PHY2061
R. D. Field
Exam 1 Solutions
Page 1 of
10
February 12, 2002
PHY 2061 Exam 1 Solutions
Problem 1 (10 points):
Circle true or false for following (1 point each).
(a) (True or False)
Gravity is much stronger than the electrostatic force.
(b) (True or False)
Electric field line do not begin or end in chargefree regions (except
at °infinity±).
(c) (True or False)
Where electric field lines are dense the electric field is weak.
(d) (True or False)
An electric dipole produces an electric field that is not zero anywhere
except at °infinity±.
(e) (True or False)
Two positive point charges separated by a distance d produce an
electric field that is not zero anywhere except at °infinity±.
(f) (True or False)
A uniform infinite sheet of charge produces an electric field that has
the same magnitude everywhere in space.
(g) (True or False)
The vector function
y
x
x
y
y
x
E
²
²
)
,
(
−
=
r
represents a possible
electrostatic field with magnitude given by
2
2
)
,
(
y
x
y
x
E
+
=
.
Note that
0
²
2
≠
−
=
×
∇
z
E
r
r
and hence this vector function cannot represent an electrostatic field
since all electrostatic fields have
0
=
×
∇
E
r
r
.
(h) (True or False)
The scalar function V(x,y,z) = xyz represents a possible electric
potential for an electrostatic field with magnitude given by
2
2
2
2
2
2
)
,
,
(
y
x
z
x
z
y
z
y
x
E
+
+
=
.
(i) (True or False)
The kinetic energy and the electric potential energy are always
positive.
(j) (True or False)
In static equilibrium the electric field is always zero inside and on the
surface of a conductor.
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PHY2061
R. D. Field
Exam 1 Solutions
Page 2 of
10
February 12, 2002
Problem 2 (15 points):
Four identical point charges
+Q
are located at the corners
of a square with sides of length
L
that lies in the xyplane
with a
°2Q
point charge at its center and with its center
at the origin as shown in the
Figure
.
Part A (3 points):
What is the electric field vector,
)
(
z
E
r
,
at a point
P
on the
zaxis a distance
z
from the origin? (
Express your answer
in terms of K, Q, L, and z and use vector notation
)
Answer:
z
z
L
z
z
KQ
E
²
2
)
2
/
(
4
2
2
/
3
2
2
−
+
=
r
Solution:
The electric field at the point
P
is the superposition of the electric fields from
each of the five charges as follows:
z
z
L
z
z
KQ
z
z
KQ
z
r
KQ
E
²
2
)
2
/
(
4
²
2
²
cos
4
2
2
/
3
2
2
2
2
−
+
=
−
=
θ
r
,
where I used cos
θ
= z/r and
2
/
2
2
L
z
r
+
=
.
Part B (3 points):
Which of the following gives the zcomponent of the electric field on the zaxis a
distance
z = L
from the origin?
(
circle one answer and show your work
)
(a)
2.18 KQ/L
2
(b)
4.0 KQ/L
2
(c)
1.43 KQ/L
2
(d)
2.0 KQ/L
2
(e)
1.73 KQ/L
2
(f)
0.18 KQ/L
2
(g)
5.66 KQ/L
2
(h)
none
Solution:
From Part A we have
2
2
/
3
2
2
2
/
3
2
2
18
.
0
2
)
2
/
3
(
4
2
)
2
/
(
4
L
KQ
L
KQ
z
L
z
z
KQ
E
L
z
z
=
−
→
−
+
=
=
,
Part C (3 points):
What is the magnitude of the electric field on the zaxis when
z
becomes very large
compared to the length,
L
, of the sides of the square?
(
Hint
: take the limit of the electric
field when
z >> L
)
Answer:
2KQ/z
2
Solution:
From Part A we have
2
2
2
/
3
2
2
2
2
)
2
/
(
4
z
KQ
z
L
z
z
KQ
E
L
z
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