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23.84:
a) We have
:
So
).
3
(
)
,
,
(
2
2
2
z
y
x
A
z
y
x
V
+

=
k
j
i
k
j
i
E
ˆ
2
ˆ
6
ˆ
2
ˆ
ˆ
ˆ
Az
Ay
Ax
z
V
y
V
x
V

+

=
∂
∂

∂
∂

∂
∂

=
b) A charge is moved in along the
z
axis. So the work done is given by:
∫
∫
=
⇒
+
=

=
⋅
=
0
0
2
0
2
0
0
0
)
(
)
2
(
ˆ
z
z
qz
W
A
z
Aq
dz
Az
q
dz
q
W
k
E
.
m
V
640
m)
250
.
0
)(
C
10
5
.
1
(
J
10
00
.
6
2
2
6
5
=
×
×
=


A
c)
.
ˆ
m
V
320

ˆ
m)
250
.
0
(
)
m
V
640
(
2
)
250
.
0
,
0
,
0
(
2
k
k
E
=

=
d) In every plane parallel to the
y
z
x
,
plane

is constant, so:
,
)
,
,
(
2
2
2
2
2
R
A
C
V
z
x
C
Az
Ax
z
y
x
V
≡
+
=
+
⇒

+
=
which is the equation for a circle since
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