PS121
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics
8.02
Fall 2008
Label with your name, table number and group. Turn in at the homework box.
Problem Set 12
Due: Tuesday, December 2
nd
at 11:30 am
Analytic Problems…
Problem 1: Coaxial Cable and Power Flow
A coaxial cable consists of two concentric long
hollow cylinders of zero resistance; the inner
has radius
a
, the outer has radius
b
, and the
length of both is
l
, with
l
>>
b
, as shown in
the figure. The cable transmits DC power from
a battery to a load.
The battery provides an
electromotive force
ε
between the two
conductors at one end of the cable, and the load
is a resistance
R
connected between the two
conductors at the other end of the cable.
A current
I
flows down the inner conductor and back
up the outer one. The battery charges the inner conductor to a charge
Q
−
and the outer
conductor to a charge
Q
+
.
(a) Find the direction and magnitude of the electric field
E
G
everywhere.
(b) Find the direction and magnitude of the magnetic field
B
G
everywhere.
(c) Calculate the Poynting vector
S
G
in the cable.
(d) By integrating
S
G
over appropriate surface, find the power that flows into the coaxial cable.
(e) How does your result in (d) compare to the power dissipated in the resistor?
Problem 2: Electromagnetic Plane Wave
An electromagnetic plane wave is propagating in vacuum has a magnetic field given by
0
10
1
ˆ
()
(
)
0
u
B f ax bt
f u
else
<
<
⎧
=+
=
⎨
⎩
Bj
G
The wave encounters an infinite, dielectric sheet at
x
= 0 of such a thickness that half of the
energy of the wave is reflected and the other half is transmitted and emerges on the far side of
the sheet.
(a)
What condition between
a
and
b
must be met in order for this wave to satisfy Maxwell’s
equations?
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(b)
What is the magnitude and direction of the
E
G
field of the incoming wave?
(c)
What is the magnitude and direction of the energy flux (power per unit area) carried by the
incoming wave, in terms of
B
0
and universal quantities?
(d)
What is the pressure (force per unit area) that this wave exerts on the sheet while it is
impinging on it?
Problem 3: Formation of a Standing Wave
Two waves traveling in opposite directions produce a standing wave, as discussed in Section
13.5 of the
Course Notes
. Let the individual wave functions be
12
(5.0 cm)sin(3.0
2.0 ),
(5.0 cm)sin(3.0
2.0 )
yx
t
t
=−
=+
where
x
and
y
are measured in centimeters.
(a)
By applying superposition principle, find the resultant wave function.
(b) What is the amplitude of the resultant simple harmonic motion at
3.0 cm
x
=
?
(c) Find the positions of the nodes (the points of zero amplitude) and the antinodes (the
maximum amplitude points).
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 Fall '08
 SCIOLLA
 RLC

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