(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
. Find the work done by F in moving an object from a point P1 along a path to a point P2 in terms of the distances d1 and d2 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 4 in Section 13.1. 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 × 108 km from the sun) to perihelion (at a minimum distance of 1.47 × 108 km). (Use the values m = 5.97 × 1024 kg, M = 1.99 × 1030 kg, and G = 6.67 × 10–11 N·m2/kg2.)
(c) Another example of an inverse square field is the electric force field
F
=
ε
q
Q
r
/
|
r
|
3
discussed in Example 5 in Section 13.1. 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 from the electron. Use part (a) to find the work done by the electric force field. (Use the value ε = 8.985 × 109.)