-
-------------------------Terminal
Examination Paper A, Electromagnetic Field and
Waves-------------------------
Beijing
Jiaotong University T
erminal
Examination (Paper A)
(the second term,
2011-2012 academic year)
Course Name:
electromagnetic fields and waves
Teacher: Wei Y
an
Class__________________Student
ID____________Name___________
Question
Number
1
11
2
12
3
13
4
14
5
15
6
16
7
17
8
18
9
19
10
20
Total
Scores
Scores
Question
Number
Scores
---
--------------------------------------------------
--------------------------------------------------
-------------
Formula
of
operations
on
scalar
and
vector
fields
in
cylindrical
coordinates and
spherical coordinates
?
f
?
?
f
?
?
a
?
?
< br>1
?
f
?
?
?
?
f
?
?
a
?
?
?
f
?
z
a
z
?
?
f
?
r
2
< br>a
r
?
1
?
f
r
?
?
1
?
r
?
r
2
2
a
?
?
1
?
f
r
sin
?
?
?
1
2
< br>a
?
?
?
f
?
?
1
1
2
2
?
f
?
2
1
?
?
?
?
(
?
)
?
< br>1
?
f
2
?
2
?
?
2
?
?
f
?
z
2
?
(
r
2
?
f
?
r
)
?
< br>r
sin
?
?
< br>?
(sin
?
)
?
?
f
2
2
r
sin
?
?
?
?
?
A
?
1
?
(<
/p>
?
A
?
)
?
?
?
a
?
?
1
?
A
?
?
?
?
a
z
?
?
z
B
z
?<
/p>
?
A
z
?
z
?
1
?
(
r
A
r
)
r
2
?
r
r
a
?
?
?
?
rB
?
?
1
r
sin
?
?
(
A
?
sin
?
)
?
?
?
?
A
?
r
sin
?
?
?
a
?
?
?
< br>?
a
r
?
1
2
r
sin
?
a
?
?
?
?
r
sin
?
B
?
?
?
B
?
1
?
B
?
?
B
r
?
?
?
< br>r
sin
?
?
< br>r
?
B
?
-------------------------------------
--------------------------------------------------
-----------------------------
There are
4 choices marked A, B, C, and D in every question
and only
one choice is correct. Please
write the symbol of the choice which one
you
think
is
right
in
the
blank
in
each
question.
In
the
same
time,
you
must write down the process to obtain
the solution
. (5×
20=100
marks)
- 1 -
-------------------------Terminal
Examination Paper A, Electromagnetic Field and
Waves-------------------------
[1-3] A
point charge
q
is enclosed
in a linear, isotropic, and homogeneous
dielectric
medium
of
infinite
extent.
The
medium
has
a
relative
permittivity
ε
r
.
Suppose
the
point
charge
is
located
at
the origin
of
the
spherical coordinate
system.
1. The electric field intensity
E
can be expressed as
(A)
q
4
π
r
2
a
r
< br>
(B)
q
4
πε
a
r
0
r
2
a
r
(C)
q
4
πε
0
ε
r
r
2
Y
ou select (
).
Solution:
2. The polarization vector
P
can be expressed as
(A)
q
4
πε
2
(
ε
r
?
1)
a
r
< br>
(B)
q
0
r
4
πε
r
?
1)
a
r
r
r
2
(
ε
(C)
?
q
4
πε
(
ε
r
?
1)
a
r
(D)
?
q
r
r
2
4
πε
2
(
ε
r
?
1)
a
r
0
r
Y
ou
select (
).
Solution:
- 2
-
(D)
?
q
4
πε
0
ε
2
a
r
r
r
-------------------------Terminal
Examination Paper A, Electromagnetic Field and
Waves-------------------------
3.
Determine the total bound charge
Q
sb
, which
is on
the surface of the
dielectric next to
the point charge q.
(A)
?
q
(
ε
r
-1)/ε
r
(B)
?
q
/
ε
r<
/p>
(C)
q
(
ε
r
?
1)
(D)
q
Y
ou
select (
).
Solution:
[4].
The
plane
z=0
marks
the
boundary
between
free
space
and
a
dielectric
medium
with
a
relative
permittivity
of
ε
r
.
The
Electric
field
intensity
next
to
the
interface
in
free
space
is
E
< br>?
E
x
a
x
?
E
y
a
y
?
E
z
a
z
.
Determine the Electric field intensity
on the other side of the interface.
(A)
(C)
E
x
a
x
?
E
y<
/p>
a
y
?
E
z
a
z
(B)
(D)
E
x
/
ε
r
a
x<
/p>
?
E
y
a
y
?
E
z
a
z
E
x
a
x
?
E
y
a
y
?
E
z
/
ε
r<
/p>
a
z
E
x
a
x
?
E
y
/
ε
r
a
y
?
E
z
a
z
Y
ou select (
).
Solution:
- 3 -
-------------------------Terminal
Examination Paper A, Electromagnetic Field and
Waves-------------------------
[5].
A
spherical
capacitor
is
formed
by
two
concentric
metallic
spheres
with inner radius
a
and outer radius
b
,
b
>
a
. The region between the
two
concentric
spherical
shells
is
filled
with
a
dielectric
medium
with
a
relative permittivity of
ε
r
. Find the
capacitance of the capacitor.
(A)
4
πε
0
ab
b
?
a
(B)
4
πε
0
ε
r
ab
b
?
a
(C)
4
πε
0
ε
r
ab
b
?
a<
/p>
(D)
4
πε
0
ab
b
?
a
Y
ou select (
).
