Question

Two electric dipoles, $A$ and $B$ with respective dipole moments $p_{A}=-5 q a \hat{i}$ and $p_{B}=-3 q a \hat{i}$ 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

The potential at a point $P$ at distance $r$ due to dipole is given as
$V =\frac{\overrightarrow{ p } \cdot \hat{ r }}{4 \pi \varepsilon_{0} r ^{2}}$.
For a point at which the potential is equal,
$\frac{5 q a}{4 \pi \varepsilon_{0} r^{2}}=\frac{3 q a}{4 \pi \varepsilon_{0}(r-R)^{2}}$
$\therefore \frac{5}{r^{2}}=\frac{3}{(r-R)^{2}}$
$\therefore \sqrt{5}(r-R)=\sqrt{3} r$
$\therefore \sqrt{5} r-\sqrt{3} r=\sqrt{5} R$
$\therefore r=\frac{\sqrt{5} R}{(\sqrt{5}-\sqrt{3})}$
$=\frac{\sqrt{5}(\sqrt{5}+\sqrt{3})}{5-3} R$
$=(1.12 \times 3.97) R$
$=4.45 \,R$