Document 7921750
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Transcript Document 7921750
Basic Properties of
Electromagnetic Waves
Electromagnetic waves consist
of oscillating electric and
magnetic fields.
(site 1)
1. E( x , y , z, t ) B( x , y , z, t )
2. Both E and B are to the direction of wave motion. At any
point in space at any time the direction of wave motion is parallel to E B
3. The wave speed in vacuum is c = 2.9979 108 m / s
Regardless of the wavelength or state of motion of the source or observer.
4. B E / c at each point in space at each time
The energy density in a light wave (energy
per unit volume) is given by
u( x, y, z, t ) 0 E( x, y, z, t )
2
3
J
/
m
where 0 = 8.854 10-12
V / m 2
This expression includes both the electric and magnetic field energy
Example:
An electromagnetic plane wave is moving in
the +y direction. It’s electric field is
polarized along the x-axis and has an
amplitude of 103 V/m. The wavelength is
550nm.
A. Find an expression for the electric field
We know that
2
2
7
k
1142
.
10
rad / m
9
550 10 m
and that
2c
2 f
k c 1.142 107 rad / m3 108 m / s
3.426 1015 rad / s
so we can write
E (y, t) = (103 Vm ) cos( k y t ) x
B. Give an expression for the magnetic field of this wave
103 mV
-6
B = E/c =
3.33
10
T
8 m
3 10 s
When E points along the + x - axis and the wave is moving
in the + y direction, we must have B pointing in the - z direction.
Therefore
B = 3.33 10-6 T cos( k y t ) ( z )
3.33 10-6 T cos( k y t ) z
C. How much energy is contained in a cube of length 10nm
centered at the location x = 124 0 nm, y = 1180nm, z = 3412 nm
at time t = 1.5 10 -15 s ?
The energy density is
u( x , y , z, t ) 0 E( x , y , z, t )
2
2
3
J
/
m
V
3
8.854 10 -12
10
2
m
V / m
cos2 (1142
.
10 7
rad
m
1180 10 9 m - 3.426 1015
rad
s
1.5 10 -15 s)
J
= 1.907 10
m3
so the total energy is u volume =
-3
J
3
8
-27
-8
1.907 10
10
m
1.907
10
J
=
1.190
10
eV
3
m
-3
Consider an electromagnetic wave whose electric field is
given by E = E 0 cos( kx t ) y
The average energy density at any point in space is the average value of
u( x , y , z, t ) 0 E( x , y , z, t )
2
0 E 20 cos2 ( kx t )
1
The average value of cos ( ), averaged over one or more periods, is
2
1
1
2
2
cos ( )
so that cos ( kx t )
2
2
2
u 0 E 20 cos2 ( kx t ) 21 0 E 20
Since the coordinates don' t enter into this expression,
it holds for every electromagnetic plane wave.
For example, for the wave discussed in the previous problem,
the average energy density is
2
3
1
J
/
m
3 V
-12
-6 J
u 8.854 10
4.427 10
2 10
V / m
2
m
m3
The average energy in this box is
average energy = ( 0 E ) A c t
1
2
2
0
The rate at which energy passes through the surface is
average energy
P =
( 21 0 E 20 ) A c
t
The average power per unit area is the intensity
P 1
I 2 0 c E 20
A