Two short electric dipoles **A** and **B** having dipole moment $p_1$ and $p_2$ respectively are placed with their axis mutually perpendicular as shown in the figure. The resultant electric field at a point $x$ is making an angle of $60^\circ$ with the line joining points $O$ and $x$. The ratio of the dipole moments $p_2/p_1$ is ______.
Answer: (B) $2\sqrt3$
Let the point $x$ be at distance $r$ from $O$ on the $x$-axis.
Dipole A lies along the $x$-axis, so $x$ is on its axial line: $E_1=\dfrac{2kp_1}{r^3}$, along the $x$-axis.
Dipole B lies along the $y$-axis, so $x$ is on its equatorial line: $E_2=\dfrac{kp_2}{r^3}$, parallel to the $y$-axis.
The angle $\theta$ of the resultant with the line $Ox$ satisfies
$$\tan\theta=\frac{E_2}{E_1}=\frac{p_2}{2p_1}=\tan60^\circ=\sqrt3\ \Rightarrow\ \frac{p_2}{p_1}=2\sqrt3$$
Solution by Sreeraj P, M.Sc Physics