Detailed electromagnetic analysis of the wake-field accelerator is presented. Particular attention is given to the two-dimensional dynamics model to understand how the wake-field affects the self-consistent radial dynamics of the beam. The corrent fields, in zeroth and first order, are obtained by using Fourier transform in the radial field. The self-focusing of the driving beam in a plasma is a result of the transverse wake. The ratio of the transverse wake-field to the longitudinal wake-field within the beam is calculated for small and large radii beams.