Fundamental EM field quantities:

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ICE2341

Electromagnetics Wave

2006.
Sep. 06

Chapter 7

Electromagnetic Wave

06/09/06

Prof. Seong
-
Ook Park


ICE2341

Electromagnetics Wave

2006.
Sep. 06

7.
Time
-
varying fields and Maxwell’s Equations (contents)

7
-
1
Introduction

7
-
2 Faraday’s Law of Electromagnetic Induction


Fundamental postulate for electromagnetic induction

7
-
2.1 A Stationary Circuit in a Time
-
Varying Magnetic Field


transformer emf (electromotive force),

Lenz’s law


7
-
2.2 Transformers


ideal transformer, real transformer, coefficient of coupling, eddy current

7
-
2.3 A Moving Conductor in a Static Magnetic Field


flux cutting emf or
motional emf

7
-
2.4 A moving Circuit in a Time
-
Varying Magnetic Field


general forms

of Faraday’s law

7
-
3 Maxwell’s Equations

Displacement current
, Maxwell’s Equation

7
-
4 Potential Functions


7
-
5 Electromagnetic Boundary Conditions


general statements
, interface between two lossless medias, interface between dielectric and conductor

Plane Wave Electromagnetic

Boundary Conditions

ICE2341

Electromagnetics Wave

2006.
Sep. 06

3

ELECTROMAGNETIC FIELDS AT BOUNDARIES

Maxwell’s Equations

Integral Form

Differential Form

s v
s
c A
c A A
D nda=
ρ
dv
B nda=0
dt
E ds= B nda
d
d
H ds= J nda D nda
dt


  
   
 

 
  
D
ρ
B 0
E B t
H J D t
  
 
  
     
Gauss’s Law

Gauss’s Law

Faraday’s Law

Ampere’s Law

ˆ
n
S
da
dv
V
A
da
ˆ
n
C
ds
ICE2341

Electromagnetics Wave

2006.
Sep. 06

4

GAUSS’S LAW
-

EXAMPLES

point

charge: q

volume

charge

density:
ρ
o


(coulombs/m
3
)

surface

charge

density:
σ
s


(coulombs/m
3
)

There is no free magnetic charge of any kind, so

s
B nda 0
 

always

S V
D nda
ρ
dv
 
 
2
π
2
π
2
r
0 0
2
r r
2
ε
E r nr sin
θ
d d q
q
ε
E 4
π
r =q E =
4
π
ε
r
   

 
2 2 r
π
2
π
2 2
r o
0 0 0 0 0
3
2 r
o
o r o o
3
o o
r o
2
ε
E r nr sin
θ
d
θ
d =
ρ
r sin
θ
drd
θ
d
ρ
r r
4
π
ε
r E 4
π
ρ
E (for r < R ) ( D
ρ
)
3 3
ε
R
ρ
E (for r
> R ) ( D 0)
3r
ε
 
  
    
   
    
s
o s
S V S
s
n s n
2
ε
E nda=
ρ
dv=
σ
da
σ
1
2
ε
E A=
σ
A E =
2
ε
 
   
  
z
r
sin

r
sin
 
r d

rd

d

ˆ
n
E

o
o
R
da
E

ˆ
n
+ + + + + + + + + + + + + + +


E
E

s
ˆ
n
A
ICE2341

Electromagnetics Wave

2006.
Sep. 06

5

FIELDS PERPENDICULAR TO BOUNDARIES

Using Gauss’s Law

 
 
( Dnda
ρ
dv):
S V
   


 
S V
1n 2n S
1 1n 2 2n S
D nda
ρ
dv (lim 0)
(D D )A=
σ
A
ε
E
ε
E =
σ
More formally:

  
  
1 2 S
1 2 2 S
n (D D )
σ
n (
ε
E
ε
E )
σ
The jump in normal D

is equal to the free

surface charge density

Likewise:

  
   
1 2
1 2
1 2
n (B B ) 0
n ( H H ) 0
Normal B is continuous

 
 
 

S
1n 2n
1n 2n
B nda 0
(B B )A 0
(B B ) 0
++++++
++++++++
A
ˆ
n
s
surface charge density

surface S

1
D
2
D
1

2

ICE2341

Electromagnetics Wave

2006.
Sep. 06

6

AMPERE’S LAW
-

EXAMPLES

line current: I

(amps)

current

density: J

(amps/m
2
)

surface current

density: J
s

(amps/m)



 

H 2
π
r I
I
H
2
π
r




      
 

  

 
r 2
2
o o
0 0
o
o o
2
o o
o
r
ˆ
H 2 r J zrdrd J 2
2
J
H r (for r < R ) ( H=J )
2
J R
H (for r > R ) ( H 0)
2 r
  
