468 Chapter 16Vectors
EXAMPLE 16.20Force and potential energy
By a generalization to three dimensions of the discussion of conservative forces
in Section 5.7 (see also Example 16.13), the components of a conservative force are
derivatives of a potential-energy function V,
The force is therefore (minus) the gradient of V,
(16.69)
0 Exercises 48–50
EXAMPLE 16.21Coulomb forces
The potential energy of interaction of charges q
1and q
2separated
by distance risV 1 = 1 q
1q
224 πε
0r(see Example 5.19). If ris the
position of q
2relative to q
1(Figure 16.31) then the force acting
onq
2due to the presence of q
1is
The unit vector fromq
1toq
2is Therefore
(16.70)
The force has strength , and acts along the line fromq
1toq
2for like charges,
fromq
2toq
1for unlike charges (see also Example 5.17). In addition, the force per
unit charge acting at point rdue to the presence of chargeq
1isE 1 = 1 F 2 q
2. Then
r
12024 πε
F
r
==r
r
r
1203120244 ππεε
ˆ
ˆr.
r
=
r
=
r
12034
r
πε
=++
=
qq
xr
y
r
z
r
r
x
12033 312034
4π
πε
εijk ( i+++yzjk)
Fi=−∇ =− j
∂
∂
+
∂
∂
+
∂
∂
V
xr yr
1204
11
πε zzr
1
k
Fijk=−∇ =−
∂
∂
∂
∂
∂
∂
V
V
x
V
y
V
z
F
V
x
F
V
y
F
V
z
xyz=−
∂
∂
,=−
∂
∂
,=−
∂
∂
q
1q
2r
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Figure 16.31