Figure 13.4.7
Period
13-17
At time
t
=
0
,
E
y
(0,0)
=
0
. Let
T
be the time that it takes the electric field to complete
one oscillation on the plane
x
=
0
. Then
E
y
(0,
T
)
=
!
E
0
sin(2
"
cT
/
#
)
=
0
. This
condition is satisfied when
T
=
!
c
,
(13.4.43)
then
E
y
(0,
T
)
=
!
E
0
sin(2
"
)
=
0
. The time
T
is called the
period.
The
frequency
f
is
defined as the inverse of the period,
f
!
1
T
.
(13.4.44)
The
angular frequency
!
is defined by
!
"
2
#
f
=
2
#
T
=
2
#
c
$
.
(13.4.45)
The SI units of angular frequency are
[rad
!
s
-1
]
.
With these definitions we can rewrite our electric and magnetic fields as
!
E
=
E
y
(
x
,
t
)
ˆ
j
=
E
0
sin(
kx
!
"
t
)
ˆ
j
!
B
=
B
z
(
x
,
t
)
ˆ
k =
B
0
sin(
kx
!
"
t
)
ˆ
k
.
(13.4.46)
In empty space, the wave propagates at the speed of light
c
. The characteristic behavior
of the sinusoidal electromagnetic wave is illustrated in Figure 13.4.8.
Figure 13.4.8
Plane electromagnetic wave propagating in the
+
x
-
direction.
We see that the
E
!
and
B
!
fields are always in phase (attaining maxima and minima at the
same time.) To obtain the relationship between the field amplitudes
0
E
and
0
B
, we make
use of Eqs. (13.4.6) and (13.4.12). We first calculate the following the partial derivatives
13-18
!
E
y
!
x
=
kE
0
cos(
kx
"
#
t
)
,
(13.4.47)
!
B
z
!
t
=
"
#
B
0
cos(
kx
"
#
t
)
,
(13.4.48)
Eq. (13.4.6) then implies that
0
0
E k
B
!
=
, which we rewrite as
0
0
E
c
B
k
!
=
=
.
(13.4.49)
Because the sinusoidal behavior of the fields are the same, the magnitudes of the fields at
any instant are related by
E
c
B
=
.
(13.4.50)
Let us summarize the important features of electromagnetic waves described in Eq.
(13.4.38):
1.
The wave is transverse since both
E
!
and
B
!
fields are perpendicular to the direction
of propagation, which points in the direction of the cross product
!
E
B
!
!
.
2.
The
E
!
and
B
!
fields are perpendicular to each other. Therefore, their dot product
vanishes,
0
!
=
E B
!
!
.
3.
The speed of propagation in vacuum is equal to the speed of light, and
0
0
1/
c
μ
!
=
.
4.
The ratio of the magnitudes and the amplitudes of the fields is
c
=
E
B
=
E
0
B
0
=
!
k
=
"
T
.
13.5
Standing Electromagnetic Waves
Let us examine the situation where there are two sinusoidal plane electromagnetic waves,
one traveling in the positive
x
-direction, with
E
1
y
(
x
,
t
)
=
E
10
sin(
k
1
x
!
"
1
t
),
B
1
z
(
x
,
t
)
=
B
10
sin(
k
1
x
!
"
1
t
)
(13.5.1)
13-19
and the other traveling in the negative
x
-direction, with
E
2
y
(
x
,
t
)
= +
E
20
sin(
k
2
x
+
!
2
t
),
B
2
z
(
x
,
t
)
=
"
B
20
sin(
k
2
x
+
!
2
t
)
.
(13.5.2)
For simplicity, we assume that these electromagnetic waves have the same amplitudes,
E
0
!
E
10
=
E
20
and
B
0
!
B
10
=
B
20
,
and
wavelengths,
!
"
!
1
=
!
2
.
Using
the
superposition principle, the electric field and the magnetic fields can be written as
E
y
(
x
,
t
)
=
E
1
y
(
x
,
t
)
+
E
2
y
(
x
,
t
)
=
E
0
[sin(
kx
!
"
t
)
+
sin(
kx
+
"
t
)]
,
(13.5.3)
B
z
(
x
,
t
)
=
B
1
z
(
x
,
t
)
+
B
2
z
(
x
,
t
)
=
B
0
[sin(
kx
!
"
t
)
!
sin(
kx
+
"
t
)]
.
(13.5.4)
Using the identities
sin(
!
±
"
)
=
sin(
!
)cos(
"
)
±
cos(
!
)sin(
"
)
(13.5.5)
The above expressions may be rewritten as
E
y
(
x
,
t
)
=
E
0
[(sin(
kx
)cos(
!
t
)
"
cos(
kx
)sin(
!
t
))
+
(sin(
kx
)cos(
!
t
)
+
cos(
kx
)sin(
!
t
))]
=
2
E
0
sin(
kx
)cos(
!
t
),
(13.5.6)
and
B
z
(
x
,
t
)
=
B
0
[(sin(
kx
)cos(
!
t
)
+
cos(
kx
)sin(
!
t
))
"
(sin(
kx
)cos(
!
t
)
"
cos(
kx
)sin(
!
t
))]
=
2
B
0
cos(
kx
)sin(
!
t
),
(13.5.7)
One may verify that the total fields
E
y
(
x
,
t
)
and
B
z
(
x
,
t
)
still satisfy the wave equation
stated in Eqs. (13.4.15) and (13.4.17) even though they no longer have the form of
functions of
k
(
x
±
ct
)
=
(
kx
±
!
t
)
. The waves described by Eqs. (13.5.6) and (13.5.7) are
standing waves
, which do not propagate but simply oscillate independently in space and
time.
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- Magnetic Field, ........., electromagnetic waves
