Wave Motion & EM Waves
(IV)
Chih-Chieh Kang
Electrooptical Eng.Dept. STUT
email:kangc@mail.stut.edu.tw
Electromagnetic wave

Radiation field far from the antenna
Maxwell’s Equations
Differential form
Integral form
B
 E  
t
D  
C E  dl  
D
 H  J 
t
B  0
D E

B  ds

S
t
S D  ds  V dv
cH  dl  S J  ds 
S B  ds  0
H
B

d
D  ds

S
dt
Maxwell’s Equations
E ：electric field intensity [V/m]
：permittivity [s2C2/ m3kg]
D ：electric flux density [C/m2]
：permeability [mkg / C2]
B ：magnetic flux density [Wb/m2] ：charge density [C/m(2)]
H ：magnetic field intensity [A/m] J：current density [A/m2]
in Cartesian coordinates
E  Ex i  E y j  Ez k
H  Hxi  H y j  Hz k
del operator




i
j k
x
y
z
3-D Wave equations for EM
waves

In free space, Maxwell’s equations  wave equations
2 E
1 2 E
2
 E  0 0 2  0
 E 2 2 0
t
c t

2
 H
1 2 H
2
2
 H  0 0 2  0
 H 2
0
2
t
c t
phase velocity c  1 / 0 0  3  108 m/s
2
0 
1
 109 s2  C2 /m 3  kg,
36
0  4  107 m  kg/C 2
Solutions of 3-D Wave
Equations for EM Waves



Every component of EM field E x , E y , E z , H x , H y , H z obeys the 3-D
scalar differential wave equation
 2  2  2
1  2

 2  2
0
2
2
2
x
y
z
c t

Solutions for time-harmonic plane waves propagating in the +k
direction



ˆ  x, y , z; t   ˆ r, t   Aˆ ei t k r   Aˆ e
  x, y , z; t   Reˆ r, t   Re Aˆ ei t k r   A cost  k  r    
  x, y , z; t   Imˆ r, t   Im Aˆ ei t k r   A sin t  k  r    
i t  k x x  k y y  k z z




r  xi  y j  z k
k  r  kx x  k y y  kz z
Solutions of 3-D Wave
Equations for EM Waves

Take


 r, t   Reˆ r, t   Re Aˆ ei k r t   A cosk  r  t    
E x r, t , E y r, t , E z r, t , H x r, t , H y r, t , H z r, t  can be expressed

as the form : Re Aˆ ei t k r 










 
 
 



 E x r, t   Re Eˆ x0ei t k r   Re Eˆ x0e k r eit  Re Eˆ x z eit

 i t  k r 
 Re Eˆ y0e k r eit  Re Eˆ y z eit
 E y r, t   Re Eˆ y 0e
 E r, t   Re Eˆ  ei t k r   Re Eˆ  e k r eit  Re Eˆ  z eit
z0
z0
z
 z



phasor representa tion : Eˆ x r   Eˆ x0e i k r , Eˆ y r   Eˆ y0e i k r , Eˆ z r   Eˆ x0e i k r
Let Eˆ r   Eˆ x r a x  Eˆ y r a y  Eˆ z r a z  Eˆ x0e i k r a x  Eˆ y0e i k r a y  Eˆ x0e i k r a z


 Eˆ x0 a x  Eˆ y0 a y  Eˆ x0 a z e i k r  Eˆ 0e i k r
Eˆ r   Eˆ 0e i k r



E  x, y , z; t   E r, t   Re Eˆ r eit  Re Eˆ 0ei t k r 

TEM Waves
Eˆ r, t   Eˆ 0ei t k r  ,
  Eˆ  0 
Bˆ r, t   Bˆ 0ei t k r 
i k  Eˆ  0 
k  Eˆ
  Bˆ     Hˆ  0    Hˆ  0 
k  Bˆ r 
 Eˆ and Bˆ are perpendicu lar to the direction of propagatio n k
 Transverse waves (TEM waves )
ˆ

B
  Eˆ  
t

i k  Eˆ  i Bˆ

1 k ˆ ˆ
E  B
v k
 Eˆ and Bˆ are perpendicu lar to each other : Eˆ  Bˆ
Relation between E and H in a
Uniform Plane Wave

In general, a uniform plane
wave traveling in the +z
direction may have both xand y-components
Eˆ  z   Eˆ x  z a x  Eˆ y  z a y
Hˆ  z   Hˆ   z a  Hˆ   z a
x
x
y
y
E x 0：amplitude of E-field,
phase constant
H y 0 ：amplitude of H-field
    0 0
intrinsic impedance
0  E x 0 / H y 0   0 /  0
Energy Transport by EM
Waves

Poynting theorem

 vS  ds  v J  Edv  vwe  wm dv
t



1
2
we   E
2
1
2
wm   H
2
J / m 
J / m 
electric energy density
3
magnetic energy density
Poynting vector determines the direction of
energy flow
S  EH
W / m 
2
3
Energy Transport by EM
Waves

Poynting theorem : Electromagnetic power flow into a closed
surface at any instant equals the sum of the time rates of
increase of the stored electric and magnetic energies plus the
ohmic power dissipated (or electric power generated, if the
surface enclosed a source) within the enclosed volume.
Energy Transport by EM
Waves
 E cost  k  r   
E  Re E e 
i t  k r  
0

0
1 k
1 k
B
E 
 E 0 cost  k  r    
v k
v k
1 k
H
 E 0 cost  k  r    
v k
1
k
S  EH 
E 0   E 0 cos2 t  k  r    
v
k
1
2 k

E0
cos2 t  k  r    
v
k
 A cos2 t  k  r    
Energy Transport by EM
Waves

Time-averaged Poynting vector
1 T
1 T
2



S
dt

A
cos

t

k

r


dt


0
0
T
T
A T
2




cos

t

k

r


d t 

0
T
A T 1




1

cos
2

t

k

r


d t 

0
T
2
A
A
 
sin 2T  k  r      sin 2 k  r    
2 4T
A
1
v
2 k
2 k
 
E0

E0
T  1
2 2 v
k
2
k
S 



 : frequency of the light wave
T : the response time of the detector t o the light wave

Energy Transport by EM
Waves

Irradiance I : average energy per unit area
per unit time
I S 
 I  E0

1
2 v
E0 
2
v
2
E 0  vwe
2
2
The intensity of light wave is proportional to the
square of the amplitude of the (electric) field.
Radiation Pressure &
Momentum
F
p
A
w
d
 ,
w  Fd ,
v
t
t
F w / d t / d  / v
p 


A
A
A
d
I
p
c
References



E. Hecht, Optics, Addison-Wesley.
F. T. Ulaby, Fundamentals of Applied
Electromagnetics, Prentice Hall.
J. D. Cutnell, and K. W. Johnson, Physics, Wiley.