Solution:
[6].
A charged semicircular
ring of radius
b
extending
from
φ
=0 to
φ
=
π
lies in the x-y plane and is centered
at origin. If the charge distribution is
ksin(
φ
)
,
compute the electric field intensity at P(0,0,h).
(A)
kb
4
πε
0
(
h
?
b
)
kb
2
2
3
/
2
?
1
?
?
π
b
a
?
2
h
a
y
< br>z
?
?
?
2
?
?
1
?
?
π
b
a
?
2
h
a
y
z
?
?
?
2
?
< br>
(B)
(D)
kb
4
πε
0
(
h
?
b
)
kb
4
πε
0
(
h
?
b
)
2
2
3
/
2
2
< br>2
3
/
2
?
1
?
π
b
a
?
2
h
a
y
z
?
?
?
2
?
?
1
?
< br>?
π
b
a
?
2
h
a
y
z
?
?
?
2
?
(C) <
/p>
2
πε
0
(
h
?
b
)
2
2
3
/
2
Y
ou select (
).
Solution:
- 4
-
-------------------------Terminal
Examination Paper A, Electromagnetic Field and
Waves-------------------------
[7-8].
Charge
is uniformly
distributed inside an
infinite
long cylinder of
radius
a
. The volume charge density
is
ρ
v
.
7. Calculate the electric
field intensity at all points inside and outside
the
cylinder.
?
?<
/p>
v
?
?
2
ε
a
?
(
?
?
a
)
?
0
E
?
?
2
?
?
v
a
a
(
?<
/p>
?
a
)
?
2
ε
?
?
?
0
?
?
v
?
?
4
ε
a
?
(
?
?
a
)
?<
/p>
0
E
?
?
2
?
?
v
a
a
(
?
?
a
)
?
4
ε
?
?
?
0
E
?
(A
)
(B)
E
?
?
v
?<
/p>
2
ε
0
(C)
a
?
(D)
?
v
a
2
4
ε
0
?
a
?
Y
ou select
(
).
Solution:
8. Take
?
?
?
as the zero electric potential point.
Compute the electric
potential at all
points inside the cylinder.
(A)
(C)
?
v
< br>a
2
?
0
2
ln
?
?
?
v
4
?
0<
/p>
2
?
a
2
?
?
2
?
(B)
?
v
a
2
?
0
2
ln
?
?
v
2
?
0
?
v
4
?
0
?
a
?
?
2
?
(D)
?
v
a
2
?
0
2
ln
?
?
?
a
2
?
?
2
?
Y
ou select
(
).
Solution:
- 5
-
-------------------------Terminal
Examination Paper A, Electromagnetic Field and
Waves-------------------------
[9-10] A
charged ring of radius
a
carries a uniform charge distribution.
The linear charge density is
ρ
l
.
9.
The electric potential at point P (0, 0,
z
) on the axis of the ring
is
(A)
?
l
< br>a
4
??
0
a
?
z
2
2
(B)
?
l
a
2
?
0
a<
/p>
?
z
2
2
(C)
?
l
z
4
?
0
a
?
z
2
2
(D)
?
l
z
2
??
0
a
< br>?
z
2
2
Y
ou select (
).
Solution:
10. The electric field
intensity at point P (0, 0,
z
) on the axis of the ring
is
(A)
(C)
?
2
2
3
/
2
2
?
0
?
(
a
?
z<
/p>
)
?
l
a
?
z
?
?
a
z
?
(B)
(D)
?
2
2
3
/
2
2
?
0
?
(
a
?
z
)
?
< br>l
a
?
a
?
?
a
z
?
?
2
2
3
/
2
2
??
0
?
(
a
?
z
)
?
l
a
?
z
?
?
a
z
?
?<
/p>
2
2
3
/
2
4
?
0
?
(
a
?
z
)
?
l
a
?
z
?
?
a
z
?
Y<
/p>
ou select (
).
Solution:
- 6 -
-------------------------Terminal
Examination Paper A, Electromagnetic Field and
Waves-------------------------
[11-12]
We define an electric dipole as a pair of equal
charges of opposite
signs
that
are
very
close
together.
Assume
that
the
magnitude
of
each
charge
is
q
and
the
separation
between
them
is
d.
If
the
charges
are
symmetrically placed
along the z axis, and the point of observation P
(r,
θ,
φ
) is quite far away so that
r>>d, as illustrated in Figure P11.
Figure P11
11. The electric
potential at point P can be written as
(A)
qd
cos
?
< br>4
πε
0
r
2
(B)
qd
< br>cos
?
2
πε
0
r
2
(C)
q
d
cos
?
4
πε
0
r
(D)
qd
sin
?
4
πε
0
r
2
Y
ou select (
).
Solution:
12. Calculate the electric
field intensity at point P.
(A)
< br>(C)
E
?
qd
4
πε
0
r
< br>qd
4
πε
0
< br>r
2
2
[
a
r
2
cos
?
?
a
?
sin
?
]
(B)
(D)
- 7 -
E
?
qd
4
πε
0
r
qd
2
πε
0
r
3
3
[
a
r
2
co
s
?
?
a
?<
/p>
sin
?
]
<
/p>
E
?
[
a
r
cos
?
?
a
?
2
sin
?
]
E
?
[
a
r
2
cos
?
?
a
?
sin
?
]
Y
ou select
(
).
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