Question

A total charge Q is broken in two parts ${Q_1}$ and ${Q_2}$ and they are placed at a distance R from each other. The maximum force of repulsion between them will occur, when

Answer 1

Hint:
The force between the two charges can be calculated by applying Coulomb’s law. This relation between the force and the charge gives is an inverse relationship and it can be used to calculate the Coulomb force due to another charge at a distance.

Complete step-by-step solution:
It is given that the charge Q is broken in two parts and one part contains charge ${Q_1}$ and another contains ${Q_2}$
Now, we will calculate the force between the two charges ${Q_1}$ and ${Q_2}$ .

The mathematical expression for the Coulomb’s force between two charges is given as follows,
$F = \dfrac{{k{Q_1}{Q_2}}}{{{R^2}}}............{\rm{(1)}}$
Here, $k$ is the Coulomb’s constant and $R$ is the distance between the objects.
As we know that charge ${Q_1}$ and charge${Q_2}$ are the broken part of charge $Q$ , so we can write,
$
Q = {Q_1} + {Q_2}\\
\Rightarrow {Q_2} = Q - {Q_1}................{\rm{(2)}}
$
Substitute the value of \[{Q_2}\] in equation (1),
$
\Rightarrow F = \dfrac{{k{Q_1}\left( {Q - {Q_1}} \right)}}{{{R^2}}}\\
\Rightarrow F = \dfrac{{k\left( {{Q_1}Q - Q_1^2} \right)}}{{{R^2}}}
$
As we know at the maximum value or top of a curve of a function can be determined by taking the slope of the function equal to zero.
The slope remains zero where the curve flattens and at that point the value of function is maximum.
Now, we differentiate the force in the respect of ${Q_1}$ ,
\[\dfrac{{dF}}{{d{Q_1}}} = \dfrac{d}{{d{Q_1}}}\left( {\dfrac{{k\left( {{Q_1}Q - Q_1^2} \right)}}{{{R^2}}}} \right)
\]
Now equate the above expression equal to zero in order to get the maximum value and Evaluate further,
\[
\Rightarrow \dfrac{d}{{d{Q_1}}}\left( {\dfrac{{k\left( {{Q_1}Q - Q_1^2} \right)}}{{{R^2}}}} \right) = 0\\
\Rightarrow \dfrac{{k\left( {Q - 2{Q_1}} \right)}}{{{R^2}}} = 0\\
\Rightarrow Q - 2{Q_1} = 0
\]
Now we can get the value of ${Q_1}$ from the above expression
\[
Q - 2{Q_1} = 0\\
\Rightarrow {Q_1} = \dfrac{Q}{2}
\]
Here, we have the value of charge ${Q_1} = \dfrac{Q}{2}$ .
Now we can calculate the value of charge ${Q_2}$ by substituting the value of charge ${Q_1}$ in the equation (2).
$
{Q_2} = Q - \dfrac{Q}{2}\\
\Rightarrow {Q_2} = \dfrac{Q}{2}
$
Therefore, the maximum force of repulsion between them will occur, when ${Q_1} = {Q_2} = \dfrac{Q}{2}$.

Note:
The Coulomb force can be attractive or repulsive, depending on the nature of charge. If the charges are similar, they will repulse; otherwise, the attraction force will act there between them.