Two electric dipoles $A$, $B$ with respective dipole moments $\vec d_A = -4qa\,\hat i$ and $\vec d_B = -2qa\,\hat i$ are placed on the $x$-axis with a separation $R$, with $A$ at the origin and $B$ at $x = R$. The distance from $A$ at which both of them produce the same potential is:
Answer: (B) $\dfrac{\sqrt2R}{\sqrt2-1}$
On the axis of a dipole $V = \dfrac{k\,\vec p\cdot\hat r}{r^2}$. Both moments point along $-x$.
Between $A$ and $B$, $\hat r$ from $A$ is $+\hat i$ and from $B$ is $-\hat i$, so the two potentials have opposite signs and cannot be equal. So the point is beyond $B$, at $x > R$, where both potentials are negative:
$$\frac{4qa}{x^2} = \frac{2qa}{(x-R)^2} \;\Rightarrow\; x = \sqrt2\,(x-R) \;\Rightarrow\; x = \frac{\sqrt2R}{\sqrt2-1}$$
Solution by Sreeraj P, M.Sc Physics