Question

Two electric dipoles, $A$ and $B$ with respective dipole moments $p_A=-5qa\hat{i}$ and $p_B=-3qa\hat{i}$ are placed on the $X$ - axis with a separation $R$, (as shown in figure). The distance from A at which both of them produce the same potential is $NR$. Determine $N$. (Take $\sqrt{5}=2.24$ and $\sqrt{3}=1.73$ )

Answer 1

Answer: 4.45

The potential at a point $P$ at distance $r$ due to dipole is given as
$V=\frac{\overrightarrow{p}\cdot \hat{r}}{4\pi {\epsilon }_0{r}^2}$.
For a point at which the potential is equal,
$\frac{5qa}{4\pi {\epsilon }_0{r}^2}=\frac{3qa}{4\pi {\epsilon }_0(r-R{)}^2}$
$\therefore \frac{5}{r^2}=\frac{3}{(r-R{)}^2}$
$\therefore \sqrt{5}(r-R)=\sqrt{3}r$
$\therefore \sqrt{5}r-\sqrt{3}r=\sqrt{5}R$
$\therefore r=\frac{\sqrt{5}R}{(\sqrt{5}-\sqrt{3})}$
$=\frac{\sqrt{5}(\sqrt{5}+\sqrt{3})}{5-3}R$
$=(1.12\times 3.97)R$
$=4.45R$