The <mean energy of a two-dimensional Fermi gas> follows from its constant <density of states> $g$ for nonrelativistic quadratic dispersion. At zero <temperature>, states below $\epsilon_F$ are filled, so
$$
N=g\epsilon_F,\qquad U=\int_0^{\epsilon_F}g\epsilon\,d\epsilon
=\frac12g\epsilon_F^2,\qquad
\boxed{\frac UN=\frac{\epsilon_F}{2}}.
$$

Solution

Past exam of the physics A course of the University of Cambridge / 2014 / Paper 2 / Section A / 4 / Solution

 Solution (source code)