An electric dipole is placed at the origin as shown in figure. A, B, C and D are four points where the electric fields are perpendicular to the dipole axis. Then which of the following figures is correct?

At any arbitrary point P defined by ($r, \mathrm{\theta }$), direction of electric field $\overrightarrow{\mathrm{E}}$ is defined by $\tan \alpha =\frac{1}{2}\mathrm{tan\theta }$ Evidently when $\overrightarrow{\mathrm{E}}$ is perpendicular to $\overrightarrow{\mathrm{P}}$, $\mathrm{\theta }+\mathrm{\alpha }={90}^0$,

$\begin{array}{l}\therefore \tan \left({90}^{\mathrm{p}}-\mathrm{\theta }\right)=\frac{1}{2}\mathrm{tan\theta }\Rightarrow \mathrm{tan\theta }=\pm \sqrt{2}\\ \therefore \mathrm{\theta }={\tan }^{-1}\left(\sqrt{2}\right),\left(\mathrm{\pi }-{\tan }^{-1}\sqrt{2}\right),\left(\mathrm{\pi }+{\tan }^{-1}\sqrt{2}\right),\left(2\mathrm{\pi }-{\tan }^{-1}\sqrt{2}\right)\end{array}$

hence points A, B, C and D symmetrically lie in the 1st, 2nd, 3rd and 4th quadrants respectively. Electric fields at C and D are mirror images of those at A and B.