Question

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

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

Answer 1

The correct option is A $\frac{\sqrt{2}R}{\sqrt{2}+1}$

As, dipole moments of dipoles point in the same direction, potential of both dipoles can be same at some point between $A$ and $B$. Let potentials be same at $P$, which is at distance $x$ from $B$ as shown below



Then , potential due to each dipole at $P$ we get ,

$\frac{4qa}{(R-x{)}^2}$ = $\frac{2qa}{(x{)}^2}$

$2x^2=(R-x{)}^2$

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

Distance from $A$ is

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

Hence, option (a) is the correct answer.