Question #43867

A rigid, smooth vertical circular wire of radius $R$ is fixed in space. A small bead of mass $m$ and positive charge $Q$ can slide freely on the wire. At the center of the circular wire, a short electric dipole is fixed. The dipole consists of two point charges $+q$ and $-q$ connected by a light insulating rod of length $L_0$ (where $L_0 \ll R$) at an initial temperature $T_0$. The rod has a coefficient of linear thermal expansion $\alpha$. The dipole is oriented such that its dipole moment vector $\vec{p}$ points horizontally to the left. Let $\theta$ be the angle specifying the position of the bead on the wire, measured counterclockwise from the downward vertical. At temperature $T_0$, the bead is in stable equilibrium at $\theta = \pi/4$. If the temperature of the dipole's rod is increased by a small amount $\Delta T$ (while the rest of the system remains unaffected), the new stable equilibrium position of the bead shifts to $\theta = \pi/4 + \Delta \theta$. Assuming $\Delta T$ is small enough that higher-order terms can be neglected, what is the value of $\Delta \theta$?