Absolutely. Electrons are accelerated by the field until they hit something. The average velocity of all the electrons due to the competing acceleration and col

Absolutely. Electrons are accelerated by the field until they hit something. The average velocity of all the electrons due to the competing acceleration and collisions is the drift velocity and it's very slow. On the other hand, electric fields move quickly. At some large proportion of the speed of light depending on the particular materials of the wire and some other factors.