Question #42491

A parallel plate capacitor with rectangular plates of length $L$ and width $w$, separated by a distance $d$, is connected to a constant voltage source $V$. A dielectric slab of mass $m$, length $L$, width $w$, and thickness $d$ is attached to a relaxed spring of spring constant $k$. The other end of the spring is fixed. The slab is partially inserted into the capacitor by a distance $x$ (where $x=0$ corresponds to the spring being relaxed and the slab just at the edge of the capacitor). The dielectric constant of the slab varies along its length as $K(y) = 1 + \alpha y^2$, where $y$ is the distance from the leading edge of the slab. Ignore fringe effects and assume parameters allow a non-zero equilibrium position $0 < x_0 < L$. Determine the nature of the equilibrium at $x_0$ and the effective force constant $k_{\text{eff}} = \left. \frac{d^2 U_{\text{eff}}}{dx^2} \right|_{x_0}$.