Question #84419

A classical dilute gas consisting of $N$ non-interacting particles, each of mass $m \text{ (kg)}$ and positive charge $q \text{ (C)}$, is maintained at an absolute temperature $T \text{ (K)}$. The gas is confined in the half-space $z \ge 0 \text{ (m)}$ and is subjected to an external potential energy field given by $U(x,y,z) = \frac{1}{2}\alpha(x^2 + y^2) + qE_0 z$, where $\alpha \text{ (J m}^{-2}\text{)}$ and $E_0 \text{ (V m}^{-1}\text{)}$ are positive constants. If $k_B \text{ (J K}^{-1}\text{)}$ is the Boltzmann constant, what is the average total energy per particle of this gas in thermal equilibrium?