Question

A charge of magnitude $q$ is divided into two parts such that force between resulting two charges is maximum when separated through some distance $r$. The division of charges would be

• A. $\frac{3q}{8}$, $\frac{5q}{8}$
• B. $\frac{2q}{4}$, $\frac{2q}{4}$
• C. $\frac{q}{2}$, $\frac{q}{2}$
• D. $\frac{3q}{6}$, $\frac{3q}{6}$

Answer 1

The correct option is C $\frac{q}{2}$, $\frac{q}{2}$

$\text{Step 1: Apply Coulomb's Law}$

Let $q$ is divided into $q_1$ and $(q-q_1)$

Force between two charges: $F=\frac{1}{4\pi {\in }_0}\frac{q_1(q-q_1)}{r^2}\Rightarrow F=\frac{1}{4\pi {\in }_0}\left(\frac{q_{1}q-q_1^2}{r^2}\right)$

$\text{Step 2: Maximum force condition}$

The force, $F$, will be maximum if,

$\frac{dF}{d{q}_1}=0$

$\Rightarrow \frac{1}{4\pi {\in }_0{x}^2}\frac{d}{d{q}_1}(q_{1}q-q_1^2)=0$

$\Rightarrow q-2q_1=0\Rightarrow q_1=\frac{q}{2}$

The charges would be $\frac{q}{2},\frac{q}{2}$.

Hence, option $\left(C\right)$ is correct.