We solved numerically the Vlasov-Maxwell system of equations for a bounded cylindrical and radial inhomogeneous plasma which is confined by a strong magnetic field directed along the axis cylinder. Through this solution we found numerically the radial structure of the axial electric field corresponding to the high frequency fundamental transverse magnetic mode propagating in the cylindrical wave guide. Our result shows that the intensity of the electric field tends to be higher in those regions where the plasma is denser and also the field presents oscillations with intensities that decrease and vanish at the radial plasma boundary. This behavior could be relevant in the design of efficient modern plasma based particle accelerators that use the axial electric field to achieve this task.