Question

A charge $Q$ is divided into two parts of $q$ and $Q-q$. If the Coulomb repulsion between them when they are separated, is to be maximum, the ratio of $\dfrac{Q}{q}$ should be:
$\begin{aligned} & A.\,2 \\\ & B.\,{1}/{2}\; \\\ & C.\,4 \\\ & D.\,{1}/{4}\; \\\ \end{aligned}$

Answer 1

We can use Coulomb law to find the electric force of a charge and to get the maximum coulomb repulsion between them when they are separated, it can be calculated after differentiating the force equation and then equating it to zero, which will give us the point of maxima where force is maximum.

Formula used:
$F=k\dfrac{{{q}_{1}}{{q}_{2}}}{{{r}^{2}}}$, where
$F$ is electric force,
$k$ is Coulomb constant,
${{q}_{1}},{{q}_{2}}$ are charges, and
$r$ is distance of separation.

Complete Step-by-Step solution:
We know that repulsive force between 2 charges can be calculated using Coulomb law.
According to Coulomb's Law, Repulsive force can be written as:
$F=k\dfrac{{{q}_{1}}{{q}_{2}}}{{{r}^{2}}}$
In the above equation, $k$ and $r$ are constant, and
${{q}_{1}}=q\,\And \,{{q}_{2}}=(Q-q)$
Now, the above repulsive force equation can be written as:
$F=k\dfrac{q(Q-q)}{{{r}^{2}}}$
To get the point of maxima of $F$, for which $F$ is maximum, we will differentiate $F$ with respect to $q$ and then equate it to zero.
$\dfrac{dF}{dq}=0$
Now, differentiating the force equation and equating it to zero, we get –
$\begin{aligned} & \dfrac{d}{dq}(k\dfrac{q(Q-q)}{{{r}^{2}}})=0 \\\ & \dfrac{k}{{{r}^{2}}}\dfrac{d}{dq}(q(Q-q))=0 \\\ & q(-1)+1.(Q-q)=0 \\\ & Q-2q=0 \\\ & q=\dfrac{Q}{2} \\\ \end{aligned}$
Hence, $q=\dfrac{Q}{2}$ is the point of maxima, where Force is maximum.
Since,
$\begin{aligned} & q=\dfrac{Q}{2} \\\ & \dfrac{Q}{q}=2 \\\ \end{aligned}$

Therefore, the correct answer is Option (A).

Note:
Coulomb Law is a powerful Law to get the repulsive force between charges. It takes both the charges into consideration and the distance between them. It is also advisable to differentiate the equation, whenever you need to find the maximum of a value or point of maxima. Differentiating the equation and equating it to zero is the most effective way of getting the answer in this type of question.