 

 
c s
s
s
ˆ
ˆ
Hds J nda
H 2L J L
J
H
2
c s
H ds J nda
   
 
z
I
r
H

rd
θ
dr
H
z
y
x
C
L
H
s
J
o
J
r
o
R
H

ICE2341

Electromagnetics Wave

2006.
Sep. 06

7

BOUNDARY CONDITIONS FOR PARALLEL FIELDS

Using Faraday’s Law:

 
 
ˆ
c A
d
Eds Bnda
dt
   
    

 
c A
1 2
1 2
d
ˆ
E ds B nda
dt
(E E )L 0 (A 0 as 0)
Therefore E E and
  
1 2
ˆ
n (E E ) 0
Using Ampere’s Law:

 
  
c A A
d
Hds Jda Dda
dt
 
    
  
c A A
1 2 s a
d
Hds J da Dda
dt
ˆ
(H H )L (J n )L (lim 0)
Therefore:

  
1 2 s
ˆ
n (H H ) J
(
There is no sheet

displacement current.)

ˆ
n
2
E
1
E
1
E
2
E
1

2

a
ˆ
n

ds
A
L
ICE2341

Electromagnetics Wave

2006.
Sep. 06

8

FIELDS INSIDE PERFECT CONDUCTORS

Electric Fields Inside Perfect conductors

Magnetic Fields Inside Perfect Conductors

  




m
If and E 0 :
But if J:
But if H:
But w cannot:
Since E 0 inside:
  
     
   
2
m m
Then J E
Then H since H J D t
Then W H 2 and w
Therefore
Therefore
E 0 inside
0 inside

 
perfect conductors
since E
  
And therefore:

Since

E 0

and

E - B t,
   
B t 0
  
therefore

B 0 inside

perfect conductors

(
unless constant and there

since the beginning of time)

ICE2341

Electromagnetics Wave

2006.
Sep. 06

9

BOUNDARY COMDITIONS, PERFECT CONDUCTORS

General Boundary Condition:

1 2 s
1 2
ˆ
n (D D )
ˆ
n (B B ) 0
   
  
1 2
ˆ
n (E E ) 0
  
1 2 s
ˆ
n (H H ) J
  
2 2 2
D B E 0
  
inside
σ

= ∞ :

1 s
1
1
1 s
ˆ
n D
ˆ
n B 0
ˆ
n E 0
ˆ
n H J
  
 
 
 
H is parallel to perfect conductors
E is perpendicular to perfect conductors


Only surface charges and surface currents

++++
1 1
E,D
2 2
E,D
1 1
H,B
2 2
H,B
ˆ
n
s


s
J
ICE2341

Electromagnetics Wave

2006.
Sep. 06

10

0
E
k
E
2
o
2





o
o
o
k




In Cartesian coordinates:

0
E
k
E
x
2
o
x
2
2
2
















2
2
2
z
y
x
Consider a uniform plane wave which is independent of
x
and
y
,


i.e.,

0
E
E
x
2
x
2






2
2
y
x
we have

0
E
k
E
x
2
o
x


2
2
dz
d


z
jk
-
o
z
jk
-
o
x
o
o
e
E
e
E
z
E




Plane wave in lossless media :

Free
-
space wavenumber

:


PLANE WAVE

ELECTROMAGNETICS

ICE2341

Electromagnetics Wave

2006.
Sep. 06

11

Consider the term



z
o
x
e
E
z
E
o
-j k



in the real situation





]
e
z
E
[
t
z
E
t
j
x
x




Re
,


z
k
t
E
o
o




cos
which represents a wave propagating to the +z
-
direction.

For a particular point, we may calculate the phase velocity
(velocity of propagation of an equip
-
hase front) as follows:

constant
z
k
t
o



c
k
z
o
o
o









dt
d
u
p
c: velocity of light

ICE2341

Electromagnetics Wave

2006.
Sep. 06

12





]
e
z
E
[
t
z
E
t
j
x
x




Re
,


z
k
t
E
o
o




cos
constant
z
k
t
o



ICE2341

Electromagnetics Wave

2006.
Sep. 06

13

note:

o
o
o
o
2
c
f
2
c
k











k
k
o
o




2
2




0

=
free space wavelength



=
wavelength traveling in lossless media

z
jk
e
E
0
0


Co
-
sinusoidal wave traveling in

z direction
with same velocity
c
.

* E
-

= 0 if concerned only wave traveling in

+z direction only

* In discontinuities medium, reflected waves

in opposite direction must considered.

