Project PHYSNET Physics Bldg. Michigan State University East Lansing, MI
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路
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MISN-0-210
ELECTROMAGNETIC WAVES FROM
MAXWELL鈥橲 EQUATIONS
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1
ELECTROMAGNETIC WAVES FROM MAXWELL鈥橲 EQUATIONS
by
Peter Signell
Michigan State University
1. Introduction
a. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
b. Waves in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
c. Vector Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
d. Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2. Vector Derivatives
a. The Gradient Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
b. The Divergence Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
c. The Curl Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3. Maxwell鈥檚 Equations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
4. Electromagnetic Waves
a. No Charge, No Current; Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 3
b. Plane-Polarized Monochromatic Waves . . . . . . . . . . . . . . . . . . 4
c. Production by a Radio Transmitter . . . . . . . . . . . . . . . . . . . . . . 5
d. How The Waves Manifest Themselves . . . . . . . . . . . . . . . . . . . 6
Acknowledgments
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6
Glossary
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
A. The Curl as a Determinant
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
B. Proof of a Vector Identity
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2
ID Sheet: MISN-0-210
Title:
Electromagnetic Waves from Maxwell鈥檚 Equations
Author: P. Signell, Michigan State University
Version: 10/18/2001
Evaluation: Stage 0
Length: 1 hr; 20 pages
Input Skills
:
1. Vocabulary: charge density (MISN-0-147), current density (MISN-
0-118), displacement (MISN-0-25), sound waves, wave frequency,
wavelength, waves on strings, wave speed (MISN-0-202), wave
equation (MISN-0-201).
2. Take derivatives of transcendental functions (MISN-0-1).
3. Take scalar and vector products of vectors using Cartesian unit
vectors (MISN-0-2).
Output Skills (Knowledge)
:
K1. Vocabulary: propagation (of a wave), polarization (direction of),
plane-polarized (wave), monochromatic (wave).
K2. Given Maxwell鈥檚 Equations, the 鈥渃url-curl鈥 vector identity, and
the definitions of the gradient, divergence, and curl operators, de-
rive the wave equations for electric and magnetic field vectors at
chargeless currentless space-points.
Output Skills (Rule Application)
:
R1. Given the definitions of the gradient, divergence, and curl opera-
tors, verify that a given electromagnetic wave, consisting of cou-
pled electric and magnetic waves, satisfies Maxwell鈥檚 Equations.
R2. Given the direction of polarization, direction of propagation, fre-
quency and amplitude of a monochromatic plane-polarized elec-
tromagnetic wave, write down the electric and magnetic fields in
vector form. Sketch the situation.
Post-Options
:
1. 鈥淓nergy and Momentum in Electromagnetic Waves,鈥 (MISN-0-
211).
2. 鈥淏rewster鈥檚 Law and Polarization,鈥 (MISN-0-225).
3
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ADVISORY COMMITTEE
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c
掳
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4
MISN-0-210
1
ELECTROMAGNETIC WAVES FROM
MAXWELL鈥橲 EQUATIONS
by
Peter Signell
Michigan State University
1. Introduction
1a. Background.
The transport of energy through mechanical sys-
tems via the collective motion of the particles that make up the system is
a familiar phenomenon, spectacularly demonstrated, for example, when
the voice of a soprano shatters a glass across the room from her. This
energy is carried by the 鈥渄isplacement鈥 waves that can be made to propa-
gate through the system (the 鈥渄isplacement鈥 referring to the displacement
from equilibrium of the particles of the system or of the pressure in the
gas).
1b. Waves in Space.
It is also possible for electric and magnetic
fields to propagate as waves in empty space, the electric and magnetic
field vectors playing the same role in electromagnetic waves as the trans-
verse displacement of the particles of a string do in waves along a stretched
string, or the pressure displacement associated with the propagation of
sound waves in air. The important difference is that there is no medium
through which this electromagnetic wave propagates, although perhaps
one could say the 鈥渕edium鈥 is the vacuum! Electric and magnetic fields
may exist in space without a material medium being present, and if they
vary in space and time in the appropriate way, the spatial variation will
propagate as a wave, transporting energy. This module deals with the
propagation of energy through a vacuum via electromagnetic disturbances
whose space and time variation satisfy the conditions for wave propaga-
tion.
