Chapter 16.3, Problem 36E

(a) Suppose that F is an inverse square force field, that is,

$F\left(r\right)=\frac{cr}{{\left|r\right|}^3}$

for some constant c, where $r=xi+yj+zk$.

(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 $F=-{\left(mMG\right)r/\left|r\right|}^3$ 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 $1.52\times {10}^8$ km from the sun) to perihelion (at a minimum distance of $1.47\times {10}^8$ km). (Use the values $m=1.52\times {10}^{24}$ kg, $M=1.52\times {10}^{30}$ kg, and $G=6.67\times {10}^{-11}$ N·3m²/kg².)

(c) Another example of an inverse square field is the electric force field $F=-\epsilon qQ{r/\left|r\right|}^3$ discussed in Example 16.1.5. Suppose that an electron with a charge of $-1.6\times {10}^{-19}$ 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 $\epsilon =8.985\times {10}^9$.)