Question

Charges –q, Q and –q are placed at an equal distance on a straight line. If the total potential energy of the system of three charges is zero, then find the ratio $\frac{Q}{q}$.

Answer 1

Total potential energy of the system is given by:
$U_{sys}=U_{qQ}+U_{Qq}+U_{qq}$

$=\frac{k(-q)Q}{r}+\frac{kQ(-q)}{r}+\frac{k(-q)(-q)}{2r}$
$=\frac{k(-q)Q}{r}+\frac{kQ(-q)}{r}+\frac{k{q}^2}{2r}$
$=\frac{kq}{r}[-Q-Q+\frac{q}{2}]$
$\Rightarrow 2Q-\frac{q}{2}=0$
$\Rightarrow 2Q=\frac{q}{2}$
$\Rightarrow \frac{Q}{q}=\frac{1}{4}$