˆ
4
ˆ
R
Q
a
R
Q
a
E
R
r
R




,

2
4
ˆ
R
Q
a
D
R

,

2
4
1
1
ˆ
R
Q
a
P
r
R

R
R
r
R
o
o
dR
R
Q
dR
R
Q
V
2
0
2
0
4
4




R
R
Q
r
o
r


1
1
1
1
4
0

R
<
R
i
:
2
0
4
ˆ
R
Q
a
E
R


,
2
4
ˆ
R
Q
a
D
R

,

0

P

,

R
R
i
i
dR
R
Q
R
R
V
V
2
0
4


R
R
R
Q
i
r
o
r
1
1
1
1
1
1
1
4
0



Boundary
c
onditions for
m
agnetic
f
ields:

Eg. Show that

μ
1
H
1n
=
μ
2
H
2n

and
J
H
H
a
n

)
(
ˆ
2
1
2
.

(Proof
)



0
S
d
B
0
2
1

S
B
S
B
n
n
,
B
1n
=
B
2n

μ
1
H
1n
=
μ
2
H
2n

I
l
H

d
w
J
w
H
w
H
l
H
sw
abcda

)
(
d
2
1

sw
t
t
J
H
H

2
1
J
H
H
a
n

)
(
ˆ
2
1
2

If
J
=0, then
H
1t
=
H
2t

Eg. Two magnetic media with permeabilities

μ
1

a
nd

μ
2

have a common boundary.
The magnetic field intensity in medium 1 at the point
P
1

has a magnitude
H
1

and
makes an angle
α
1

with the normal. Determine the magnitude and the direction
of the magnetic field intensity at point
P
2

in medium 2.

(Sol.)
1
2
1
2
1
1
2
2
1
1
1
2
2
2
tan
tan
sin
sin
cos
cos

H
H
H
H

)
tan
(
tan
1
1
2
1
2

2
1
2
1
2
1
1
2
1
2
2
2
2
2
2
2
2
2
2
2
)
cos
(
sin
)
cos
(
)
sin
(

H
H
H
H
H
H
n
t

Eg.

Consider a plane boundary (
y
=0) between air (region 1,

μ
r1
=1
) and iron
(region 2,
μ
r2
=5000
).

(a)
Assuming

y
x
B
ˆ
10
ˆ
5
.
0
1

(
mT
), find

2
B

and
the

angle
that

2
B

makes with the interface.

(b)
Assuming
y
x
B
ˆ
5
.
0
ˆ
10
2

(
mT
), find

1
B

and the angle that

1
B

makes with the normal to the interface.

(
Sol.
)

(a)
y
x
B
ˆ
10
ˆ
5
.
0
1

,
y
B
x
B
B
y
x
ˆ
ˆ
2
2
2

,
2500
5
.
0
5000
2
1
2
2

x
o
x
o
x
x
B
H
B
H

10
1
2

y
y
B
B

y
x
B
ˆ
10
ˆ
2500
2

,
250
500
tan
tan
1
1
1
1
2
2

y
x
B
B

(b)

y
x
B
ˆ
5
.
0
ˆ
10
2

,
y
B
x
B
B
y
x
ˆ
ˆ
1
1
1

,
2
2
2
1
1
1

x
x
x
x
B
H
B
H

002
.
0
5000
10
1
2
1

x
BE
x
B
B

,
5
.
0
2
1

y
y
B
B
,

y
x
B
ˆ
5
.
0
ˆ
002
.
0
1

,

004
.
0
5
.
0
002
.
0
tan
1
1
1

y
x
B
B

Magnetic flux lines round a cylindrical bar magnet
:

Eg
.
A very large slab of material of thickness
d

lies perpendicularly to uniform
magnetic field intensity
0
0
ˆ
H
z
H

. Ignoring edge
effect, determine the magnetic
field intensity in the slab:

(a) if the slab material has a permeability

μ
, (b) if the
slab permanent magnet having a magnetization vector
i
i
M
z
M
ˆ

.
[

]

(Sol.)

(a)
2
2
2
H
B

,
o
o
z
z
H
H
B
B

2
1
2
,
o
o
H
z
H
z
H

ˆ
ˆ
2
2

(b)
)
(
2
2
i
o
M
H
B

,
z
z
B
B
1
2

)
(
ˆ
)
(
2
2
i
o
o
o
i
o
M
H
z
H
H
M
H

Eg. Assume that
N

turns of wire are wound around a toroidal core of a
ferromagnetic material with permeabilit
y

μ
.
The core has a mean radius
r
0
, a
a

(
a

<<

r
0
), and a narrow air gap of length
l
g
, as
shown in Figure. A steady current
I
0

flows in the wire. Determine (a) the
magnetic flux density
B
f

in the ferromagnetic core; (b) the magnetic field
intensity
H
f

in the cor
e; and, (c) the magnetic field intensity
H
g

in the gap. [

]

(Sol.)

