Question

If $q_1+q_2=q$, then the value of the ratio $\frac{q_1}{q}$, for which the force between $q_1$ and $q_2$ is maximum is

• A. $0.25$
• B. $0.75$
• C. 1
• D. $0.5$

Answer 1

Answer: (D)

Let $r$ be the distance between $q_1$ and $q_2$.
According to Coulomb's law, the force between them is
$F=\frac{1}{4\pi {\epsilon }_0}\frac{q_1{q}_2}{r^2}$
$=\frac{1}{4\pi {\epsilon }_0}\frac{q_1\left(q-q_1\right)}{r^2}$
$\left(\because q_1+q_2=q\right)$
For $F$ to be maximum $\frac{dF}{d{q}_1}=0$
$\therefore \frac{d}{d{q}_1}\left[\frac{1}{4\pi {\epsilon }_0{r}^2}\left(q_{1}q-q_1^2\right)\right]=0$
$\Rightarrow q-2q_1=0$
$\Rightarrow \frac{q_1}{q}=\frac{1}{2}=0.5$
$\left[\text{ As }\frac{1}{4\pi {\epsilon }_0{r}^2}\ne 0\right]$