Answer:
a. [tex]F'=\frac{1}{12}F[/tex]
b. [tex]d'=\frac{d}{2}[/tex]
Explanation:
a. The magnitude of the electric force is given by the Coulomb's law:
[tex]F=\frac{kq_1q_2}{d^2}[/tex]
In this case, we have [tex]q_1'=\frac{q_1}{3}[/tex] and [tex]d'=2d[/tex]:
[tex]F'=\frac{kq_1'q_2}{d'^2}\\F'=\frac{k(\frac{q_1}{3})q_2}{(2d)^2}\\F'=\frac{kq_1q_2}{3*4d^2}\\F'=\frac{1}{12}\frac{kq_1q_2}{d^2}\\F'=\frac{1}{12}F[/tex]
b. In this case, we have [tex]q_1'=\frac{q_1}{4}[/tex]:
[tex]F=\frac{kq_1'q_2}{d'^2}\\\frac{kq_1q_2}{d^2}=\frac{k(\frac{q_1}{4})q_2}{d'^2}\\\frac{1}{d^2}=\frac{1}{4d'^2}\\d'^2=\frac{d^2}{4}\\d'=\frac{d}{2}[/tex]