o
C
NI
l
H

d
,
f
g
f
B
a
B
B

ˆ

,
o
g
o
f
g
o
f
NI
l
B
l
r
B

)
2
(

g
g
o
o
o
o
f
g
g
o
o
o
o
f
l
l
r
NI
a
H
l
l
r
NI
B

)
2
(
ˆ
)
2
(

g
g
o
o
o
g
l
l
r
NI
a
H

)
2
(
ˆ

2
-
-
state Currents

Differential
c
urrent:

s
v
Nq
t
t
s
a
v
Nq
t
Q
I
n

ˆ

Current
d
ensity:

v
v
Nq
J

(
A
/
m
2
),


s
S
d
J
I

(
A
)

Let
E
E
v
J
E
v



,

mobility
:

ty
conductivi
:

h
h
e
e
u
u

electrons holes

Eg. An
emf

V

is applied across a parallel
-
plate capacitor of area
S
. The sp
ace
between the conducting plates is filled with two different lossy dielectrics of
thicknesses
d
1

and
d
2
, permittivitie
s
ε
1

and

ε
2
, and conductivities

σ
1

and

σ
2
,
respectively. Determine (a) the current density between the plates, (b) the electric
field intensities in both dielectrics. [

]

(Sol
.
)

2
1
1
2
2
1
2
2
1
1
2
2
1
1
2
1
d
d
V
d
d
V
S
I
J
I
S
d
S
d
I
R
R
V

2
2
1
1
2
2
1
1
,
E
E
J
d
E
d
E
V

,
2
1
1
2
2
1
d
d
V
E

,
2
1
1
2
1
2
d
d
V
E

Eg. Assume a rectangular conducting sheet of conductivity

σ
, width
a
, and height
b
. A potential difference is applied to the side edges. Find (a) the potential
distribution, (b) the current density everywhere
within the sheet. [

]

(Sol.)

(a)
V
(
x
)=
Cx
,
V
(
a
)=
C
a
=
V
0

V
(
x
)=
x
a
V
o

(b)
a
V
x
x
V
E
o
ˆ
)
(



a
V
x
E
J
o

ˆ

Equation of
c
ontinuity:

0





t
J
dv
J
dv
dt
d
dt
dQ
S
d
J
I
v
s

If
0
,

t
t
p
E
E
J

,
ρ
=
t
e

0

Eg. Lightning strikes a lossy dielectric sphere

ε
r
=1.2
,
σ
=10
S
/
m
m

at
time
t
=0, depositing uniformly in the sphere a total charge
1
mC
. Determine for
all
t

for

(a) the electric field intensity both inside and outside
the sphere, (b) the
current density in the sphere, (c) calculate the time it takes for the charge density
in the sphere to diminish to

1
%
of its initial value, (d) calculate the charge in the
electrostatic energy stored in the sphere as the charge density
diminished from
the initial value to
1
%
of the value. What happens to this energy? (e) determine
the electrostatic energy stored in the space outside the sphere. Does this energy
charge with time?

(Sol
.
)

3
3
0
239
.
0
3
4
m
C
b
Q

,
t
e

0

0
:
....
Re
10
5
.
7
ˆ
:
10
ˆ
4
ˆ
:
....
10
5
.
7
ˆ
3
ˆ
4
3
4
ˆ
:
0
10
42
.
7
10
6
2
2
0
0
10
42
.
7
9
0
2
3
10
10
J
b
R
a
E
J
b
R
b
m
V
R
Q
a
R
Q
a
E
b
R
m
V
e
R
a
e
R
a
R
R
a
E
b
R
a
t
R
i
i
R
R
t
R
t
R
R
i







J
dR
R
E
W
e
W
W
e
dv
E
W
d
s
n
t
e
c
v
i
i
t
i
i
t
3
0
2
2
0
0
0
4
2
0
2
'
2
12
0
10
5
.
4
4
2
10
01
.
0
2
10
88
.
4
100
01
.
0
'

Boundary
c
ond
i
tions f
or
c
urrent
d
ensit
ies
:

Governing Equations for Steady Current Density

Differential Form

Integral Form

0

J

s
s
d
J
0

0

J

c
d
J
0
1

2
1
2
1
2
1
0
0

t
t
n
n
J
J
J
J
J
J

E
g.

Two lossy dielectric media with permittivities and conductivities

(
ε
1
,
σ
1
)

and

(
ε
2
,
σ
2
)

are in contact. An electric field with a magnitude

1
E

is incident from
medium 1 upon the interface at an angle

α
1

an
d
measured

from the common
normal
, as in Fig
ure
.

(a)

Find the
magnitude

and direction of

2
E

in medium 2.

(b)

Find the surface charge density at the interface.

(
Sol.
)

(a)
t
t
E
E
2
1

=>
2
1
sin
sin
2
1

E
E

,
n
n
J
J
2
1

=>
2
2
1
1
cos
cos
2
1

E
E

=>

1
2
1
1
2
1
2
tan
tan
)
cos
(
sin
1
2
2
2
1

E
E

(b)
s
n
n
D
D

1
2

=>

s
E
E
n
n

1
1
2
2
,
ρ
s
=

cos
1
1
2
2
1
E

Eg. Two conducting media with conductivities

σ
1

and

σ
2

are separated by an
interface. The steady current density in medium 1 at point
P
1

has a magnitude
J
1

and makes an angle

α
1

with the normal. Determine the magnitude and direc
tion
of the current density at point
P

2

in Medium 2. [

]

(Sol
.
)

1
2
1
2
2
2
1
1
1
2
2
2
1
1
tan
tan
sin
sin
,
cos
cos

J
J
J
J

2
2
2
2
2
2
2
2
2
2
2
cos
sin

J
J
J
J
J
n
t

2
1
2
1
1
2
1
1
1
2
cos
sin

J
J

n
n
s
s
n
n
s
n
n
n
n
n
n
E
E
E
E
D
D
E
E
J
J
1
2
1
2
1
2
2
1
2
1
2
2
1
1
2
1
2
2
1
1
2
1

. If
n
n
s
D
E
1
1
1
1
2



.

