Question

A positive point charge $q$ is fixed at origin. a small dipole with a dipole moment $\overrightarrow{p}$ is placed along the x-axis far away from the origin with $\overrightarrow{p}$ pointing along positive $x$-axis. Find the kinetic energy of the dipole when it reaches a distance $d$ from the origin.

• A. $\frac{qp}{4\pi {ϵ}_0{d}^2}$
• B. $\frac{2qp}{\pi {ϵ}_0{d}^2}$
• C. $\frac{qp}{2\pi {ϵ}_0{d}^2}$
• D. $\frac{4qp}{\pi {ϵ}_0{d}^2}$

Answer 1

The correct option is A $\frac{qp}{4\pi {ϵ}_0{d}^2}$

Applying Energy Conservation Principle,
$\left(\begin{array}{c}\text{Increase in}\\ \text{kinetic energy of}\\ \text{dipole}\end{array}\right)=\left(\begin{array}{c}\text{decrease in}\\ \text{electrostatic potential}\\ \text{energy of the dipole}\end{array}\right)$
Kinetic energy of dipole at distance $d$ from origin
$=U_i-U_f\Rightarrow KE=0-(-\overrightarrow{p}.\overrightarrow{E})=\overrightarrow{p}.\overrightarrow{E}=\left(p\hat{i}\right).\left(\frac{1}{4\pi {ϵ}_0}\frac{q}{d^2}\hat{i}\right)$
$\Rightarrow KE=\frac{qp}{4\pi {ϵ}_0{d}^2}$