Two identical electric point dipoles have dipole moments

Question:

Two identical electric point dipoles have dipole moments $\overrightarrow{\mathrm{P}_{1}}=\mathrm{P} \hat{i}$ and $\overrightarrow{\mathrm{P}_{2}}=-\mathrm{P} \hat{i}$ and are held on the $x$ axis at distance 'a' from each other. When released, they move along $x$-axis with the direction of their dipole moments remaining unchanged. If the mass of each dipole is ' $\mathrm{m}$ ', their speed when they are infinitely far apart is :

1. (1) $\frac{\mathrm{P}}{a} \sqrt{\frac{1}{\pi \varepsilon_{0} m a}}$

2. (2) $\frac{\mathrm{P}}{a} \sqrt{\frac{1}{2 \pi \varepsilon_{0} m a}}$

3. (3) $\frac{\mathrm{P}}{\mathrm{a}} \sqrt{\frac{2}{\pi \varepsilon_{0} \mathrm{ma}}}$

4. (4) $\frac{\mathrm{P}}{a} \sqrt{\frac{2}{2 \pi \varepsilon_{0} m a}}$

Correct Option: , 2

Solution:

(2) Let $v$ be the speed of dipole.

Using energy conservation

$K_{i}+U_{i}=K_{f}+U_{f}$

$\Rightarrow 0-\frac{2 k \cdot p_{1}}{r^{3}} p_{2} \cos \left(180^{\circ}\right)=\frac{1}{2} m v^{2}+\frac{1}{2} m v^{2}+0$

$\because$ Potential energy of interaction between dipole

$\left.=\frac{-2 p_{1} p_{2} \cos \theta}{4 \pi \epsilon_{0} r^{3}}\right)$

$\Rightarrow m v^{2}=\frac{2 k p_{1} p_{2}}{r^{3}} \Rightarrow v=\sqrt{\frac{2 k p_{1} p_{2}}{m r^{3}}}$

When $p_{1}=p_{2}=p$ and $r=\mathrm{a}$

$v=\frac{p}{a} \sqrt{\frac{1}{2 \pi \epsilon_{0} m a}}$