Two ideal electric dipoles $A$ and $B$, having their dipole moment $p_1$ and $p_2$ respectively are placed on a plane with their centres at $O$ as shown in the figure. At point $C$ on the axis of dipole $A$, the resultant electric field is making an angle of $37^\circ$ with the axis. The ratio of the dipole moment of $A$ and $B$, $\frac{p_1}{p_2}$ is: (take $\sin37^\circ = \frac{3}{5}$)
Answer: (C) $\frac{2}{3}$
Let $OC = r$. Point $C$ is on the axis of dipole $A$ and on the equatorial line of dipole $B$.
$E_A = \dfrac{2kp_1}{r^3}$ (along the axis), $E_B = \dfrac{kp_2}{r^3}$ (perpendicular to the axis).
$$\tan37^\circ = \frac{E_B}{E_A} = \frac{p_2}{2p_1} = \frac{3}{4} \Rightarrow \frac{p_1}{p_2} = \frac{2}{3}$$
Solution by Sreeraj P, M.Sc Physics