This is an edited version of an unpublished 1979 EFI (U. Chicago) preprint: The U(N) lattice gauge theory in 2-dimensions can be considered as the statistical mechanics of a Coulomb gas on a circle in a constant electric field. The large N limit of this system is discussed and compared with exact answers for finite N. Near the fixed points of the renormalization group and especially in the critical region where one can define a continuum theory, computations in the thermodynamic limit $(N rightarrow infty)$ are in remarkable agreement with those for finite and small N. However, in the intermediate coupling region the thermodynamic computation, unlike the one for finite N, shows a continuous phase transition. This transition seems to be a pathology of the infinite N limit and in this simple model has no bearing on the physical continuum limit.
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