2
-
4

Maxwell’s Equations

and Plane EM Waves

Note:

t
D

is equival
ent to a current density, called
the
displacement current density.

Eg.

A voltage source
V
0
sin
(
ω
t
)
, is connected across a parallel
-
plate capacitor
C
.
Find the displacement current in the capacitor.

(Sol.)
t
V
d
A
t
CV
dt
dv
C
i
C
C

cos
cos
0
0

t
d
V
D
E
D

sin
0

,
C
A
D
i
t
V
d
A
S
d
t
D
i



cos
0

Lorent
z

c
ondition
:

t
V
A



=0

)
(
)
(
)
(
)
(
2
2
2
2
2
2
t
V
A
J
t
A
A
t
A
t
V
J
A
A
t
A
V
t
J
A
t
D
J
B











If
Lorent
z

Condition holds,

we have

J
t
A
A



2
2
2





)
(
)
(
)
(
2
2
t
V
t
V
A
t
V
t
A
V
E
D



2
2
2
t
V
V

Effective

permittivity
:

E
j
E
j
j
E
j
E
t
E
J
H
C





)
(
'
'
'
'
'


j
j
C
. Similarly,
'
'
'

j

Loss

tangent:



'
'
'
tan
C

Eg.

A sinusoidal electric intensity of amplitude 250
V
/
m

and frequency 1
GHz

exists in a lossy dielectric medium that has a relative permittivity of

2.5 and loss
tangent of 0.001. Find the average power dissipated in the medium per cubic
meter.

(Sol.)

)
/
(
34
.
4
250
)
10
39
.
1
(
2
1
2
1
2
1
)
/
(
10
39
.
1
)
5
.
2
)(
36
10
)(
10
2
(
001
.
0
,
001
.
0
tan
3
2
4
2
4
9
9
0
m
W
E
JE
p
m
S
r



Maxwell’s Equations in the
s
ource
-
free
r
egions
:

t
H
E

,
t
E
H

,
0

E

,
0

H

Phasor

representations:

Eg.
)
(
ˆ

z
j
Ae
x
,
)
(
)
5
4
ˆ
5
3
ˆ
(

z
j
e
y
x
, etc.

Instantaneous representations:

Eg.

]
ˆ
Re[
)
cos(
ˆ
)
(
t
j
z
j
e
Ae
x
z
t
A
x

, etc.

In case

E

and
H

are proportional to
e
jωt
, we have
H
j
t
H
E



and
E
j
t
E
H



.

Eg.

Given that
)
10
6
sin(
)
15
cos(
2
ˆ
9
z
t
x
y
H

in air, find
E

and
β
.

(
Sol
.)
Phasor
:
z
j
e
x
y
H

)
15
cos(
2
ˆ

,

2
.
13
400
)
15
(
2
0
0
2
2
2

z
j
e
x
j
z
x
x
H
j
E



)]
15
sin(
180
ˆ
)
15
cos(
158
ˆ
[
1
0

]
)
,
(
Re[
)
,
,
(
t
j
e
z
x
E
t
z
x
E

Eg.

Given that
)
10
6
sin(
)
10
cos(
1
.
0
ˆ
9
z
t
x
y
E

in air,
find
H

and
β
.

(
Sol.
)
Phasor
:

z
j
e
x
y
E

)
10
cos(
1
.
0
ˆ

,

3
10
400
)
10
(
2
0
0
2
2
2

z
j
e
x
z
x
x
j
E
j
H




)]
10
cos(
)
10
(
1
.
0
ˆ
)
10
cos(
1
.
0
ˆ
[
1
0
0

]
)
,
(
Re[
)
,
,
(
t
j
e
z
x
H
t
z
x
H

Eg.

T
he electric field intensity of a spherical wave in free space is
)
cos(
sin
ˆ
0
kR
t
R
E
a
E
R

.

Determi
ne the magnetic field intensity
.

(
Sol.
)
Phasor:

jkR
R
e
R
E
a
E

sin
ˆ
0

jkR
jkR
e
R
E
a
H
e
R
E
jk
a
RE
R
R
a
E
H
j



sin
ˆ
sin
)
(
ˆ
)
(
1
ˆ
0
0
0
0
0
0

Plane EM
w
aves
e
xcited by a
c
urrent
s
heet
:

Given
)
(
ˆ
)
(
t
J
x
t
J

at

z
=0, the field components of the EM plane wave excited by
the current density are
)
(
2
ˆ
)
,
(
p
v
z
t
J
x
t
z
E

and
)
(
2
1
ˆ
)
,
(
p
v
z
t
J
y
t
z
H

,
respectively.

