Question

A very short electric dipole is placed along the $\text{x-}$axis at the origin $\text{O}$. A point $\text{P}$ is at a distance of $20\text{cm}$ from this origin such that $\text{OP}$ makes an angle $\frac{\pi }{3}$ with the $\text{x}$-axis. If the electric field at $\text{P}$ makes an angle $\theta$ with the $\text{x}$-axis, the value of $\theta$ would be

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{3}+{\tan }^{-1}\left(\frac{\sqrt{3}}{2}\right)$
• C. $\frac{2\pi }{3}$
• D. ${\tan }^{-1}\left(\frac{\sqrt{3}}{2}\right)$

Answer 1

The correct option is B $\frac{\pi }{3}+{\tan }^{-1}\left(\frac{\sqrt{3}}{2}\right)$

Let $\alpha$ be the angle between resultant electric field vector and position vector.

From the figure, we can say angle of resultant electric field with $\text{x-}$axis will be $\alpha +\frac{\pi }{3}$

We know that,
$\tan \alpha =\frac{1}{2}\tan \frac{\pi }{3}$

$\Rightarrow \alpha ={\tan }^{-1}\frac{\sqrt{3}}{2},$

Now, from figure, the angle $\theta$ between resultant electric field and $\text{x-}$axis will be
$\theta =\frac{\pi }{3}+\alpha$

Substituting the value of $\alpha$, we get

$\theta =\frac{\pi }{3}+{\tan }^{-1}\frac{\sqrt{3}}{2}$

Hence, option (c) is the correct answer.