Question

Two electric dipoles, $A,B$ with respective dipole moments ${\overrightarrow{d}}_A=-4qa\hat{i}$ and ${\overrightarrow{d}}_B=-2qa\hat{i}$ are placed on the $x-$axis with a separation $R$, as shown in the figure


The distance from $A$ at which both of them produce the same potential is
$\left[\text{Mains-2019, Jan 10th, Shift 1}\right]$

Answer 1

Given:
Dipole moment of $A,{\overrightarrow{d}}_A=-4qa\hat{i}$ Dipole moment of $B,{\overrightarrow{d}}_B=-2qa\hat{i}$
Seperation distance $=R$

Let at a distance ${}^{\prime }{x}^{\prime }$ from point $B$, both the dipoles produce same potential

$\therefore \frac{4qa}{(R+x{)}^2}=\frac{2qa}{\left(x^2\right)}$

$\Rightarrow \sqrt{2}x=R+x\Rightarrow x=\frac{R}{\sqrt{2}-1}$

Distance from $A,=\frac{R}{\sqrt{2}-1}+R=\frac{\sqrt{2}R}{\sqrt{2}-1}$

Hence, option $\left(D\right)$ is correct.