Physics Question on electrostatic potential and capacitance

Question

A charge q is placed at the centre of the line joining two exactly equal positive charges The system of three charges will be in equilibrium, if q is equal to

Answer 1

Solution

For equilibrium of charge Q, the force of repulsion due to similar charges Q should be balanced by the force of attraction due to charge q and
$\frac{1}{ 4 \pi \varepsilon_0 } \times \frac{ Q q}{ ( r / 2)^2 } + \frac{1}{ 4 \pi \varepsilon_0 } \times \frac{ Q^2 }{ r^2 } = 0$
or $4 \times \frac{ Q}{ r^2 } q = - \frac{ Q^2 }{ r^2 }$ or $4q = - Q$ or $q = - \frac{ Q}{ 4}$ .