(a) Suppose that F is an inverse square force field, that is,
F
(
r
)
=
c
r
|
r
|
3
for some constant c, where
r
=
x
i
+
y
j
+
z
k
.
(a) Find the work done by F in moving an object from a point
P
1
along a path to a point
P
2
in terms of the distances
d
1
and
d
2
from these points to the origin.
(b) An example of an inverse square field is the gravitational field
F
=
−
(
m
M
G
)
r
/
|
r
|
3
discussed in Example 16.1.4. Use part (a) to find the work done by the gravitational field when the earth moves from aphelion (at a maximum distance of
1.52
×
10
8
km from the sun) to perihelion (at a minimum distance of
1.47
×
10
8
km). (Use the values
m
=
1.52
×
10
24
kg,
M
=
1.52
×
10
30
kg, and
G
=
6.67
×
10
−
11
N·3m2/kg2.)
(c) Another example of an inverse square field is the electric force field
F
=
−
ε
q
Q
r
/
|
r
|
3
discussed in Example 16.1.5. Suppose that an electron with a charge of
−
1.6
×
10
−
19
C is located at the origin. A positive unit charge is positioned a distance
10
−
12
m from the electron and moves to a position half that distance iron: the electron. Use part (a) to find the work done by the electric force field. (Use the value
ε
=
8.985
×
10
9
.)