PHY2061
R. D. Field
Exam 1 Solutions
Page 1 of
8
October 1, 2002
PHY 2061 Exam 1 Solutions
Problem 1 (20 points):
Circle true or false for following (2 points each).
(a) (True or False)
Unlike gravity the electrostatic force can be both attractive and
repulsive.
(b) (True or False)
Electric field lines never cross.
(c) (True or False)
Positively charged particles move from low to high potential.
(d) (True or False)
In a region of space where the electric potential is constant there can
be no charged particles.
(e) (True or False)
The vector function
z
x
y
y
x
z
z
y
x
E
ˆ
ˆ
ˆ
)
,
,
(
+
+
=
r
represents a possible
electrostatic field with magnitude given by
2
2
2
)
,
,
(
z
y
x
z
y
x
E
+
+
=
. Note that
0
=
×
∇
E
r
r
and hence this vector function can represent an electrostatic field.
(f) (True or False)
The scalar function V(x,y,z) = x
2
+y
2
+z
2
represents a possible electric
potential for an electrostatic field with magnitude given by
2
2
2
2
)
,
,
(
z
y
x
z
y
x
E
+
+
=
.
(g) (True or False)
The scalar function V(x,y,z) = x
2
yz represents a possible electric
electrostatic field in a vacuum. Note that
0
2
0
≠
∇
−
=
V
ε
ρ
and hence these is a non zero
charge density which cannot be true in a vacuum.
(h) (True or False)
The net electric flux through a closed surface is always positive or
zero.
(i) (True or False)
The divergence of the electric field is always zero.
(j) (True or False)
The electric field has energy but no mass.
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View Full DocumentPHY2061
R. D. Field
Exam 1 Solutions
Page 2 of
8
October 1, 2002
Problem 2 (20 points):
Three point charges are placed on the yaxis with
Q
at
y = d/2,
Q
at
y = d/2
, and
+2Q at y = 0
, as shown in the
Figure
.
Part A (3 points):
What is the electric potential,
V(x)
, on the xaxis a distance
x
from
the origin?
(
Express your answer in terms of K, Q, d, and x
.
and
take V = 0 at infinity
)
Answer:
2
2
)
2
/
(
2
2
)
(
d
x
KQ
x
KQ
x
V
+
−
=
Solution:
The electric potential at the point
x
on the xaxis is the
superposition of the electric potentials from each of the three
charges as follows:
2
2
)
2
/
(
2
2
2
)
(
d
x
KQ
x
KQ
r
KQ
r
KQ
x
KQ
x
V
+
−
=
−
−
=
,
where I used and
2
2
)
2
/
(
d
x
r
+
=
.
Part B (3 points):
Which of the following gives the value of the electric potential on the xaxis a point
A
a
distance
x = d
from the origin?
(
circle one answer and show your work, and take V = 0
at infinity
)
(a)
0.67 KQ/d
(b)
0.21 KQ/d
(c) 0.67 KQ/d
(d)
0.21 KQ/d
(e)
1.5 KQ/d
(f)
KQ/d
(g)
zero
(h)
none
Solution:
From Part A the electric potential at the point
A
on the xaxis is given by:
d
KQ
d
KQ
d
d
KQ
d
KQ
d
x
V
21
.
0
5
2
1
2
)
2
/
(
2
2
)
(
2
2
=
−
=
+
−
=
=
,
Part C (3 points):
What is the electric potential,
V(y)
, on the yaxis a distance
y > d/2
from the origin?
(
Express your answer in terms of K, Q, d, and y
.
and take V = 0 at infinity
)
Answer:
)
)
2
/
(
(
2
)
(
2
2
2
d
y
y
KQd
y
V
−
−
=
Solution:
The electric potential at the point
y
on the yaxis is the superposition of the
electric potentials from each of the three charges as follows:
)
)
2
/
(
(
2
)
2
/
(
2
2
)
2
/
(
)
2
/
(
2
)
(
2
2
2
2
2
d
y
y
KQd
d
y
KQy
y
KQ
d
y
KQ
d
y
KQ
y
KQ
y
V
−
−
=
−
−
=
+
−
−
−
=
.
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