  We consider $N$ particles in the plane influenced by a general external
potential that are subject to the Coulomb interaction in two dimensions at
inverse temperature $\beta$. At large temperature, when scaling $\beta=2c/N$
with some fixed constant $c>0$, in the large-$N$ limit we observe a crossover
from Ginibre's circular law or its generalization to the density of
non-interacting particles at $\beta=0$. Using several different methods we
derive a partial differential equation of generalized Liouville type for the
crossover density. For radially symmetric potentials we present some asymptotic
results and give examples for the numerical solution of the crossover density.
These findings generalise previous results when the interacting particles are
confined to the real line. In that situation we derive an integral equation for
the resolvent valid for a general potential and present the analytic solution
for the density in case of a Gaussian plus logarithmic potential.
