Why the electrons don't fall down when we cut a current-carrying wire?

When we cut a current-carrying wire, the potential difference becomes zero across the conductor as the circuit becomes incomplete, also the electric field vanishes. Now, there is no more electric force on electrons. So, the electrons present in whichever atom they are when the wire is cut off, they stay there. To jump to the next atom and move they need more energy or pressure (voltage).

 The electrons do not fall down because the attractive force between the nucleus and electrons of an atom of a conductor is more than the attraction force of the earth (gravitational force) on the electron.