(i) Derivation of expression of electric field on the equatorial line of the dipole
(ii) Depiction of orientation for stable and unstable equilibrium
(i) Let the point ' $\mathrm{P}$ ' be at a distance ' $\mathit{r}$ ' from the mid point of the dipole.
${\mathit{E}}_{+\mathit{q}}=\frac{\mathit{q}}{4\mathit{\pi }{\mathit{\epsilon }}_0\left({\mathit{r}}^2+{\mathit{a}}^2\right)}$
${\mathit{E}}_{-\mathit{q}}=\frac{\mathit{q}}{4\mathit{\pi }{\mathit{\epsilon }}_0\left({\mathit{r}}^2+{\mathit{a}}^2\right)}$
Both are equal and their directions are as shown in the figure. Hence net electric field
${\mathit{E}}_{+\mathit{q}}=\left[-\left({\mathrm{E}}_{+\mathit{q}}+{\mathrm{E}}_{-\mathit{q}}\right)\cos \mathit{\theta }\right]\hat{\mathit{p}}$
${\mathit{E}}_{\mathit{p}}=-\frac{2\mathit{q}\mathit{a}}{4\mathit{\pi }{\mathit{\epsilon }}_0{\left({\mathit{r}}^2+{\mathit{a}}^2\right)}^{\frac{3}{2}}}\mathrm{}\hat{\mathit{p}}$
(ii) Stable equilibrium, $\mathit{\theta }=0^{\circ }$