PX157 - A3 - principle of super position

total force, ${\vec{F}}_{0}$ , on charge, $q_{0}$ :

{\vec{F}}_{0} = \sum_{i = 1}^{N} {\vec{F}}_{i 0} = \frac{q_{0}}{4 π ϵ_{0}} \sum_{i = 0}^{N} q_{i} \frac{{\vec{r}}_{0} - {\vec{r}}_{i}}{| {\vec{r}}_{0} - {\vec{r}}_{i} |^{3}}

eg:
$q_{1} = q_{2} = q$
${\vec{r}}_{1} = d \hat{y}$ and ${\vec{r}}_{2} = - d \hat{y}$
${\vec{r}}_{Q} = x \hat{x}$
$\begin{align} \vec F_{Q} &= \frac{Q q_{1}}{4\pi\epsilon_{0}} \frac{\vec r_{Q} - \vec r_{1}}{|\vec r_{Q} - \vec r_{1}|^{3}} + \frac{Q q_{2}}{4\pi\epsilon_{0}} \frac{\vec r_{Q} - \vec r_{2}}{|\vec r_{Q} - \vec r_{2}|^{3}} \\$
\therefore \vec F_{Q} &= \frac{Q q_{1}}{4\pi\epsilon_{0}} \frac{2x\hat x}{(x^{2}+d^{2})^{\frac{3}{2}}}
\end{align}$$