Question

How does the random motion of free electrons in a conductor get affected when a potential difference is applied across its ends?

Answer 1

An electric charge will experience a force if a potential difference is applied across its ends. Such a force would provide directionality to the electrons.

Complete solution: Consider first the case when no electric field is present. The weakly bound electrons of an atom would be knocked off due to the thermal effect. These electrons would then collide with the fixed ions. These electrons would eventually approach the ions due to the local potential differences. But the direction of motion of these electrons and their collision would be random. In addition, an electron colliding with an ion emerges with the same speed as before the collision. All these imply that at a given time, there is no preferential direction for the velocities of the electrons. Thus on the average, the number of electrons travelling in any direction will be equal to the number of electrons travelling in the opposite direction. So, there will be no net electric current.
But when potential difference is applied, the electrons will drift towards the region of higher potential. The random motion of free electrons thus becomes almost a directed motion.

Note: In some materials, the electrons will be bound, i.e., they will not accelerate even if an electric field is applied. These are the insulators that lack such free electrons or thermal electrons. In other materials, notably metals, some of the electrons are practically free to move within the bulk material. In order to maintain a steady electric current, the ends of a metal conductor can be supplied with fresh charges to make up for the neutralised charges. Cells and batteries can maintain a steady electric current.