1c. Vector Derivatives.
To deal with electromagnetic waves in space
it is far easier to use Maxwell鈥檚 equations in
derivative
form than in
inte-
gral
form. In electricity and magnetism we are dealing with scalar fields
like the charge density
蟻
and vector fields like
~
E
, and we will find that
we must deal with three kinds of derivative operators: the 鈥済radient鈥
operator that operates on a scalar field and produces a vector one, the
鈥渄ivergence鈥 operator that operates on a vector field and produces a scalar
one, and the 鈥渃url鈥 operator that operates on a vector field and produces
5
MISN-0-210
2
another vector one.
1d. Partial Derivatives.
When taking derivatives of field quantities
we generally use 鈥減artial derivatives,鈥 denoted
鈭
, to remind us that the
spatial coordinates are not functions of time. That is, whereas a particle
has a single value of, say,
x
, at any particular time, a field has values at a
continuum of values of
x
. Otherwise, partial derivatives are like ordinary
derivatives:
鈭
鈭倄
隆
x
3
y
4
z
5
垄
= 3
x
2
y
4
z
5
,
鈭
鈭倄
(sin 3
x
cos
y
) = 3 cos 3
x
cos
y .
2. Vector Derivatives
2a. The Gradient Operator.
The gradient operator operates on a
scalar function and produces a vector function that is the steepest 鈥渦p-
hill鈥 slope at any point where the vector function is evaluated. That is,
the gradient of a function, evaluated at some space-point, points in the
direction that is most steeply 鈥渦p hill鈥 in that function at that point. The
magnitude of the gradient is the value of the slope in the 鈥渟teepest ascent鈥
direction at that space-point. As an example, suppose the gradient of a
field scalar field
f
is the vector field
~g
. In Cartesian coordinates this is:
~g
=
~
鈭
f
鈮
藛
x
鈭俧
鈭倄
+ 藛
y
鈭俧
鈭倅
+ 藛
z
鈭俧
鈭倆
.
(1)
陇
Show that
~
鈭
(
x
3
y
4
) = (3
x
2
y
4
)藛
x
+ (4
x
3
y
3
)藛
y
.
2b. The Divergence Operator.
The 鈥渄ivergence鈥 operator operates
on a vector function, say
~g
(
x, y, z
), to give a scalar function
f
(
x, y, z
):
f
=
~
鈭 路
~g
鈮
鈭俫
x
鈭倄
+
鈭俫
y
鈭倅
+
鈭俫
z
鈭倆
.
(2)
The divergence of a vector function, evaluated at some space-point,
gives the extent to which the function has a source or sink at that point.
For example, the divergence of an electric field gives the charge density
at that point (positive charges are sources for the field, negative charges
are sinks).
陇
Show that:
~
鈭 路
隆
x
3
y
4
z
5
垄
藛
z
= 5
x
3
y
4
z
4
.
6
MISN-0-210
3
2c. The Curl Operator.
The 鈥渃url鈥 operator operates on a vector
function, say
~g
(
x, y, z
), to give another vector function:
~
鈭 脳
~g
鈮
碌
鈭俫
y
鈭倄
鈭
鈭俫
x
鈭倅
露
藛
z
+
碌
鈭俫
z
鈭倅
鈭
鈭俫
y
鈭倆
露
藛
x
+
碌
鈭俫
x
鈭倆
鈭
鈭俫
z
鈭倄
露
藛
y .
(3)
The curl of a function, evaluated at some space-point, gives the greatest
鈥渃irculation鈥 at that point, where by 鈥渃irculation鈥 one means the line
integral of the function around a loop of infinitesimal radius. The direc-
tion of the curl is normal to the plane of the loop with the greatest line
integral.
1
陇
Show that:
~
鈭 脳
隆
x
3
y
4
z
5
垄
藛
z
=
隆
4
x
3
y
3
z
5
垄
藛
x
鈭
隆
3
x
2
y
4
z
5
垄
藛
y
.
