Question
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 $\_\_\_\_$ .
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Solution
Sol. [Key Concept: Axial field $\propto \ 2p_{1}$, equatorial $\propto \ p_{2}$; use $\tan{\ 60}^{\circ} = p_{2}/(2p_{1})$.]
${\overrightarrow{E}}_{A} = \frac{2{kP}_{1}}{x^{3}}( - \widehat{i})\{$ Axial point $\}{\overrightarrow{E}}_{B} = \frac{{kP}_{2}}{x^{3}}( - \widehat{j})$ {Equatorial point}
Now : $tan\theta = \frac{\left| {\overrightarrow{E}}_{B} \right|}{\left| {\overrightarrow{E}}_{A} \right|}$
$${\Rightarrow tan60^{\circ} = \frac{P_{2}}{2P_{1}} }{\Rightarrow \frac{P_{2}}{P_{1}} = 2\sqrt{3} }$$
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