negatively charged paddle touches the
electroscope. (c) The negatively charged
paddle is removed.
(a)
(b)
(c)
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692
Chapter 21
Electrostatics
connection is removed in Figure 21.11d. Now, when the paddle is moved away from the
electroscope in Figure 21.11e, the electroscope is still positively charged (but with a smaller
deflection than in Figure 21.11b). The same process also works with a positively charged
paddle. This process is called
charging by induction
and yields an electroscope charge that
has the opposite sign from the charge on the paddle.
21.5
Electrostatic Force—Coulomb’s Law
The law of electric charges is evidence of a force between any two
charges at rest. Experiments show that for the electrostatic force
exerted by a charge
q
2
on a charge
q
1
,
F
2
1
→
, the force on
q
1
points
toward
q
2
if the charges have opposite signs and away from
q
2
if the
charges have like signs (Figure 21.12). This force on one charge due
to another charge always lies on a line between the two charges.
Coulomb’s Law
gives the magnitude of this force as
F
k
q q
r
=
1
2
2
,
(21.6)
where
q
1
and
q
2
are electric charges,
r
r
r
=
1
2
–
is the distance
between them, and
k
=
10
N m
C
9
2
2
8 99
.
⋅
(21.7)
is
Coulomb’s constant
.
You can see that one Coulomb is a
very
large charge. If two charges
of 1 C each were at a distance of 1 m apart, the magnitude of the force they would exert on
each other would be 8.99 billion N. For comparison, this force equals the weight of 450 fully
loaded space shuttles!
The relationship between Coulomb’s constant and another constant,
0
, called the
elec-
tric permittivity of free space
,
is
k
=
1
4
0
.
(21.8)
Consequently, the value of
0
is
0
12
8 85 10
=
C
N m
2
2
.
.
–
⋅
(21.9)
An alternative way of writing equation 21.6 is then
F
q q
r
=
1
4
0
1
2
2
.
(21.10)
As you’ll see in the next few chapters, some equations in electrostatics are more convenient
to write with
k
, while others are more easily written in terms of 1/(4
0
).
Note that the charges in equations 21.6 and 21.10 can be positive or negative, so the
product of the charges can also be positive or negative. Since opposite charges attract and
like charges repel, a negative value for the product
q
1
q
2
signifies attraction and a positive
value means repulsion.
Finally, Coulomb’s Law for the force due to charge 2 on charge 1 can be written in vec-
tor form:
F
k
q q
r
r
r
k
q q
r
r
2
1
1
2
3
2
1
1
2
2
21
→
−
=
=
–
(
)
–
ˆ .
(21.11)
In this equation,
r
ˆ
21
is a unit vector pointing from
q
2
to
q
1
(see Figure 21.13). The negative
sign indicates that the force is repulsive if both charges are positive or both charges are
negative. In that case,
F
2
1
→
points away from charge 2, as depicted in Figure 21.13a. On the
other hand, if one of the charges is positive and the other negative, then
F
2
1
→
points toward
charge 2, as shown in Figure 21.13b.

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