Distribution of charge along the tower and lightning channel when lightning strikes a tall tower

We derive expressions to calculate the charge density along the tower and lightning channel based on the assumptions of the transmission line model of the lightning strikes to a tall tower and a series point current source placed at the tower top. These expressions are used to calculate the very close-range electric fields in the monopole (continuity equation) technique in terms of the retarded current and charge density along the tower and lightning channel and their results are compared with those calculated from the traditional dipole (Lorentz condition) technique in terms of the retarded current along the tower and lightning channel. Alternative explanations are provided to the inversion of polarity of the vertical electric field at very close range based on distribution of charge density along the tower and lightning channel.