If it is a s
inusoidal EM plane wave
,

)
cos(
ˆ
)
(
0
t
J
x
t
J

at

z
=0
.

We have

)
cos(
2
ˆ
)
,
(
0
kz
t
J
x
t
z
E

,
)
cos(
2
ˆ
)
,
(
0
kz
t
J
y
t
z
H

.

Electromagnetic
w
ave
s
pectrum
:

2
-
5
Plane EM
w
aves

in a simple,

nonconducting and source
-
free

region

I
n a
s
imple,

nonconducting and s
ource
-
free

r
egion
:

t
H
E

,
t
E
H

,
0

E

,
0

H

E
E
E
t
E
H
t
E

2
2
2
2
)
(
)
(




0
2
2
2

t
E
E


.

V
elocity

of the plane EM wave
:

v
=

1

In vacuum,

μ
0
=
4
π
×
10
-
7
,

ε
0
=

36
1
×
10
-
9
)
/
(
10
3
1
8
0
0
s
m
c

.

Wave
n
umber
:

k
=
ω
/
v
=



2
/
2

f
v

Assume

0
2
2

E
k
E
e
E
t
j

(drop
e
jωt

factor)

Suppose

jkz
jkz
e
E
e
E
z
E
E
k
dz
z
E
d
z
E
E

0
0
2
2
2
)
(
0
)
(
)
(

Traveling
w
ave in

+
z
-
direction
:

)
cos(
]
Re[
)
,
(
0
0
0
kz
t
E
e
e
E
t
z
E
t
j
jkz

Let

ωt
-
kz
=
constant

Phase
v
elocity
:

v
p
=
k
dt
dz

If
)
ˆ
ˆ
ˆ
(
),
(
ˆ

z
y
x
x
H
z
H
y
H
x
j
E
z
E
x
E


)
(
1
)
(
)
(
1
)
(
,
0
z
E
z
E
jk
j
z
H
H
H
x
x
y
z
x


,
w
here

η
=



k
,

a
nd

η
0
=120π

377Ω

in free space
.

TEM w
aves (
Transverse electromagnetic
w
aves)
:
E

and
H

direction of
propagation (
n
a
ˆ
)

R
a
jk
R
k
j
z
jk
y
jk
x
jk
n
z
y
x
e
E
e
E
e
E
z
y
x
E
R
E

ˆ
0
0
0
)
,
,
(
)
(
,

where

z
z
y
y
x
x
R
ˆ
ˆ
ˆ

,
k
a
k
n
ˆ

,
and



2
2
2
2

z
y
x
k
k
k

R
a
jk
n
R
a
jk
z
y
x
R
a
jk
R
a
jk
R
a
jk
R
a
jk
n
n
n
n
n
n
e
a
E
jk
e
k
z
k
y
k
x
j
E
e
E
e
E
E
e
e
E
E

ˆ
0
ˆ
0
ˆ
0
ˆ
0
0
ˆ
ˆ
0
)
ˆ
(
)
ˆ
ˆ
ˆ
(
)
(
)
(
)
(
0

n
n
a
E
E
a
ˆ
0
ˆ
0
0

(
TE
)
. Similarly,
n
a
H
H
ˆ
0
0

(
TM
)

Relation between E
-
field and H
-
field
of the
p
lane

EM
w
ave
:

)
(
ˆ
)
(
)
(
ˆ
)
(
1
)
(
1
)
(
R
H
a
R
E
R
H
a
jk
j
R
H
j
R
E
n
n



,

where

η
=



k

)
(
ˆ
1
)
(
1
)
(
R
E
a
R
E
j
R
H
n


n
n
a
H
R
E
a
R
H
ˆ
)
(
ˆ
1
)
(

Eg. The instantaneous expression fo
r the magnetic field intensity of a uniform
plane wave propagating in the +
y

direction in air is given by
)
4
10
cos(
10
4
ˆ
0
7
6

y
k
t
z
H

A/m
. (a)

Determine
k
0

and the location where

z
H

vanishes at
t
=3
ms
.

(b) Write the instantaneous expression

for
E

.

(S
ol.
)

30
10
3
10
10
8
7
0
7

c
k
,
y
a
n
ˆ
ˆ

(a)
]
2
)
1
2
cos[(

n
=0

)
4
1
10
3
(
30
2
1
2
4
30
10
3
10
4
3
7
n
y
n
y

(b)
)
,
(
ˆ
)
,
(
0
t
z
H
a
t
z
E
n

,

)
4
30
10
cos(
10
480
ˆ
)
,
(
7
6

y
t
x
t
z
E

Eg. A
100M
Hz

uniform plane wave
x
E
x
E
ˆ

propagate
s

in the +
z

direction.
Suppos
e

ε
r
=4
,

μ
r
=1
,
σ
=0
,
and it has a maximum value of
10
-
4
V
/
m

at
t

=0

and
z
=0.125
m
. (a)
Write the instantaneous expression
s

for
E

and
H

. (b)
Determine
the location
where

E

is a positive maximum when

t
=10
-
8
sec
.

