MAXWELL’S EQUATIONS AND
ELECTROMAGNETIC WAVES
29
E
XERCISES
Section 29.2 Ambiguity in Ampère’s Law
13.
I
NTERPRET
In this problem we are asked to find the displacement current through a surface.
D
EVELOP
As shown in Equation 29.1, Maxwell’s displacement current is
0
0
0
(
)
E
d
d
d EA
dE
I
A
dt
dt
dt
ε
ε
ε
Φ
=
=
=
E
VALUATE
The above equation gives
2
12
2
2
0
(8.85
10
C /N
m
)(1 cm
)(1.5 V/m
s)
1.33 nA
d
dE
I
A
dt
ε
μ
−
=
=
×
⋅
⋅
=
A
SSESS
Displacement current arises from changing electric flux and has units of amperes (A), just like ordinary
current.
14.
The electric field is approximately uniform in the capacitor, so
0
( / )
, and
/
E
D
EA
V d A
I
t
E
φ
ε
φ
=
=
=
∂
∂ =
12
2
0
(
/
)
/
(8.85
10
F/m)(10 cm)
(220 V/ms)/(0.5 cm)
3.89 A.
A d dV dt
ε
μ
−
=
×
=
Section 29.4 Electromagnetic Waves
15.
I
NTERPRET
We are given the electric and magnetic fields of an electromagnetic wave and asked to find the
direction of propagation.
D
EVELOP
The direction of propagation of the electromagnetic wave is the same as the direction of the cross
product
G
G
.
E
B
×
E
VALUATE
When
E
G
is parallel to
and
ˆ
j
B
G
is parallel to
the direction of propagation is parallel to
ˆ
,
i
,
E
B
×
G
G
or
ˆ
늿
j
i
k
×
.
= −
A
SSESS
For electromagnetic waves in vacuum, the directions of the electric and magnetic fields, and of wave
propagation, form a right-handed coordinate system.
16.
(a)
The peak amplitude is the magnitude of
which is
늿
(
E i
j
+
),
2.
E
Note that
늿
ˆ
2 ,
i
j
n
+
=
where
is a unit vector
45
°
between the positive
x
and
y
axes.
(b)
When
ˆ
n
E
G
is parallel to
(for
ˆ
n
sin(
)
kz
t
ω
−
positive)
points 45
°
into the
second quadrant (so that
Thus,
B
G
, and
is in the
direction).
E
B
E
B
z
⊥
×
+
G
G
G
G
B
G
is parallel to the unit
vector
늿
(
)
/
2.
i
j
− +
0
Section 29.5 Properties of Electromagnetic Waves
17.
I
NTERPRET
This problem is about measuring the distance between the Sun and the Earth using light-minutes.
D
EVELOP
A light-minute (abbreviated as c-min) is approximately equal to
8
1
1 c-min
(3
10
m/s)(60 s)
1.8
10
m
=
×
=
×
On the other hand, the mean distance of the Earth from the Sun (an Astronomical Unit) is about
11
1.5
10
m.
SE
R
=
×
E
VALUATE
In units of c-min,
can be rewritten as
SE
R
11
10
1c-min
(1.5
10
m)
8.33 c-min
1.8
10
m
SE
R
=
×
=
×
29.1

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