I've come across an instructive paper which explains qualitatively what happens inside wires of a simple circuit during the transient state, leading to the steady state where there is an uniform E field everywhere inside the wires.
Here are two figures from the paper which are pretty much self explanatory :
What I don't understand is that sentence from figure 6 :
"pile-up of surface charge will continue until the net electric field does point to the left"
Why does the pile up of charges continues even after the net field becomes 0 ? Why charges keep piling up until the E field reverses ? Shouldn't the pile up stops when the net E field reaches 0 ?