(S
ol.
)

3
4
0
0

r
r
k
,
z
a
n
ˆ
ˆ

,

60
0
0

r
r

(a)
)
10
2
cos(
10
ˆ
ˆ
)
,
(
8
4

kz
t
x
E
x
t
z
E
x

has the
maximum i
n case of

0
10
2
8

kz
t
6

)
6
3
4
10
2
cos(
10
ˆ
)
,
(
8
4

z
t
x
t
z
E

,

)
6
3
4
10
2
cos(
60
10
ˆ
)
,
(
ˆ
1
)
,
(
8
4

z
t
y
t
z
E
a
t
z
H
n

(b)
1
)
2
cos(

n
,

2
3
8
13
2
6
3
4
)
10
(
10
2
max
max
8
8
n
z
n
z

Polarization

of
the EM
w
ave
:

T
he

direction

of electric field

of the EM wave
.

In the following text, we
assume
a
ll

EM
wave
s
to be
z
-
propagated

if we do not
spe
cify them
.

L
inear

polariz
ation
s

in the
x

and the
y
-
direction
,
respectively
:

)
(
ˆ

kz
j
x
e
E
x
E

,
)
(
ˆ

kz
j
y
e
E
y
E

L
inear

polariz
ation

in general case:

)
(
)
(
ˆ
ˆ

kz
j
y
kz
j
x
e
E
y
e
E
x
E

,
where
E
x

and
E
y

are
in
phase (we can assume the
both
to be
real
)
.

R
ight

hand circular

polariz
ation
:

)
(
0
)
(
0
ˆ
ˆ

kz
j
kz
j
e
jE
y
e
E
x
E

L
eft

hand circular

polariz
ation
:

)
(
0
)
(
0
ˆ
ˆ

kz
j
kz
j
e
jE
y
e
E
x
E

R
ight

hand elliptical

polariz
ation
:

)
(
20
)
(
10
ˆ
ˆ

kz
j
kz
j
e
jE
y
e
E
x
E

(
20
10
E
E

)

L
eft

hand elliptical

polariz
ation
:

)
(
20
)
(
10
ˆ
ˆ

kz
j
kz
j
e
jE
y
e
E
x
E

(
20
10
E
E

)

We

can
/transmit

line
arly
-
polariz
ed

EM waves by a linear
dipole antenna.

We

can
/transmit

circularly
-
polariz
ed

EM waves by a
circular reflector antenna.

I
nstantaneous
E
xpression for
E

of right

hand elliptical

polarization

(drop
phase factor
e
-

)
:

)
,
(
ˆ
)
,
(
ˆ
)
sin(
ˆ
)
cos(
ˆ
]
ˆ
ˆ
[
Re
)
,
(
2
1
20
10
20
10
t
z
E
y
t
z
E
x
kz
t
E
y
kz
t
E
x
e
e
jE
y
e
E
x
t
z
E
t
j
jkz
jkz

10
1
)
,
0
(
)
cos(
E
t
E
t

,
1
]
)
,
0
(
[
]
)
,
0
(
[
)
,
0
(
)
sin(
2
20
2
2
10
1
20
2

E
t
E
E
t
E
E
t
E
t

,
)
,
0
(
)
,
0
(
tan
1
2
1
t
E
t
E
t

1.
)
ˆ
ˆ
(
2
1
)
ˆ
ˆ
(
2
1
ˆ
y
x
y
x
x
jE
y
E
x
jE
y
E
x
E
x

:

A linearly polarized plane wave can be
resolved into a righ
t

hand and left

hand elliptically
-

or circularly
-
polarized waves
.

2.

)
2
ˆ
2
ˆ
(
)
2
ˆ
2
ˆ
(
ˆ
ˆ
1
0
1
0
0
1
1
0
.
0
0
E
E
j
y
E
E
x
E
E
j
y
E
E
x
jE
y
E
x

:

A circularly

polarized plane wave can be resolved into two
opposite

elliptically

polarized waves
.

3.
)
2
ˆ
2
ˆ
(
)
2
ˆ
2
ˆ
(
ˆ
ˆ
2
1
2
1
2
1
2
1
.
2
1
E
E
j
y
E
E
x
E
E
j
y
E
E
x
jE
y
E
x

:

A
n

elliptically

polarized plane wav
e can be resolved into two opposite
circularly

polarized

waves
.

Eg. The

E

field of a uniform plane wave propagating in a dielectric medium is
given by
)
3
10
sin(
ˆ
)
3
10
cos(
2
ˆ
)
,
(
8
8
z
t
y
z
t
x
z
t
E

V/m
.

(a)
Determine the
frequency and wavelength of the wav
e.

(b)
What is the dielectric constant of the
medium?

(c)
Describe the polarization of the wave.

(d)
Find the corresponding

H

field.

(Sol.
)

Phasor:
3
/
3
/
ˆ
2
ˆ
jz
jz
je
y
e
x
E

(a)
Hz
f
7
8
10
59
.
1
10

,

3
2
2
3
1

k
k

(b)
3
/
1
10
3
0
0
8

r
r
v

(c) It is the left

hand elliptically
-
polarized wave propagating along +
z

direction.

