Question

A charge $Q$ is divided into two parts $q_1$ and $q_2$ such that they experience maximum force of repulsion when separated by certain distance. The ratio of $Q,q_1$ and $q_2$ is :

• A. 1 : 1 : 2
• B. 1 : 2 : 2
• C. 2 : 2 : 1
• D. 2 : 1 : 1

Answer 1

Answer: (D)

$\text{Step 1: Divide Q}$

$Q$ is divided into two parts $q_1$ and$q_2$

$\therefore$ $Q=q_1+q_2$ $....\left(1\right)$

$\text{Step 2: Force of repulsion between}{q}_1\text{and}{q}_2$

Let $r$ be the distance between charges.

Using Coulomb's law, $F=\frac{k{q}_1{q}_2}{r^2}$$=\frac{k{q}_1(Q-q_1)}{r^2}$ $....\left(2\right)$

$\text{Step 3: Condition for force to be maximum}$

The repulsive force between the charges will be maximum if,

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

$\Rightarrow \frac{k}{r^2}(Q-2q_1)=0$

$\Rightarrow q_1=\frac{Q}{2}$

Equation $\left(1\right)\Rightarrow$ $q_2=Q-\frac{Q}{2}=\frac{Q}{2}$

$\text{Step 4: Ratio of}Q:q_1:q_2$

$Q:q_1:q_2=Q:\frac{Q}{2}:\frac{Q}{2}=1:\frac{1}{2}:\frac{1}{2}$

$\Rightarrow Q:q_1:q_2=2:1:1$

Hence, Option (D) is correct.