Question

The force between two charges at distance r is F. When they place at……distance, the force would be F/3.
(a) $\dfrac{r}{2}$
(b) $\sqrt 3$
(c) $\sqrt 3 r$
(d) $2r$

Answer 1

To determine the required distance, we will use the coulombs expression of force. Here two different magnitudes of the force are given for two different distances to obtain two forces equations. One equation will contain a given distance, and another will contain the required distance, which needs to be determined, using these two equations for the correct answer.

Complete step by step answer:
We will consider that the required distance that needs to be determined is x and the magnitude of the two charges is ${q_1}$ and ${q_2}$.
We know that the magnitude of the force developed by the two charges separated by some distance can be determined with the help of coulomb's law. In the first condition, the force's magnitude is F, and the separation distance between two charges is r. So from coulomb's law, write the expression of the force for the first condition
$F = \dfrac{{K{q_1}{q_2}}}{{{r^2}}}$...... (1)
Here $K$ is the electrostatic constant.
In the second condition, the force's magnitude becomes F/3, and we consider that the required distance between the two charges is x. So, write the expression of force for the second condition.
Therefore, we get
$\dfrac{F}{3} = \dfrac{{K{q_1}{q_2}}}{{{x^2}}}$...... (2)
We will then divide the equation (2) by equation (1) so that the required distance can be determined.
Therefore, we get
$\begin{array}{l} \dfrac{{\dfrac{F}{3}}}{F} = \dfrac{{\dfrac{{K{q_1}{q_2}}}{{{x^2}}}}}{{\dfrac{{K{q_1}{q_2}}}{{{r^2}}}}}\\\ \dfrac{1}{3} = \dfrac{{{r^2}}}{{{x^2}}}\\\ {x^2} = 3{r^2}\\\ x = \sqrt 3 r \end{array}$
Therefore, the force would be F/3 when the charges are placed at $\sqrt 3 r$, and option (c) is correct.

Note: Here, the magnitude of the two charges remains the same when the variation in the magnitude of force and detachment distance occurs. Remember that according to Coulomb's law, the charges' products relate directly with force, and the detachment distance among the charges related inversely with force.