Electrical breakdown of a gap between a wire (modeled as a hyperboloid) and a plane has been investigated numerically by solving the two-dimensional form of the diffusion flux equations for the charged particle number densities and Poisson's equation for the self-consistent electric field. Electron impact ionization, thermal ionization, and three-body recombination have been considered as the charged particle production and loss mechanisms. The electrode surfaces are considered to be absorbing and the initial density of the particles is small, but nonzero, A gap length of 0.5 mm is investigated and the gas medium is air or argon at atmospheric pressure. The temporal development of the profiles of ion and electron number densities, potential and electric field, and current growth on both the electrodes are presented when the applied voltage is 1500 and 2500 V for both positive and negative wires. When the wire is negatively biased, the peaks in the radial distribution of both of the charged particle densities near the wire occur off the axis except during the very early part of the breakdown. With positive polarity, the electron density maximum always occurs on the discharge axis, while for ions it moves away from the axis, later in the transient, due to the reverse particle drift in the electric field from the negative polarity case, The discharge spreads farther out into the ambient (almost two times the gap length) when the wire is negatively biased than with positive polarity. The effect of charge separation on the externally applied electric field is significant at voltages 2500 V and higher. Ionization is greater in argon than in air for a fixed potential difference between the electrodes.