Question #43865

A small dielectric cylinder of length $2L$ and cross-sectional area $A$ is placed along the $x$-axis from $x = -L$ to $x = L$. It is subjected to a uniform external electric field $\vec{E} = E_0 \hat{i}$. The material's electric susceptibility follows a Curie-like temperature dependence, $\chi_e = \frac{C}{T}$, where $C$ is a constant and $\chi_e \ll 1$ (so the internal field is approximately $E_0 \hat{i}$). A temperature gradient is maintained across the cylinder such that $T(x) = T_0 (1 + \beta x)$, where $\beta L \ll 1$. Let $p_0$ be the dipole moment of the cylinder if it were at a uniform temperature $T_0$. Find the electric potential $V(d)$ at a point $x = d$ on the axis ($d \gg L$). Retain the leading order term and the first non-vanishing correction term involving $\beta$.