3. Maxwell鈥檚 Equations
Here are the famous Maxwell鈥檚 Equations in differential form:
~
鈭 路
~
E
= 4
蟺k
e
蟻 ,
(4)
~
鈭 路
~
B
= 0
,
(5)
~
鈭 脳
~
E
=
鈭
鈭 ~
B
鈭倀
,
(6)
~
鈭 脳
~
B
= 4
蟺k
m
j
+
c
鈭
2
鈭 ~
E
鈭倀
.
(7)
Gauss鈥檚 law is the integral form of Eq. (4). Ampere鈥檚 law is the integral
form of Eq. (7) for the case where the electric field does not vary with
time. The Ampere-Laplace-Biot-Savart law is derived from a combination
of Eqs. (5) and (7), also for the case where the electric field does not vary
with time. The Faraday-Henry law of magnetic induction is the integral
form of Eq. (6).
4. Electromagnetic Waves
4a. No Charge, No Current; Waves.
For the case where there is
no charge or current at a point in space, it is easy to show that waves can
exist there. We set
蟻
= 0 and
j
= 0 in Eqs. (4)-(7), then take the time
derivative of both sides of Eqs. (6) and (7):
Help: [S-3]
1
c
2
鈭
2
~
E
鈭倀
2
=
~
鈭 脳
鈭 ~
B
鈭倀
,
1
For another way to remember the definition of the curl, see Appendix A.
7
MISN-0-210
4
鈭
2
~
B
鈭倀
2
=
鈭
~
鈭 脳
鈭 ~
E
鈭倀
.
We now put Eqs. (6) and (7) into the right sides of the above two equations
to get:
鈭
1
c
2
鈭
2
~
E
鈭倀
2
=
~
鈭 脳
(
~
鈭 脳
~
E
)
,
(8)
鈭
1
c
2
鈭
2
~
B
鈭倀
2
=
~
鈭 脳
(
~
鈭 脳
~
B
)
,
(9)
To further reduce the above equations, we make use of the identity:
2
~
鈭 脳
(
~
鈭 脳
~
A
) =
~
鈭
(
~
鈭 路
~
A
)
鈭
(
~
鈭 路
~
鈭
)
~
A
=
~
鈭
(
~
鈭 路
~
A
)
鈭 鈭
2
~
A .
(10)
where
~
A
is any vector field and:
鈭
2
鈮
~
鈭 路
~
鈭
=
鈭
2
鈭倄
2
+
鈭
2
鈭倅
2
+
鈭
2
鈭倆
2
.
Finally, using that identity and Eqs. (4) and (5) for our chargeless case,
we get these two
wave equations
for waves traveling with velocity
c
:
1
c
2
鈭
2
~
B
鈭倀
2
=
鈭
2
~
B ,
(11)
1
c
2
鈭
2
~
E
鈭倀
2
=
鈭
2
~
E .
(12)
4b. Plane-Polarized Monochromatic Waves.
We can write down
solutions to the wave equations, Eqs. (11) and (12), for the case where
the field vectors lie entirely in a plane and where the solutions contain
only one frequency. Of course one must show that the solutions we write
down really are solutions by substituting them into Eqs. (11) and (12) and
showing that those equations are satisfied. For the electric field vector the
plane-polarized monochromatic solution is:
~
E
=
~
E
0
cos(
~k
路
~r
鈭
蠅t
)
,
(13)
2
This identity is proved in Appendix B. Physicists often remember the rule,
~
A
脳
(
~
B
脳
~
C
) =
~
B
(
~
A
路
~
C
)
鈭
~
C
(
~
A
路
~
B
)
,
as the words 鈥滲AC minus CAB鈥 along with the positions of the parentheses. However,
with
~
A
鈮
~
B
鈮
~
鈭
, we must rearrange the order on the right side of the equation to
keep the operators to the left of the functions they operate upon. Thus we wind up
with Eq. (10).
8
MISN-0-210
5
where the
direction
of
~k
gives the direction of propagation of the electric
wave and the
magnitude
of
~k
is 2
蟺
divided by the wave鈥檚 wavelength and
is related to the wave鈥檚 frequency through its velocity:
3
k
=
2
蟺
位
;
蠅
= 2
蟺f
;
f
=
1
T
=
c
位
,
where
蠅
is the wave鈥檚 angular frequency,
f
is its frequency,
T
is its period,
c
is its speed, and
位
is its wavelength.
