Question 257 of 311

Two identical electric point dipoles have dipole 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 :

A$\dfrac{P}{a}\sqrt{\dfrac{1}{{\pi \epsilon }_{0}ma}}$B$\dfrac{P}{a}\sqrt{\dfrac{1}{2{\pi \epsilon }_{0}ma}}$C$\dfrac{P}{a}\sqrt{\dfrac{2}{{\pi \epsilon }_{0}ma}}$D$\dfrac{P}{a}\sqrt{\dfrac{3}{2{\pi \epsilon }_{0}ma}}$

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Answer: (B) $\dfrac{P}{a}\sqrt{\dfrac{1}{2{\pi \epsilon }_{0}ma}}$

${k}_{i}+{u}_{i}={k}_{f}+{u}_{f}$
$0-\dfrac{2kp}{{r}^{3}}{p}_{2}\cdot \cos \left(180^{\circ}\right)=\dfrac{1}{2}m{v}^{2}+\dfrac{1}{2}m{v}^{2}+0$
$m{v}^{2}=\dfrac{2k{p}_{1}{p}_{2}}{{r}^{3}}$
$\Rightarrow \theta =\sqrt{\dfrac{2k{p}_{1}{p}_{2}}{m{r}^{3}}}$
{p}_{1}={p}_{2}=p&r=a
$v=\dfrac{p}{a}\sqrt{\dfrac{1}{2\pi {ϵ}_{0}ma}}$