CHAPTER 3
3.1. An empty metal paint can is placed on a marble table, the lid is removed, and both parts are discharged
(honorably) by touching them to ground. An insulating nylon thread is glued to the center of the lid,
and a penny, a nickel, and a dime are glued to the thread so that they are not touching each other. The
penny is given a charge of
+
5 nC, and the nickel and dime are discharged. The assembly is lowered
into the can so that the coins hang clear of all walls, and the lid is secured. The outside of the can is
again touched momentarily to ground. The device is carefully disassembled with insulating gloves and
tools.
a) What charges are found on each of the five metallic pieces? All coins were insulated during the
entire procedure, so they will retain their original charges: Penny:
+
5
nC; nickel: 0
; dime: 0
. The
penny’s charge will have induced an equal and opposite negative charge (5 nC) on the inside wall
of the can and lid. This left a charge layer of +5 nC on the outside surface which was neutralized
by the ground connection. Therefore, the can retained a net charge of
−
5 nC
after disassembly.
b) If the penny had been given a charge of
+
5 nC, the dime a charge of
−
2 nC, and the nickel a
charge of
−
1 nC, what would the final charge arrangement have been? Again, since the coins are
insulated, they retain their original charges. The charge induced on the inside wall of the can and
lid is equal to negative the sum of the coin charges, or
−
2 nC. This is the charge that the can/lid
contraption retains after grounding and disassembly.
3.2. A point charge of 12 nC is located at the origin. four uniform line charges are located in the
x
=
0
plane as follows: 80 nC/m at
y
= −
1 and
−
5 m,
−
50 nC/m at
y
= −
2 and
−
4 m.
a) Find
D
at
P (
0
,
−
3
,
2
)
: Note that this point lies in the center of a symmetric arrangement of line
charges, whose fields will all cancel at that point. Thus
D
arise from the point charge alone, and
will be
D
=
12
×
10
−
9
(
−
3
a
y
+
2
a
z
)
4
π(
3
2
+
2
2
)
1
.
5
= −
6
.
11
×
10
−
11
a
y
+
4
.
07
×
10
−
11
a
z
C
/
m
2
= −
61
.
1
a
y
+
40
.
7
a
z
pC
/
m
2
b) How much electric flux crosses the plane
y
= −
3 and in what direction? The plane intercepts all
flux that enters the
−
y
halfspace, or exactly half the total flux of 12 nC. The answer is thus 6 nC
and in the
−
a
y
direction.
c) How much electric flux leaves the surface of a sphere, 4m in radius, centered at
C(
0
,
−
3
,
0
)
? This
sphere encloses the point charge, so its flux of 12 nC is included. The line charge contributions
are most easily found by translating the whole assembly (sphere and line charges) such that the
sphere is centered at the origin, with line charges now at
y
= ±
1 and
±
2. The flux from the line
charges will equal the total line charge that lies within the sphere. The length of each of the inner
two line charges (at
y
= ±
1) will be
h
1
=
2
r
cos
θ
1
=
2
(
4
)
cos
sin
−
1
1
4
=
1
.
94 m
That of each of the outer two line charges (at
y
= ±
2) will be
h
2
=
2
r
cos
θ
2
=
2
(
4
)
cos
sin
−
1
2
4
=
1
.
73 m
27
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3.2c. (continued)
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 Spring '08
 Brown
 Charge, Electromagnet, Electric charge, charge density

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