# [SSM] True or false: (a) Maxwell's equations apply only to electric ...

Electronics - Devices

Oct 18, 2013 (4 years and 6 months ago)

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[SSM]

True or false:

(
a
)

Maxwell’s equations apply only to
electric and magnetic
fields that are constant over time.

(
b
)

The
electromagnetic
wave equation can be derived from Maxwell’s equations.

(
c
)

Electromagnetic waves are transverse waves.

(
d
)

The el
ectric and magnetic fields of an electromagnetic wave in free space are in phase.

(
a
) False. Maxwell’s equations apply to both time
-

-

E
b
) True. One can use Faraday’s law and the modified version of Ampere’s law to

E
c
) True. Both the electric and magnetic fields of an electromagnetic wave oscillate
at right angles to the direction of propagation of the wave.

(
d
) True.

8

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Determine the Concept

We can that
has units of W/m
2

by
substituting the SI units of
,

and
and simplifying the resulting
expression.

30

An electromagnetic wave has an intensity of 100 W/m
2
. Find its

(
a
) rms electric field strength, and (
b
) rms magnet
ic field strength.

Picture the Problem

We can use
P
r

=
I
/
c

to find the radiation pressure. The
intensity of the electromagnetic wave is related to the rms values of its electric
and magnetic field strengths according to
I

=
E
rms
B
rms
/

0
, where
B
rms

=
E
rm
s
/
c.

(
a
) Relate the intensity of the
electromagnetic wave to
E
rms

and
B
rms
:

or, because
B
rms

=
E
rms
/
c
,

Solv
ing

for
E
rms

yields
:

32

The rms value of an electromagnetic wave’s e
lectric field strength is
400 V/m. Find the wave’s (
a
) rms magnetic field strength, (
b
) average energy
density, and (
c
) intensity.

Picture the Problem

Given
E
rms
, we can find
B
rms

using
B
rms

=
E
rms
/
c
. The
average energy density of the wave is given by
u
av

=
E
rms
B
rms
/

0
c

and the
intensity of the wave by
I

=
u
av
c

.

(
a
) Express
B
rms

in terms of
E
rms
:

Substitute numerical values and
evaluate
B
rms
:

(
b
) The average energy density
u
av

is given by:

Substitute numerical values and
evaluate
u
av
:

(
c
) Express the intensity as the
product of the average energy
density and the speed of light in a
vacuum:

Substitute numerical values and
eva
luate
I
:

37

••

[SSM]

An electromagnetic plane wave has an

electric field
that is
parallel to the
y
axis, and has a
Poynting vector
that
is given by
, where
x
is in meters,
k

= 3.00

10
9

ra
d/s, and
t
is in seconds
. (
a
) What is the direction of propagation of
the wave? (
b
) Find the wavelength and frequency of the wave. (
c
) Find the electric
and magnetic fields of the wave as functions of
x

and
t
.

Picture the Problem

We can determine the dire
ction of propagation of the
wave, its wavelength, and its frequency by examining the argument of the cosine
function. We can find
E

from
and
B

from
B

=
E
/
c
. Finally, we can
use the definition of the Poynting vector and the given expr
ession for
to find
and
.

(
a
) Because the argument of the cosine function is of the form
, the wave
propagates in the +
x

direction.

(
b
) Examining the argument of
the
cosine function, we note that the
wave number
k

of the wave is:

Examining the argument of the
cosine function, we note that the
angular frequency

of the wave
is:

Solv
ing

for
f

yields
:

(
c
) Express the magnitude of
in
terms of
E
:

Substitute numerical values and evaluate
E
:

Because

and
:

where
k

= 3.00

10
9

.

Use
B

=
E
/
c

to evaluate
B
:

Because
, the direction
of
must be such that th
e cross
product of

with
is in the
positive
x

direction:

where
k

= 3.00

10
9

.

42

Show by direct substitution that Equation 30
-
8
a

is satisfied by the wave
f
unction

where
c

=

/
k
.

Picture the Problem

We can show that Equation 30
-
8
a

is satisfied by the wave
function
E
y

by showing that the ratio of

2
E
y
/

x
2

to

2
E
y
/

t
2

is 1/
c
2

where
c

=

/
k
.

Differentiate

wi
th respect to
x
:

Evaluate the second partial
derivative of
E
y

with respect to
x
:

(1)

Differentiate

with respect to
t
:

Evaluate the second partial

derivative of
E
y

with respect to
t
:

(2)

Divide equation (1) by equation (2)
to obtain:

or

provided
c

=

/
k
.

43

Use the values of

0

and

in SI units to

compute

and show
that it is
equal to

3.00

10
8

m/s.

Picture the Problem
Substitute numerical values and evaluate
c
:

44

••

(
a
) Use Maxwell’s equations to show for a plane wave, in which

and

are independent of
y

and z, that

and
.

(
b
) Show that
E
z

and
B
y

also satisfy the wave equation.

Picture the Problem
We can use Figures 30
-
5 and 30
-
6 and a derivation simi
lar
to that in the text to obtain the given results.

In Figure 30
-
5, replace
B
z

by
E
z
.
For

x

small:

Evaluate the line integral of

around the rectangular area

x

z
:

(1)

Express the magnetic flux through
the same area:

Substitute in equation (1) to obtain:

or

In Figure 30
-
6, replace
E
y

b
y
B
y

and
evaluate the line integral of

around the rectangular area

x

z
:

provided there are no conduction
currents.