ICE2341

Electromagnetics Wave

2006.
Sep. 06

14

For the magnetic field:

H
-j
E
o




















z
y
x
o
x
H
z
H
y
H
x
-j
0
0
z
E
z
0
0
z
y
x
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ




0
0
o
jk
-













z
z
o
o
x
o
y
x
H
e
E
z
j
z
z
E
j
H
H


ICE2341

Electromagnetics Wave

2006.
Sep. 06

15







z
E
z
H
z
E
e
E
k
H
x
o
y
x
o
o
z
o
o
o
y













1
1
o
jk
-










377
120
o
o
o
o

:
intrinsic impedance of the free space

Instantaneous expression for :

H





]
[
Re
ˆ
,
t
j
y
e
z
H
y
t
z
H






z
k
t
E
y
o
o
o





cos
ˆ
(
A/m)

ICE2341

Electromagnetics Wave

2006.
Sep. 06

16


A uniform plane wave is an electromagnetic wave in which the electric and
magnetic fields and the direction of propagation are mutually orthogonal, and
their amplitudes and phases are constant over planes perpendicular to the
direction of propagation





]
e
z
E
[
t
z
E
t
j
x
x




Re
,


z
k
t
E
o
o




cos

Let us examine a possible plane wave solution given
by


ICE2341

Electromagnetics Wave

2006.
Sep. 06

17

note:

H
E



E

H

and are transverse to the propagation direction .

z
ˆ


the uniform plane wave is a transverse electromagnetic


wave or TEM wave.

General form of TEM wave



z
jk
y
jk
x
jk
-
o
z
y
x
e
E
z
y
x
E





,
,
with

2
o
2
z
2
y
2
x
k
k
k
k



Define:

wavenumber vector

z
z
y
y
x
x
r
n
k
k
z
k
y
k
x
k
z
y
x
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ









H

E

n
ˆ
ICE2341

Electromagnetics Wave

2006.
Sep. 06

18









r
n
ˆ
j k
-
r
n
ˆ
-j k
r
k
-j
ˆ
1
1
























e
E
n
r
E
j
r
H
e
E
e
E
r
E
o
o
o








k
Intrinsic impedance
of the medium :

Similarly,













r
H
n
r
H
n
jk
j
r
H
j
r
E
e
H
e
H
r
H
o
o

























ˆ
ˆ
1
1
r
n
ˆ
-j k
r
k
-j



ICE2341

Electromagnetics Wave

2006.
Sep. 06

19

ICE2341

Electromagnetics Wave

2006.
Sep. 06

20


Strictly speaking, uniform plane waves can be produced
only by sources of infinite extent.



However, point sources create spherical waves.


Locally, a spherical wave looks like a plane wave.



Thus, an understanding of plane waves is very important in
the study of electromagnetics.




Uniform Plane Wave Solutions

ICE2341

Electromagnetics Wave

2006.
Sep. 06

21

REFLECTION FROM PERFECT CONDUCTOR

A UPW reflected by a perfect conductor
(revisited


see Lecture 3).

inc refl
ˆ
ˆ
E E xcos( t kz) E xcos( t kz)
     
E 0 at z 0
 
refl inc
E E
  
inc
ˆ
E 2E xsin t sinkz
   
inc ref
0 0
E E
ˆ
ˆ
H ycos( t kz) ycos( t kz)
     
 
inc
0
E
ˆ
H 2 ycos t coskz
  

Note

s
H (z 0) 0
J H
 
 
z
=
0
H
y
t 0
 
t
  
S
z
=
0
z
E
x
t 0
 
t 2
  
t 2
  
Perfect
Conductor
ICE2341

Electromagnetics Wave

2006.
Sep. 06

22

z
x
i

i
k
r

r
k
t
k
t

c

,
 
t t
,
 
ONE WAY TO VISUALIZE SNELL’S LAW

t
i
sin
sin



PHASE MATCHING”: k
iz

= k
tz

=> k
o

sin θ
i

=k
t

sin θ
t

=

i
t
n
n
and

max
t
sin 1
 

,
so there is no real transmitted wave for

t
i
i
n
sin
n
 
For

i

greater than this

1
c t i
"critical angel" sin (n n )

 
there is total reflection.

t i t i
k k =>
θ
θ
 
t i t i
k < k =>
θ
θ

0 0
air,
k =
ω
 
(
Often called total


internal


reflection)

o
i 0 0 i c
(e.g. [
ε
=2
ε
,
μ
=
μ
] [n= 2] [
θ
=45 ])
 
glass

air


捲楴i捡l


angle”

z
x
i

i
k
r

r
k
z
k
t
k
t

,
glass
2
o
k k
t o

i t
  
 
  
ICE2341

Electromagnetics Wave

2006.
Sep. 06

23

NON
-
UNIFORM PLANE WAVE (NUPW)

Normal refraction:


Beyond the critical angle: evanescence

i c
θ
<
θ
i

glass
air
t

glass

o

o


(
free space
)
z
Line of constant phase
i

glass

z
x
                         
 
                         
 
 
                         
 
                         
 
                         
 
 
o


t
90

 
x
e



k z


t
k

t
k

i c
 

evanescent
segion
line of
constant
amplitude
ICE2341

Electromagnetics Wave

2006.
Sep. 06

24

WHAT HAPPENS WHEN THE CRITICAL ANGLE

i
θ
>
C
θ
?
2 2 2 2 2 2
t t t tz tz iz tz
k = =k +k =k +k
  
Since

we have

and

where

Thus

2 2 2
tx t iz i c
k =k k <0 for
θ
>
θ
!

