Question #43794

A diatomic molecule is modeled as two point masses $m_1 = m$ and $m_2 = 2m$ carrying charges $+q$ and $-q$ respectively. They are bound by an effective harmonic potential such that the internal restoring force is $F_{\text{int}} = -K(x - x_0)$, where $x$ is their separation and $x_0$ is the natural equilibrium length. The molecule is placed in a non-uniform electric field $\vec{E} = \alpha z \hat{k}$ (where $\alpha > 0$ is a constant). At $t=0$, the center of mass of the molecule is at $z=0$, and the masses are aligned along the z-axis with $m_1$ at a larger z-coordinate. The molecule is initially at rest with separation $x = x_0$. Assuming the molecule is very stiff ($K \gg q\alpha$) such that the separation $x$ remains close to $x_0$, find the time-dependent quasi-equilibrium shift in the separation, $\Delta x(t) = x_{\text{eq}}(t) - x_0$, to leading order in $1/K$ for $t > 0$.