陇
Show that Eq. (13) satisfies Eq. (12) by direct substitution on both sides
of Eq. (12).
Help: [S-2]
Now with Eq. (13), Eq. (6) becomes:
~
鈭 脳
~
E
=
鈭
(
~k
脳
~
E
0
) sin(
~k
路
~r
鈭
蠅t
)
,
but this equals (
鈭/鈭倀
)
~
B
which can only be true if the arguments in the
cosine functions match:
~
B
=
~
B
0
cos(
~k
路
~r
鈭
蠅t
)
.
(14)
We put the solutions, Eqs. (13) and (14), into Eqs. (11) and (12) and find:
~
B
0
=
1
蠅
(
~k
脳
~
E
0
) =
1
c
(藛
k
脳
~
E
0
)
,
so:
藛
E
0
脳
藛
B
0
= 藛
k
~
E
0
路
~
B
0
= 0
B
0
=
1
c
E
0
.
(15)
陇
Show that the picture on the cover of this module requires all three of
Eqs. (15).
4c. Production by a Radio Transmitter.
A vertical radio trans-
mitter tower is a good example of a device that produces a plane-polarized
monochromatic wave (the frequency of the wave is the frequency to which
you set the dial in order to receive the wave). A large current is sent up
and down the vertical tower, as a sine wave with a single frequency.
陇
Suppose we look at the tower during the part of the cycle when the
current is moving upward. Use Ampere鈥檚 law to show that the magnetic
3
These are general properties of waves: see 鈥淭he Wave Equation,鈥 (MISN-0-201).
9
MISN-0-210
6
field to the right of the antenna is as given on this module鈥檚 cover. Then
use Eqs. (15) tell you the direction of the electric field and the direction
of propagation of the wave.
4d.
How The Waves Manifest Themselves.
Depending on its
frequency, an electromagnetic wave may be a radio or television wave
coming through the air to your receiver, or it could be an X-ray, or a
gamma ray from a radioactive decay, or a ray of light of a particular color.
These objects are all
identical
waves except for their frequencies.
Acknowledgments
Preparation of this module was supported in part by the National
Science Foundation, Division of Science Education Development and
Research, through Grant #SED 74-20088 to Michigan State Univer-
sity.
Glossary
鈥
propagation
: motion.
鈥
polarization, direction of
:
in a plane-polarized electromagnetic
wave, the direction of the electric field vector.
鈥
plane-polarization
:
in an electromagnetic wave, the condition in
which the electric field vector always lies in the same plane (in con-
trast to, say, circular polarization where the electric field vector rotates
around the axis of propagation).
鈥
monochromatic
: in an electromagnetic wave, the condition of a wave
having a single frequency, a single wavelength (in contrast to being a
mixture of different wavelengths). In a more sophisticated view, it
means that there is only one Fourier component.
A. The Curl as a Determinant
Recall that the cross-product of two vectors
~
C
and
~
D
can be written
as the determinant
~
C
脳
~
D
=
炉
炉
炉
炉
炉
炉
藛
x
藛
y
藛
z
C
x
C
y
C
z
D
x
D
y
D
z
炉
炉
炉
炉
炉
炉
10
MISN-0-210
7
The
x
-component of vector
~
C
脳
~
D
is the term in the expanded determinant
which is proportional to 藛
x
:
(
~
鈭 脳
~
D
)
x
=
鈭侱
z
鈭倅
鈭
鈭侱
y
鈭倆
The other two components of the curl of
D
are thus:
(
~
鈭 脳
~
D
)
y
=
鈭侱
x
鈭倆
鈭
鈭侱
z
鈭倄
(
~
鈭 脳
~
D
)
z
=
鈭侱
y
鈭倄
鈭
鈭侱
x
鈭倅
B. Proof of a Vector Identity
We here prove:
~
A
脳
(
~
B
脳
~
C
) =
鈭
(
~
A
路
~
B
)
~
C
+
~
B
(
~
A
路
~
C
)
.
