Question

A charge $Q$ is divided into two parts $q$ and $Q-q$. Find the relationship between $Q$ and $q$ , if the two parts placed at a given distance $r$ apart have the maximum coulomb repulsion force?

• A. $q=\frac{Q}{2}$
• B. $q=\frac{Q}{3}$
• C. $q=\frac{2Q}{2}$
• D. $q=\frac{Q}{4}$

Answer 1

The correct option is A $q=\frac{Q}{2}$

According to the coulomb's law, the electric force between the given two parts,

$F=\frac{1}{4\pi {ϵ}_0}\frac{q(Q-q)}{r^2}....\left(1\right)$

For the force to be maximum:

$\frac{dF}{dq}=0$

Differentiating $\left(1\right)$ with respect to ${}^{\prime }{q}^{\prime }$ on both sides,

$\frac{1}{4\pi {ϵ}_0{r}^2}\frac{d}{dq}[Qq-q^2]=0$

$\Rightarrow \frac{1}{4\pi {ϵ}_0{r}^2}[Q-2q]=0...\left(1\right)$

$\Rightarrow Q-2q=0$

$\Rightarrow q=\frac{Q}{2}$

Now, differentiating equation $\left(1\right)$ again, we have

$\frac{d^{2}F}{d{q}^2}=\frac{-2}{4\pi {ϵ}_0{r}^2}$

Since, $\frac{d^{2}F}{d{q}^2}$ is negative at $q=\frac{Q}{2}$.

The repulsive force is maximum.

Hence, option (a) is the correct answer.