Question

Two charges each of $+Q$ units are placed along a line. A third charge $-q$ is placed between them. At what position and for what value of q, will the system be in equilibrium?

Answer 1

Let the charge $q$ be placed at 0, so that its distance from charge at $A=x$ and from charge $B=r-x$.

Now, the system is in equilibrium, if electrostatic force on $q$ due to charge at $A=$ electrostatic force on $q$ due to charge at $B$ ie,

$\frac{1}{4\pi {\epsilon }_0}\frac{qQ}{x^2}=\frac{1}{4\pi {\epsilon }_0}\frac{qQ}{(r-x{)}^2}$

$(r-x{)}^2=x^2$

or $x=\frac{r}{2}$

Three charges will be in equilibrium if net force on each charge is zero i.e,

$\frac{1}{4\pi {\epsilon }_0}\frac{qQ}{{\left(\frac{r}{2}\right)}^2}+\frac{1}{4\pi {\epsilon }_0}\frac{Q-Q}{r^2}=0$

or, $q=\frac{-Q}{4}$