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?

Difficult

Solution

At any arbitrary point P defined by ( $γ, θ$ ), direction of electric field $\vec{E}$ is defined by $\tan \propto = \frac{1}{2} tanθ$ Evidently when $\vec{E}$ is perpendicular to $\vec{P}$ , $θ + α = 90^{0}$ ,
$\begin{array}{l} ∴ \tan (90^{p} - θ) = \frac{1}{2} tanθ \Rightarrow tanθ = \pm \sqrt{2} \\ ∴ θ = \tan^{- 1} (\sqrt{2}), (π - \tan^{- 1} \sqrt{2}), (π + \tan^{- 1} \sqrt{2}), (2 π - \tan^{- 1} \sqrt{2}) \end{array}$
hence points A, B, C and D symmetrically lie in the 1st, 2nd, 3red and 4th quadrants respectively. Electric fields at C and D are mirror images of those at A and B.

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