We only need to prove the identity for one component since the others
will follow by cycling the subscripts. So we take the
x
-component of the
left side:
[
~
A
脳
(
~
B
脳
~
C
)]
x
=
A
y
(
~
B
脳
~
C
)
z
鈭
A
z
(
~
B
脳
~
C
)
y
=
A
y
(
B
x
C
y
鈭
B
y
C
x
)
鈭
A
z
(
B
z
C
x
鈭
B
x
C
z
)
=
鈭
(
A
y
B
y
+
A
z
B
z
)
C
x
+
B
x
(
A
y
C
y
+
A
z
C
z
)
=
鈭
(
A
x
B
x
+
A
y
B
y
+
A
z
B
z
)
C
x
+
B
x
(
A
y
C
y
+
A
z
C
z
+
A
x
C
x
)
=
鈭
(
~
A
路
~
B
)
C
x
+
B
x
(
~
A
路
~
C
)
hence:
~
A
脳
(
~
B
脳
~
C
) =
鈭
(
~
A
路
~
B
)
~
C
+
~
B
(
~
A
路
~
C
)
so substituting
~
鈭
for
~
A
and
~
B
:
~
鈭 脳
(
~
鈭 脳
~
C
) =
鈭
(
~
鈭 路
~
鈭
)
~
C
+
~
鈭
(
~
鈭 路
~
C
)
~
鈭 脳
(
~
鈭 脳
~
C
) =
鈭掆垏
2
~
C
+
~
鈭
(
~
鈭 路
~
C
) ,
and the identity is proved.
11
MISN-0-210
PS-1
PROBLEM SUPPLEMENT
陇
Warning
: First make sure you have done the first five (out of the six)
problems scattered through the text, marked like this warning. If you
skip any one of them, you will probably not be prepared for even the first
problem below.
Note: Problems 4-7 also occur in this module鈥檚
Model Exam
.
1. The electric field of a plane electromagnetic wave in vacuum is repre-
sented by:
E
x
= 0 ,
E
y
= 0
.
50 (N/C) cos
拢
2
.
09 m
鈭
1
(
x
鈭
ct
)
陇
,
E
z
= 0 .
a. Determine the wavelength, frequency, polarization, and propagation
vector of the wave.
Help: [S-4]
b. Determine
the
components
of
the
wave鈥檚
magnetic
field.
Help: [S-1]
2. Solve (a) and (b) of Problem 1 for the wave represented by:
E
x
= 0
,
E
y
= 0
.
50 (N/C) cos
拢
0
.
419 m
鈭
1
(
x
鈭
ct
)
陇
,
E
z
= 0
.
50 (N/C) cos
拢
0
.
419 m
鈭
1
(
x
鈭
ct
)
陇
.
3. Determine the components of the
~
E
- and
~
B
-fields which describe the
following electromagnetic waves that propagate along the positive
x
-
axis:
a. A wave whose plane of
~
E
-vibration makes an angles of 45
鈼
with the
positive
y
- and
z
-axes.
b. A wave whose plane of
~
E
-vibration makes an angle of 120
鈼
with the
positive
y
axis and an angle of 30.0
鈼
with the positive
z
-axis.
12
MISN-0-210
PS-2
4. Given these electric and magnetic fields:
E
x
=
E
0
cos
2
蟺
位
(
y
+
ct
),
E
y
= 0,
E
z
= 0,
B
x
= 0,
B
y
= 0,
B
z
= 0.
a. Determine whether or not these fields satisfy the wave equations.
[D]
b. Determine whether or not these fields satisfy Maxwell鈥檚 Equations.
[B]
c. If your answer to (a) and (b) is yes, what relationship must exist
between the
E
and
B
amplitudes? [A]
5. Given these electric and magnetic fields:
E
x
= 0,
E
y
=
E
0
sin[2
蟺谓
鲁
x
c
鈭
t
麓
],
E
z
= 0,
B
x
= 0,
B
y
= 0,
B
z
=
B
0
sin[2
蟺谓
鲁
x
c
鈭
t
麓
].
(a), (b), (c): Repeat Problem 1 using the above components.
Answers: (a) [G], (b) [E], and (c) [H].
