Two identical electric point dipoles have dipole moments $\vec p_1 = p\hat i$ and $\vec p_2 = -p\hat i$ and are held on the $x$-axis at distance $a$ from each other. When released, they move along the $x$-axis with the direction of their dipole moments remaining unchanged. If the mass of each dipole is $m$, their speed when they are infinitely far apart is:
Answer: (B) $\dfrac pa\sqrt{\dfrac{1}{2\pi\varepsilon_0ma}}$
The field of $\vec p_1$ on its axis at distance $a$ is $\dfrac{2kp}{a^3}\hat i$. The energy of $\vec p_2$ in it:
$$U = -\vec p_2\cdot\vec E = -(-p)\frac{2kp}{a^3} = \frac{2kp^2}{a^3} = \frac{p^2}{2\pi\varepsilon_0a^3}$$
This positive energy (they repel) becomes the kinetic energy of the two equal masses moving apart with equal speeds:
$$2\cdot\frac12mv^2 = \frac{p^2}{2\pi\varepsilon_0a^3} \Rightarrow v = \frac pa\sqrt{\frac{1}{2\pi\varepsilon_0ma}}$$
Solution by Sreeraj P, M.Sc Physics