Notes 6
Fall 2004.doc
1
The Uniform Plane Wave
Electromagnetic wave propagation
Any
wave
can be represented by a function of both space and time.
Wave motion occurs when a disturbance at one place and time is related to an effect at another place
and at a later time.
Waves carry energy and have a definite velocity
.
Starting with Maxwell’s first equation we can take the curl of both sides to get
t
∂
∂
×
∇
=
×
∇
×
∇
B

E
G
G
G
G
G
Using the vector identity (see Notes 2)
()
A
A
A
2
G
G
G
G
G
G
G
G
∇
−
⋅
∇
∇
=
×
∇
×
∇
then pulling out the
t
∂
∂
operator from the righthand side and using
H
B
G
G
o
μ
=
yields
()
t
B

E
E
∂
∂
×
∇
=
∇
−
⋅
∇
∇
G
G
G
G
G
G
G
2
()
H

E
E
G
G
G
G
G
G
G
×
∇
∂
∂
=
∇
−
⋅
∇
∇
t
μ
o
2
Noting that free space is a
sourceless
region
( )
0
,
0
=
=
ρ
c
v
J
G
, we can substitute on the righthand side
with
+
=
×
∇
c
J
G
G
G
H
t
t
o
o
∂
∂
ε
=
∂
∂
ε
E
E
G
G
from Maxwell’s second equation and also substitute on the left
hand side with Maxwell’s fourth equation
0
E
E
D
=
⋅
∇
→
=
⋅
∇
ε
=
⋅
∇
G
G
G
G
G
G
0
o
to get
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
ε
∂
∂
=
∇
−
t
t
μ
o
o
2
E

E
G
G
G
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
ε
=
∇
−
2
2
o
o
2
t
μ
E

E
G
G
G
0
1
t
2
o
o
2
2
=
∇
ε
µ
−
∂
∂
E
E
G
G
G
This is a second order partial differential equation in
E
G
involving both the spatial and temporal
dimensions.
This equation is called the 3dimensional
vector wave equation
for free space.
Note that
it was derived directly from Maxwell’s equations.
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Fall 2004.doc
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The vector wave equation is a mathematical description of the wave phenomenon.
The quantity
o
o
1
ε
µ
has an interesting meaning and history.
It has units of velocity and a quick calculation shows
that
s
m
8
o
o
10
3
1
×
=
ε
µ
which one recognizes as the measured velocity of light in free space, c.
It was
this recognition by Maxwell that finally showed visible light to be only one small part of a much larger
continuum called the electromagnetic spectrum.
With this, we can now write (for free space)
0
c
t
2
2
2
2
=
∇
−
∂
∂
E
E
G
G
G
In an analogous fashion we can derive a wave equation for the magnetic field as well:
0
c
t
2
2
2
2
=
∇
−
∂
∂
H
H
G
G
G
As a simplification we will select an electromagnetic wave that is propagating in the
direction and
that has an oscillating electric field whose vector direction is in the
direction.
It is of utmost
importance that one recognizes and distinguishes these
two
directions associated with any electric field
(or magnetic field) traveling wave.
The onedimensional wave equation becomes
z
ˆ
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 Fall '10
 Czarnecki
 Electromagnet, wave equation, free space

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