Question

Two identical electric point dipoles have dipole moments $\overrightarrow{P_1}=P\hat{i}\text{and}\overrightarrow{P_2}=-P\hat{i}$ and held on the $x-$ axis at distance ${}^{\prime }{a}^{\prime }$ 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 ${}^{\prime }{m}^{\prime }$, their speed
when they are infinitely far apart is :

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

Answer 1

The correct option is B $\frac{P}{a}\sqrt{\frac{1}{2\pi {\epsilon }_{0}ma}}$


Let $v$ be the speed of dipole.
Using energy conservation

$K_i+U_i=K_f+U_f$

$\Rightarrow 0-\frac{2k{P}_1}{r^3}{P}_2\cos \left({180}^{\circ }\right)=\frac{1}{2}m{v}^2+\frac{1}{2}m{v}^2+0$

$(\because \text{Potential energy of interaction between dipole =}\frac{-2p_1{p}_2\cos \theta }{4\pi {ϵ}_0{r}^3})$

$\Rightarrow m{v}^2=\frac{2k{P}_1{P}_2}{r^3}\Rightarrow v=\sqrt{\frac{2k{P}_1{P}_2}{m{r}^3}}$

When $P_1=P_2=P\text{and}r=a$

$\therefore v=\frac{p}{a}\sqrt{\frac{1}{2\pi {ϵ}_{0}ma}}$

Hence, option $\left(\text{B}\right)$ is correct.