(d)
3
120
0
0

r
,
z
a
n
ˆ
ˆ

)
ˆ
2
ˆ
(
120
3
)
ˆ
2
ˆ
(
ˆ
1
ˆ
1
3
/
3
/
3
/
3
/
jz
jz
jz
jz
n
je
x
e
y
je
y
e
x
z
E
a
H

)]
3
10
cos(
ˆ
)
3
10
sin(
ˆ
[
120
3
]
)
(
Re[
)
,
(
8
8
z
t
y
z
t
x
e
z
H
t
z
H
t
j

Eg.
Write down the
instantaneous
expression for the

electric
-

and magnetic
-
field
intensities

of sinusoidal time
-
varying uniform plane wave propagating in free
space and having the following characteristics: (1)
f
=10GHz; (2) direction of
propagation is the +
z

direction; (3) left
-
hand circular polarization;
(4) the initial
condition is the electric field in the
z
=0 plane and
t
=0 having an
x
-
component
equal to
E
0
and a
y
-
component equal to

√3
E
0
.

[

]

(Sol.
)

10
10
2

,
2
8
10
3
2
10
3

k
c
k
v

Phasor:
)
(
)
(
ˆ
ˆ

kz
j
kz
j
jAe
y
Ae
x
E

for
the

left
-
hand cir
cular polarization

)
sin(
ˆ
)
cos(
ˆ
]
ˆ
ˆ
[
Re
)
,
(
)
(
)
(

kz
t
A
y
kz
t
A
x
e
jAe
y
Ae
x
t
z
E
t
j
kz
j
jkz

z
=0 and
t
=0
,
)
sin(
ˆ
)
cos(
ˆ
)
0
,
0
(

A
y
A
x
E

=
0
0
3
ˆ
ˆ
E
y
E
x

θ
=
tan
-
1
(
-

√3
)
,
A
=2
E
0

)]
3
(
tan
10
3
2
10
2
sin(
2
ˆ
)]
3
(
tan
10
3
2
10
2
cos[
2
ˆ
)
,
(
1
2
10
0
1
2
10
0

z
t
E
y
z
t
E
x
t
z
E

]
ˆ
ˆ
[
120
2
]
ˆ
ˆ
[
ˆ
1
ˆ
1
)
(
)
(
0
)
(
)
(
0
0

kz
j
kz
j
kz
j
kz
j
n
je
x
e
y
E
jAe
y
Ae
x
z
E
a
H

]
)
(
Re[
)
,
(
t
j
e
z
H
t
z
H

Application of

polarization:

Liquid Crystal Display
(
LCD
)

The polarizations of incident lights are synchronized by the rotations of molecules of
liquid crystal, which were
controlled

by an AC voltage.
And
then
the output polarizer
can block the orthogonally
-
polarized li
ghts

to control the
output

optical intensities
.

Poynting v
ector
:

H
E
P

t
B
E

,

t
D
J
H

J
E
t
D
E
t
B
H
H
E
E
H
H
E

)
(
)
(
)
(

2
2
2
)
2
1
(
)
2
1
(
)
(
)
(
E
E
t
H
t
J
E
t
E
E
t
H
H






s
v
v
v
dv
E
dv
H
E
t
dv
H
E
S
d
H
E
2
2
2
)
2
2
(
)
(
)
(

H
E
P

is the electromag
netic power flow per unit area
.

Instantaneous
p
ower
d
ensity:

]
)
(
Re[
]
)
(
Re[
)
,
(
t
j
t
j
e
z
H
e
z
E
t
z
P

Set
)
(
0
)
(
0
ˆ
)]
(
ˆ
[
1
)
(
ˆ
)
(
ˆ
)
(

z
j
z
n
z
j
x
e
e
E
y
z
E
a
z
H
e
E
x
z
E
x
z
E

,

)
cos(
ˆ
]
)
(
Re[
)
,
(
0
z
t
e
E
x
e
z
E
t
z
E
z
t
j

and
)
cos(
ˆ
]
)
(
Re[
)
,
(
0

z
t
e
E
y
e
z
H
t
z
H
z
t
j

]
)
(
Re[
]
)
(
Re[
)
,
(
)
,
(
)
,
(
t
j
t
j
e
z
H
e
z
E
t
z
H
t
z
E
t
z
P

)]
2
2
cos(
[cos
2
ˆ
2
2
0

z
t
e
E
z
z
2
0
E

Average
p
ower
d
en
sity:

)
Re(
2
1
*
H
E
P
av

T
z
av
e
E
z
dt
t
z
P
T
P
0
2
2
0
cos
2
ˆ
)
,
(
1

,

where
T

is the period
. A
nd it can be

prove
d that

)
Re(
2
1
*
H
E
P
av

.

Eg.

Show that
)
,
(
t
z
P

of a circularly

polarized plane wave propagating in a
lossless medium is a constant.

(Sol
.)
Assuming right

hand circularly

polarized p
lan
e wave,

z
a
n
ˆ
ˆ

)]
sin(
ˆ
)
cos(
ˆ
[
)
,
(
0
z
t
y
z
t
x
E
t
z
E

)]
cos(
ˆ
)
sin(
ˆ
[
)
ˆ
(
1
)
,
(
0
z
t
y
z
t
x
E
E
a
t
z
H
n

2
0
ˆ
)
,
(
)
,
(
)
,
(
E
z
t
z
H
t
z
E
t
z
P

Eg.

T
he radiation electric field intensity of an antenna system is

E
a
E
a
E
ˆ
ˆ

, find the ex
pression for the average outward power flow per
unit area.

