Question #43687

A fixed ideal electric dipole of dipole moment $\vec{p} = p \hat{i}$ is located at the origin of a coordinate system. A stationary radioactive nucleus is located at point $A$ with polar coordinates $(r, \pi)$ in the $xy$-plane, where the polar angle $\theta$ is measured from the positive $x$-axis. The nucleus undergoes a reaction, emitting a charged fragment of mass $m$ and quantized charge $q = +ne$ (where $e$ is the elementary charge and $n$ is a positive integer). The heavy residual nucleus remains fixed at $A$ and is uncharged. The $Q$-value of the reaction is entirely converted into the initial kinetic energy of the emitted fragment. The fragment is constrained to move along a fixed, frictionless, non-conducting circular track of radius $r$ centered at the origin. It just reaches point $B$ with polar coordinates $(r, \pi/3)$ in the $xy$-plane, where its kinetic energy momentarily becomes zero. Assuming the only force doing work on the fragment is the electrostatic force from the dipole, which of the following expressions gives the correct value of the integer $n$?