Question #42760

A homogeneous cylindrical poorly conducting rod of length $L$ (in $\text{m}$), cross-sectional area $A$ (in $\text{m}^2$), and constant permittivity $\epsilon$ (in $\text{F}\cdot\text{m}^{-1}$) has an initial uniform electrical conductivity $\sigma_0$ (in $\Omega^{-1}\cdot\text{m}^{-1}$). Due to uniform Joule heating from a prior transient current, the momentum relaxation time of the charge carriers decreases, resulting in a new uniform steady-state conductivity $\sigma = \sigma_0 / \sqrt{3}$. Immediately after this heating phase, an AC source is connected across the rod, driving a total uniform current $I(t) = I_0 \cos(\omega t)$ (in $\text{A}$) through it, where $I_0$ is the current amplitude and $t$ is time. If the angular frequency of the AC source is $\omega = \sigma_0 / (\sqrt{3}\epsilon)$ (in $\text{rad}\cdot\text{s}^{-1}$), which of the following expressions gives the amplitude of the potential difference $V_0$ (in $\text{V}$) across the rod?