(
Sol.
)

r
n
a
a
ˆ
ˆ

,
)
ˆ
ˆ
(
)
ˆ
(
1

E
a
E
a
E
a
H
n

)
(
ˆ
2
1
)]
ˆ
*
ˆ
(
)
ˆ
ˆ
Re[(
2
1
)
Re(
2
1
2
2
*
*

E
E
a
E
a
E
a
E
a
E
a
H
E
P
r
av

Eg.
Find
P

on the surface of a long, straight conducting wire of radius
b

and
conductivit
y
σ

t
hat c
arries a direct current
I
. Verify Poynting

s theorem.

(Sol.)
2
2
ˆ
ˆ
b
I
z
J
E
b
I
z
J


,
3
2
2
2
ˆ
2
ˆ
b
I
a
H
E
P
b
I
a
H
r


R
I
b
I
b
b
I
dS
a
P
S
d
P
s
r
s
2
2
2
2
2
2
)
(
2
2
ˆ







2
-
6

Plane
EM
Wave in a
L
ossy
M
edia

E
j
E
j
j
E
j
E
E
j
J
H
c







)
(
,
'
'
'

j
j
c

.

Similarly,

'
'
'

j
c

Complex
w
ave
n
umber
:

c
c
k


.

L
oss tangent
:



'
'
'
tan
c

P
ropagation constant
:

2
1
)
1
(






j
j
j
j
jk
c
c

z
j
z
z
jk
z
e
e
e
e
E
c

I
f

the medium is
lossless,

α
=0
;

else if

the medium is

lossy
,
α
>0
.

Phase
c
onstant
:

2

2
1
2
]
1
)
(
1
[
2





,

2
1
2
]
1
)
(
1
[
2





Case 1

Low
-
loss
D
ielectric
:

1



2

,
]
)
(
8
1
1
[(
2




Intrinsic impedance
:

)
2
1
(


j
c

Phase velocity
:

]
)
(
8
1
1
[
1
1
2





c
p
v

Case 2

Good
C
on
ductor
:

1







f

2
,

and



f
j
j
c
c
)
1
(
)
(

)
1
(
j

Phase velocity
:



2

p
v

S
kin
D
epth

(depth of penetration)
:



f
1
1

.

For
a
good conductor
,

2
1
1

Eg
.
)
10
cos(
100
ˆ
)
,
(
7
t
x
z
t
E

V/m

at
z
=0 in seawater:
ε
r
=72
,
μ
r
=1
,

σ=4
S/m
. (a)
Determine
α
,
β
,
v
p
, and
η
c
. (b) Find the distance at which the amplitude of
E

is
1
%

of its value at
z
=0. (c) Write
E
(
z
,
t
) and
H
(
z
,
t
) at
z
=0.8
m
, suppose it propagates
in the +
z

direct
ion.

(
Sol.
)

7
10

,
f
=5×
10
6
Hz
,

σ
/
ωε
0
ε
r
=200>>1
,

S
eawater is a good conductor in
this case.

(a)



m
Np
f
/
89
.
8
,

f
j
c
)
1
(

s
m
v
p
/
10
53
.
3
6

,

m
707
.
0
2

,
m
112
.
0
1

(b)
m
z
e
z
518
.
0
)
100
ln(
1
01
.
0

(c)
)
cos(
100
ˆ
]
)
(
Re[
)
,
(
z
t
e
x
e
z
E
t
z
E
z
t
j

)
11
.
7
10
cos(
082
.
0
ˆ
)
8
.
0
cos(
100
ˆ
)
,
8
.
0
(
8
.
0
7
8
.
0

t
x
t
e
x
t
E
m
z

)
,
8
.
0
(
ˆ
1
)
,
8
.
0
(
t
E
a
t
H
n

,
)
61
.
1
10
cos(
026
.
0
ˆ
]
)
8
.
0
(
Re[
ˆ
)
,
8
.
0
(
7

t
y
e
E
y
t
H
t
j
c
x

Eg. The magnetic field intensity of a linearly polarized uniform plane wave
propagating in the +
y

direction in seawater

ε
r
=
80
,

μ
r
=
1,
σ
=4
S/m

is
)
3
10
sin(
1
.
0
ˆ
10

t
x
H

A/m
. (a)
Determine the attenuation constant, the phase
constant, the intrinsic impedance, the phase velocity, the wavelength, and the
skin depth.

(b)
Find the location at which the amplitude of
H

is 0.01
A/m
.

(c)
Write the expressions f
or
E
(
y
,
t
) and
H
(
y
,
t
) at
y
=0.5
m

as function of
t
.

(S
ol.
)

(a)

σ
/
ωε
=0.18<<1
,

Seawater is a low
-
loss dielectric in this case.

