Question

Two electric dipoles, \(A, B\) with respective dipole moments \(\overrightarrow{d}_A = -4 qa\hat{i}\) and \(\overrightarrow{d}_B = -2 qa\hat{i}\) 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

• A. $\frac{R}{\sqrt{2}-1}$
• B. $\frac{\sqrt{2}R}{\sqrt{2}+1}$
• C. $\frac{R}{\sqrt{2}+1}$
• D. $\frac{\sqrt{2}R}{\sqrt{2}-1}$

Answer 1

The correct option is D $\frac{\sqrt{2}R}{\sqrt{2}-1}$

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.