6. With:
E
x
=
E
0
cos(
kz
鈭
蠅t
),
E
y
=
E
0
cos(
kz
鈭
蠅t
),
E
z
= 0,
write down the space-time dependence of the components of the mag-
netic field that will result in an electromagnetic wave that satisfies
both the wave equations and Maxwell鈥檚 equations. [C]
7. A plane-polarized monochromatic electromagnetic wave of frequency
谓
has the electric field polarized in the
z
-direction and the wave propa-
gates in the negative
y
-direction. Determine the components of
~
E
and
~
B
that satisfy Maxwell鈥檚 equations and the wave equations. [F]
13
MISN-0-210
PS-3
Brief Answers
:
1. a. 3.01 m, 1
.
00
脳
10
8
Hz, polarized in the 藛
y
direction,
~k
= 2
.
09 m
鈭
1
藛
x
.
b.
B
x
= 0
,
B
y
= 0
,
B
z
= 0
.
17
脳
10
鈭
8
T cos
拢
2
.
09 m
鈭
1
(
x
鈭
ct
)
陇
.
2 a. 15.0 m, 2
.
00
脳
10
7
Hz, polarized in the
y
-
z
plane,
~k
= 0
.
419 m
鈭
1
藛
x
.
b.
B
x
= 0 ,
B
y
=
鈭
0
.
17
脳
10
鈭
8
T cos[0
.
419 m
鈭
1
(
x
鈭
ct
)] ,
B
z
= +0
.
17
脳
10
鈭
8
T cos[0
.
419 m
鈭
1
(
x
鈭
ct
)] ,
3 a.
E
x
= 0,
E
y
= +0
.
707
E
0
cos(
kx
鈭
蠅t
).
E
z
= +0
.
707
E
0
cos(
kx
鈭
蠅t
).
B
x
= 0,
B
y
=
鈭
0
.
707(
E
0
/c
) cos(
kx
鈭
蠅t
).
B
z
= +0
.
707(
E
0
/c
) cos(
kx
鈭
蠅t
).
b.
E
x
= 0,
E
y
=
鈭
0
.
500
E
0
cos(
kx
鈭
蠅t
),
E
z
= +0
.
866
E
0
cos(
kx
鈭
蠅t
).
B
x
= 0,
B
y
=
鈭
0
.
866(
E
0
/c
) cos(
kx
鈭
蠅t
).
B
z
=
鈭
0
.
500(
E
0
/c
) cos(
kx
鈭
蠅t
),
A. No such wave exists.
B. No.
C.
B
x
=
鈭
E
0
c
cos(
kz
鈭
蠅t
),
B
y
= +
E
0
c
cos(
kz
鈭
蠅t
),
B
z
= 0.
D. Yes.
E. Yes.
14
MISN-0-210
PS-4
F.
E
x
= 0,
E
y
= 0,
E
z
= +
E
0
cos
h
2
蟺谓
鲁
y
c
+
t
麓i
,
B
x
=
鈭
E
0
c
cos
h
2
蟺谓
鲁
y
c
+
t
麓i
,
B
y
= 0,
B
z
= 0.
G. Yes.
H.
B
0
=
E
0
/c
.
15
MISN-0-210
AS-1
SPECIAL ASSISTANCE SUPPLEMENT
S-1
(from PS-Problem 1)
For electromagnetic waves, the three important directions are: 藛
k
, the
direction of propagation; 藛
E
, the direction of the electric field; and 藛
B
,
the direction of the magnetic field. Any two of these may be known in
a problem and we must find the third. Since these three are mutually
perpendicular, they obey this cyclic rule:
藛
E
脳
藛
B
= 藛
k ,
藛
k
脳
藛
E
= 藛
B ,
藛
B
脳
藛
k
= 藛
E ,
where each line is obtained from the one above it by cycling the vectors
one place to the right.
S-2
(from TX-4b)
Recall that
~r
=
x
藛
x
+
y
藛
y
+
z
藛
z
and
~k
路
~r
=
k
x
x
+
k
y
y
+
k
z
z
. Then:
~
鈭
x
(
~k
路
~r
) =
鈭
鈭倄
(
~k
路
~r
) =
k
x
.
and similarly for
~
鈭
y
and
~
鈭
z
. Then:
~
鈭
(
~k
路
~r
) =
~k .
This means that:
~
鈭
cos(
~k
路
~r
鈭
蠅t
) =
鈭
~k
sin(
~k
路
~r
鈭
蠅t
)
.
