The electric field and surface charges far and close to the battery for the transmission line

We consider two long resistive straight parallel wires carrying opposite constant currents and calculate the potential and electric field everywhere in space and the surface charge densities along the wires. The problem is solved through Laplace's equation in bi-cylinder coordinates, far from the battery. We compare these calculations with previous known results that used different methods. We also calculate the numerical solution for the case in which the battery is present, and show the equipotentials and surface charges close to the battery.