Evaluate these integrals to obtain:

(
b
) Using the first result

obtained in
(
a
), find the second partial
derivative of
E
z

with respect to
x
:

or

Use the second result obtained in
(
a
) to obtain:

or, because

0

0

= 1/
c
2
,

.

Using the second result obtained in
(
a
), find the second partial
derivative of
B
y

with respect to
x
:

or

Use the first result obtained in (
a
)
to obtain:

or, because

0

0

= 1/
c
2
,

.

46

An electromagnetic wave has a frequency of 100 MHz and is traveling
in a vacuum. The magnetic field is given by
.
(
a
) Find the wavelength and the direction of propagation of this wave. (
b
) Find the
elect
ric field vector
. (
c
) Determine the Poynting vector, and use it to find
the intensity of this wave.

Picture the Problem

We can use
c

=
f

to find the wavelength. Examination of
the argument of the cosine function will reveal the di
rection of propagation of
the wave. We can find the magnitude, wave number, and angular frequency of
the electric vector from the given information and the result of (
a
) and use these
results to obtain
(
z
,
t
). Finally, we can use its
definition to find the Poynting
vector.

(
a
) Relate the wavelength of
the wave to its frequency and
the speed of light:

Substitute numerical values
and evaluate

:

From the sign of the argument of the co
sine function and the spatial dependence
on
z,

we can conclude that the wave propagates in the +
z

direction.

(
b
) Express the amplitude of
:

Find the angular frequency
and wave number of the
wave:

and

Because
is in the positive
z

direction,
must be in the negative
y

direction in
order to satisfy the Poynting vector expression:

(
c
) Use its

definition to express and evaluate the Poynting vector:

or

The intensity of the wave is the
average magnitude of the
Poynting vector. The average
value of the square of the
cosine function is 1/2:

5
0

••

A 20
-
kW beam of
electromagnetic
radiation is normal to a surface that
reflects 50 percent of the radiation. What is the force
on
this surface?

Picture the Problem

The total force on the surface is the sum
of the force due to
the reflected radiation and the force due to the absorbed radiation. From the
conservation of momentum, the force due to the 10 kW that are reflected is twice
the force due to the 10 kW that are absorbed.

Express the total force on th
e
surface:

Substitute for
F
r

and
F
a
to
obtain:

Substitute numerical values and
evaluate
F
tot
:

5
1

••

[SSM]

The electric fields of two harmonic electromagnetic waves of
angular fre
quency

1

and

2

are given by

and
by
. For the resultant of these two waves, find (
a
) the
instantaneous Poynting vector and (
b
) the time
-
averaged Poynting vector.

(
c
) Repeat Parts (
a
) and (
b
) if the
direction of propagation of the second wave is
reversed so that

Picture the Problem

We can use the definition of the Poynting vector and the
relationship between
and
to find the in
stantaneous Poynting vectors for
each of the resultant wave motions and the fact that the time average of the cross
product term is zero for

1

2
, and ½ for the square of cosine function to find
the time
-
averaged Poynting vectors.

(
a
) Because both wav
es propagate
in the
x

direction:

Express
B

in terms of
E
1

and
E
2
:

Substitute for
E
1

and
E
2
to obtain:

The instantaneous Poynting vector for the res
ultant wave motion is given by:

(
b
) The time average of the cross
product term is zero for

1

2
,
and the time average of the square
of the cosine terms is ½:

(
c
) In this case
b
ecause the wave with
k

=
k
2

propagates in the
direction. The magnetic field is then:

The instantaneous Poynting vector for the resultant wave motion is given by:

The time averag
e of the square of
the cosine terms is ½:

5
2

••

Show that

(Equation 30
-
10) follows from

(Equation 30
-
6
d

with

I

= 0) by integrating along a
suitable curve
C

and over a suitable
surface
S

in a manner that parallels the
derivation of Equation 30
-
9
.

Picture the Problem

We’ll choose the
curve with sides

x

and

z

in the
xy

plane shown in the diagram and apply
Equation 30
-
6
d

to show that

.

Because

x

is v
ery small, we
can approximate the difference
in
B
z

at the points
x
1

and
x
2

by:

Then:

The flux of the electric field through
this curve is approximately:

btain:

or

5
4

••

The intensity of the sunlight striking Earth’s upper atmosphere is

1.37 kW/m
2
. (
a
) Find the rms values of the magnetic and electric fields of this
light. (
b
) Find the average power output
of the Sun. (
c
) Find the intensity and the
radiation pressure at the surface of the Sun.

Picture the Problem

We can use
I

=
E
rms
B
rms
/

0

and
B
rms

=
E
rms
/
c

to express
E
rms

in terms of
I
. We can then use
B
rms

=
E
rms
/
c

to find
B
rms
. The average power
output o
f the
S
un is given by
where
R

is the Earth
-
Sun distance. The
intensity and the radiation pressure at the surface of the sun can be found from
the definitions of these physical quantities.

(
a
) The intensity of the radiation
is given
by:

Substitute numerical values and evaluate
:

Use

to evaluate
:

(
b
) Express the
average power
output of the Sun in terms of the
solar constant:

where
R

is the earth
-
sun distance.

Substitute numerical values and
evaluate
P
av
:

(
c
) Express the intensity at the
surface of the Sun in te
rms of
the sun’s average power output
r
:

Substitute numerical values (see
Appendix B for the radius of the
Sun) and evaluate
I

at the
surface of the Sun: