The Lorentz gauge is the preferred gauge when problems in electromagnetic scattering and diffraction are solved by a classical Maxwell equation potential function approach. The free space magnetic vector and scalar potentials subject to the Coulomb gauge are obtained in the paper. It is shown that the Coulomb gauge vector potential dyadic Green's function can be extracted from the well known Lorentz gauge free space vector potential Green's function. The electric field mixed potential integral equation (MPIE) for a straight, circular, finite-length, narrow conducting cylinder scatterer is formulated and solved in both the Lorentz and Coulomb gauges and results are presented in terms of the cylinder surface current. The merits and drawbacks of the Coulomb gauge formulation are discussed.