Now you fill in the remaining steps to get:
鈭
2
~
E
=
鈭
k
2
~
E .
and do a similar job on the other side of the wave equation.
16
MISN-0-210
AS-2
S-3
(from TX-4a)
The various partial derivatives, (
鈭/鈭倀
), (
鈭/鈭倄
), etc., are independent of
each other so can be taken in any order. Thus, for example,
~
鈭
鈭
鈭倀
f
(
~r, t
) =
鈭
鈭倀
~
鈭
f
(
~r, t
)
,
where
f
is any function.
S-4
(from PS-Problem 1)
(
~k
路
~r
) is written in the problem as (2
.
09 m
鈭
1
x
). The obvious conclusion is
that
k
y
=
k
z
= 0 and hence that
k
=
q
k
2
x
+
k
2
y
+
k
2
z
=
k
x
= 2
.
09 m
鈭
1
.
17
MISN-0-210
ME-1
MODEL EXAM
~
鈭
f
= 藛
x
鈭俧
鈭倄
+ 藛
y
鈭俫
鈭倅
+ 藛
z
鈭俬
鈭倆
~
鈭 路
~g
=
鈭俫
x
鈭倄
+
鈭俫
y
鈭倅
+
鈭俫
z
鈭倆
~
鈭 脳
~g
=
碌
鈭俫
y
鈭倄
鈭
鈭俫
x
鈭倅
露
藛
z
+
碌
鈭俫
z
鈭倅
鈭
鈭俫
y
鈭倆
露
藛
x
+
碌
鈭俫
x
鈭倆
鈭
鈭俫
z
鈭倄
露
藛
y
~
鈭 脳
(
~
鈭 脳
~g
) =
鈭掆垏
2
~g
+
~
鈭
(
~
鈭 路
~g
) ;
鈭
2
鈮
~
鈭 路
~
鈭
~
鈭 路
~
E
= 4
蟺k
e
蟻
~
鈭 路
~
B
= 0
~
鈭 脳
~
E
=
鈭
鈭 ~
B
鈭倀
~
鈭 脳
~
B
= 4
蟺k
m
j
+
c
鈭
2
鈭 ~
E
鈭倀
1. See Output Skills K1-K2 in this module鈥檚
ID Sheet
.
2. Given these electric and magnetic fields:
E
x
=
E
0
cos
2
蟺
位
(
y
+
ct
),
E
y
= 0,
E
z
= 0,
B
x
= 0,
B
y
= 0,
B
z
= 0.
a. Determine whether or not these fields satisfy the wave equations.
b. Determine whether or not these fields satisfy Maxwell鈥檚 Equations.
c. If your answer to (a) and (b) is yes, what relationship must exist
between the
E
and
B
amplitudes?
3. Given these electric and magnetic fields:
E
x
= 0,
E
y
=
E
0
sin[2
蟺谓
鲁
x
c
鈭
t
麓
],
E
z
= 0,
B
x
= 0,
B
y
= 0,
B
z
=
B
0
sin[2
蟺谓
鲁
x
c
鈭
t
麓
].
(a), (b), (c): Repeat Problem 1 using the above components.
18
MISN-0-210
ME-2
4. With:
E
x
=
E
0
cos(
kz
鈭
蠅t
),
E
y
=
E
0
cos(
kz
鈭
蠅t
),
E
z
= 0,
write down the space-time dependence of the components of the mag-
netic field that will result in an electromagnetic wave that satisfies
both the wave equations and Maxwell鈥檚 equations.
5. A plane-polarized monochromatic electromagnetic wave of frequency
谓
has the electric field polarized in the
z
-direction and the wave propa-
gates in the negative
y
-direction. Determine the components of
~
E
and
~
B
that satisfy Maxwell鈥檚 equations and the wave equations.
Brief Answers
:
1. See this module鈥檚
text
.
2. See Problem 4 in this module鈥檚
Problem Supplement
.
3. See Problem 5 in this module鈥檚
Problem Supplement
.
4. See Problem 6 in this module鈥檚
Problem Supplement
.
5. See Problem 7 in this module鈥檚
Problem Supplement
.
19
20