2

m
Np
/
96
.
83

)
2
1
(


j
c

0 2 8 3
.
0
8
.
41
j
e

]
)
(
8
1
1
[(
2




300

,
s
m
v
p
/
10
33
.
3
7

,
m
2
10
19
.
1
1

,

m
3
10
67
.
6
2

(b)
m
y
e
y
2
10
74
.
2
10
ln
1
1
.
0
01
.
0

(c)
)
3
10
sin(
1
.
0
ˆ
)
,
(
10

y
t
e
x
t
y
H
y
,

300
,
5
.
0

y

)
3
10
sin(
10
75
.
5
ˆ
)
,
5
.
0
(
10
20

t
x
t
H

)
0283
.
0
3
10
sin(
10
41
.
2
ˆ
)
,
5
.
0
(
ˆ
)
,
5
.
0
(
ˆ
ˆ
10
18

t
z
t
H
a
t
E
y
a
n
c
n

Eg. Given that the skin dep
th for graphite at 100

MHz

is 0.16
mm
, determine (a)
the conductivity of graphite, and (b) the distance that a 1
GHz

wave travels in
graphite such that its field intensity is reduced by 30
dB
.

(S
ol.
)

(a)
m
S
f
/
10
99
.
0
10
16
.
0
1
5
3



(b) At
f
=
10
9
Hz
,
m
Np
f
/
10
98
.
1
4



m
e
z
e
dB
z
4
10
10
10
75
.
1
log
5
.
1
log
20
)
(
30

Eg. Determine and compare the intrinsic impedance,
attenuation

constant
,

and
skin depth of copper

σ
cu
=
5.8×
10
7
S
/
m
,
silver

σ
ag
=6.15
×
10
7
S
/
m
,

and brass

σ
br
=1.59
×
10
7
S
/
m

at following frequencies
:

60
Hz

and

1
GHz
.

(S
ol.
)



f

,

1

,



f
,

)
1
(
j
c

C
opper: 60
Hz

6
c
10
)
1
(
02
.
2
j

,
m
Np
/
10
17
.
1
2

,

m
3
10
53
.
8

1
GHz

3
c
10
)
1
(
25
.
8
j

,
m
Np
/
10
79
.
4
5

,

m
6
10
09
.
2

Group
v
elocity
:

d
d
d
d
v
g
/
1

]
)
(
)
c o s [ (
]
)
(
)
c o s [ (
)
,
(
0
0
z
t
E
z
t
E
z
t
E

)
cos(
)
cos(
2
0
z
t
z
t
E

Let

z
t
=constant

/
1
dt
dz
v
g

d
d
d
d
/
1

Eg. Show that

d
dv
v
v
p
p
g

and

d
dv
v
v
p
p
g

(
Proof)

p
v
,

p
v

,

d
dv
v
d
d
v
p
p
g

2

,


2

,

d
d
d
d

0
,

d
dv
v
v
p
p
g

An example of
longi
tudinal

v
p
>0 but
longitudinal

v
g
=0 in barber

s pole.

Eg. A 3
GHz
,
y
-
polarized uniform plane wave propagates in the +
x

direction in a
nonmagnetic medium having a dielectric constant 2.5

and a loss tangent 10
-
2
. (a)

Determine the distance over which the amplitude of the propagating wave will be
cut in half
. (b)
Determine the intrinsic impedance, the wavelength, the phase
velocity, and the group velocity of the wave in the medium.

(c)
Assu
ming
)
3
10
6
sin(
50
ˆ
9

t
y
E

V/m

at
x
=0, write the instantaneous expression for
H

for all
t

and
x
.

(
Sol.
)

3
9
9
2
2
10
166
.
4
5
.
2
10
36
1
10
3
2
10
1
10





It
is
a
low

loss dielectric

material
:
m
/
34
.
99
]
)
(
8
1
1
[
2





)
2
1
(


j
c

=

29
.
0
238

(a)

2

=0.497
,

m
d
e
d
395
.
1
2
1
497
.
0

(b)
s
m
v
p
/
10
8973
.
1
8

,

s
m
d
d
d
d
v
g
/
10
8975
.
1
)
/
(
1
8

(c)
)
0016
.
0
3
(
497
.
0
3
497
.
0
21
.
0
ˆ
ˆ
1
50
ˆ

j
x
n
t
j
x
e
e
z
E
a
H
e
e
y
E

)
332
.
0
6
.
31
10
6
sin(
21
.
0
ˆ
)
,
(
9
497
.
0

x
t
e
z
t
x
H
x

A/m

Plasma:

Ionized gasses with equal electron and ion densit
ies
.

Ionosphere:

50~500
Km

in altitude

Simple m
odel

of plasma
:
An electron of charge

e
,
mass
m
,

position

x

E
m
e
x
x
m
dt
x
d
m
E
e

2
2
2
2

Electric dipole

E
m
e
x
e
p

2
2

Total
electric dipole moment:

E
m
Ne
p
N
P

2
2

E
E
m
Ne
P
E
D
p

)
1
(
)
1
(
2
2
0
0
2
2
0
0

,
where
0
2

m
Ne
p

is the
plasma
a
ngular frequency
.

)
1
(
)
1
(
2
2
0
2
2
0
f
f
p
p

.

Propagation constant:
)
(
1
2
2
0
f
f
j
p



Intrinsic

impedance of the plasma
:
2
0
)
(
1
f
f
p
c

where
)
(
120
0

Case 1

f
<
f
p
:

γ

is rea
l,
c

is pure imaginary

Attenuation

EM wave is in c
utoff
.

Case 2

f
>
f
p
:

γ

is pure imaginary,
c

is real

EM wave can
prop
agate

through the
plasma.