I would agree with every part of the previous answer, except the first one. The density of charge carriers is (in typical cases) determined by the material, and thus unaffected by changes in current. The change in current just means the charges move faster. The electric field lines applying a force to the charge carriers get more dense (denoting an increase in E field strength) but the charges do not get proportionally more dense with current.
Also, for the second answer, I would add the important assumption that the current is uniform and changes uniformly.