Electromagnetic Waves
Maxwell`s Equations
Lecture 23
Thursday: 8 April 2004
ELECTROMAGNETIC
WAVES
• E = Em sin(kx - w t)
• B = Bm sin(kx - w t)
k
2

2
w  2 f 
T
wave speed  c 
w
k
f
POYNTING VECTOR
S
1
0
EB
•This is a measure of power per area. Units
are watts per meter2.
•Direction is the direction in which the wave
is moving.
POYNTING VECTOR
•However, since E and B are perpendicular,
S
1
0
EB
E
and since
c
B
1 2
c 2
S
E 
B
c 0
0
INTENSITY
S
1
c 0
E sin ( kx  wt )
2
m
2
1
2
S  Intensity  I 
Em
2c 0
1 2
SI
Erms
c 0
Em
Erms 
2
RADIATION PRESSURE
p 1 U 1
F

 IA
t c t c
IA
F
(complete absorption)
c
2 IA
F
(complete reflection)
c
RADIATION PRESSURE
F
Pressure =
A
I
pr 
(complete absorption)
c
2I
pr 
(complete reflection)
c
RADIATION PRESSURE
Momentum:
U
p 
(complete absorption)
c
2 U
p 
(complete reflection)
c
Comet: Picture taken
WEDNESDAY 26 March 1997,at
2000 CST,at White Bear
Lake,Minnesota.
MAXWELL’S EQUATIONS
q
 E  dA 
0
d B
 E  ds  
dt
 B  dA  0
d E 


 B  ds   0  i   0
dt 
AMPERE’S LAW
Original:
 B  ds   0i
As modified by Maxwell:
d E 


 B  ds   0  i   0
dt 
Reasons for the Extra Term
 SYMMETRY
 CONTINUITY
SYMMETRY
A time
varying magnetic field produces an
electric field.
A time
varying electric field produces a
magnetic field.
SYMMETRY
Maxwell' s Equations in Free Space
q0
i0
d B
 E  dA  0
 E  ds  
dt
d E
 B  dA  0
 B  ds   0 0
dt
CONTINUITY
CONTINUITY
Without Maxwell' s modification:
At P1:  B  ds   0i
At P2 :  B  ds  0
At P3:  B  ds   0i
CONTINUITY
With Maxwell's modification:
At P1:  B  ds   0i
d E
At P2 :  B  ds   0 0
  0id
dt
At P3:  B  ds   0i
d E
0
 id "Displacement current"
dt
CONTINUITY
i  id
q  CV
A
C  0
V  Ed
d
A
q   0 Ed   0 AE   0 E
d
dq
d E
i
 0
 id
dt
dt
AMPERE’S LAW
AMPERE’S LAW
For path 1:  B  ds   0i
For path 2:  B  ds   0 ( i )    0i
For path 3:  B  ds   0 (id  i )  0
since id  i
ELECTROMAGNETIC
WAVES
Maxwell' s Equations in Free Space
q0
i0
d B
 E  dA  0
 E  ds  
dt
d E
 B  dA  0
 B  ds   0 0
dt
ELECTROMAGNETIC
WAVES
E
B

x
t
c
B
E
  0  0
x
t
Em E
 c
Bm B
1
8
 3.0  10 m/s
 0 0
MAXWELL’S EQUATIONS
q
 E  dA 
0
d B
 E  ds  
dt
 B  dA  0
d E 


 B  ds   0  i   0
dt 
  0 (i  id )