2 2
tx z t
k = j
α
= j k k
  
2 2 2
z i i i
2 2
t t t
k = sin
θ
k =
 
  
z tx
z
-jk z+jk x
t
o
-jk z+ x
o
E = yTE e
= yTE e

z
x
i

i
k
r

r
k
t
k
t

z t
k k

ICE2341

Electromagnetics Wave

2006.
Sep. 06

25

NON
-
UNIFORM PLANE WAVES (2)

Beyond the critical angle:

Called:

“non
-
uniform plane wave”

“evanescent wave” (no power in direction of decay)

“surface wave”

“inhomogeneous plane wave”

In general:

And, if the medium is lossless,

z
t
x- jk z
t
o
- jk r
o
E = yTE e (x<0)
= yTE e


-j(k -jk ) r
E,H e
 


where:

t
z
k =k z j x k - jk
 
 
k k 0
 
 
z
x
i

i
k
r

r
k
t
k
t

0

k

k

ICE2341

Electromagnetics Wave

2006.
Sep. 06

26

PLANE WAVES AT BOUNDARIES

Wave equation:

Simple example: UPW propagating along one axis (say z)

UPW propagation in arbitrary direction:

2
2 2
k
( )E 0
   
in source
-
free region

2 2 2
2
2
where , k=
2 2 2
c
x y z
    
      

  
2
2
0 ( k )E 0
2
x y
z
  
    
 

jkz
0
E E e


x y z
jk x jk y jk z
0
E E e
  

x y z
jk r
0
ˆ
ˆ
ˆ
k xk yk zk
E E e where
ˆ
ˆ
ˆ
r xx yy zz
 

 



 


z
wave
λ

plane phase fronts




λ
z
x
y
k
ICE2341

Electromagnetics Wave

2006.
Sep. 06

27

WAVES PROPAGATING IN THREE DIMENSIONS

Dispersion relation:

Wave vector k :
jk r 2 2
0
o
Substitute E e into wave equation ( k )E 0 w
here
 
  
2 2 2
2
2 2 2
x y z
  
    
  
2 2 2 2 2
x y z o
k k k k
    


x y z
j(k x k y k z)
x y z
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
E 0 x y z xE yE zE e
x y z
  
 
  
       
 
  
 
x y z
x y z
j(k E k E k E ) jk E 0
likewise H jk H 0
      
    
k E
k H


x
k
y
k
z
k
0
E
H
k
o
k k

E k 0
H k 0
 
 
λ
y
x
k
y
k
x

x x
2 k

 
r
x o x
o
k k cos
(where k k )


on

k r constant
phase front
 
ICE2341

Electromagnetics Wave

2006.
Sep. 06

28

CONSIDER UPW AT PLANAR BOUNDARY

Case I: TE wave

Trial solutions:


Transverse Electric”

E Plane of incidence

o i o i
x z
o r o r
t t t t
jk cos x jk sin z
jk x jk z
i
0 0
jk cos x jk sin z
r
0
jk cos x jk sin z
t
0
ˆ
ˆ
Incident: E yE e yE e
ˆ
Reflected: E y E
e
ˆ
Transmitted: E yTE e
   
 
   
   
 
 

E
k
r
E
i
H
x
k
z
k
i
k
i

i

o i
k k

r

t

t
k
t
E
i
E
r
k
o
k
  
z
k
,
 
t t
,
 
x
y
z
ICE2341

Electromagnetics Wave

2006.
Sep. 06

29

IMPOSE BOUNDARY CONDITIONS @ X=0

E is continuous at x 0

k
z
o i o r t t t
i r t
z z z
jk sin z jk sin z jk sin sin z
0 0 0
o i o r t t
k k k
E e E e TE e for all z
Therefore: k sin k sin k sin
      
   
    



z
k
2 2 2 2 2
x y z o
k k k k
    
r i
Angle of incidence equals angle of refl
ection
  
=>

and

t o t
i
i t i t
t t
phase
sin k v
n
"Snell's Law"
sin k v n
where n c v c "Refractive Index"
 

   

  
 
vacuum water
glass
n 1 n 1.3 at visible wav
elengths
n 1.5 1.66 < 9 at audio-rad
io frequencies
 
 