Conversion of potential energy of charge into kinetic energy

Question

Suppose I have a battery of voltage v which can flow $Q$ coulomb of charge through a metalic conductor

When voltage is applied across this conductor the electrons inside it get accelerated and move with drift velocity. let us assume that at first the potential energy of 1C charge is 100J Will this 100J of energy instantly get converted into kinetic energy or it will slowly convert into kinetic energy?

Another question is, will that kinetic energy increase with time because electric force is inversly proportional to square of distance, as electron move toward the positive terminal of battery force on the electron increase then kinetic energy must be increase

And when electron reached positive terminal of battery. Will there kinetic energy instantly get zero?

Pls answer me if you understand what I am trying to say

Answer 1

Will this 100j of energy instantly get converted into kinetic energy or it will slowly convert into kinetic energy?

Neither. The kinetic energy of the electrons is not at any time relevant or significant in the behavior of an ordinary circuit.

The only time that the KE of the electrons will become important is in a cathode ray tube, a particle accelerator, or other similar devices.

Answer 2

I'm not quite sure what you're asking, but I will try to answer either way. When you have a battery capable of outputting some voltage "V" and you have a conductor with resistance "R" and you connect your battery terminals with your conductor, the current "I" will start flowing. This current is constant throughout your conductor, and it's just amount of charge that passes through your conductor in some amount of time. Why is the current, i.e. the velocity of electrons, constant throughout the conductor? Because if you imagine that is not the case, i.e. if you assume electrons are faster in certain part of the conductor, that would mean that in other parts of your conductor electrons would build up, and that's not really energetically favourable because they repel each other. Regarding your question about transformation of energy : as I said, as soon as you close the circuit, the current "I" is established, this current doesn't change over time (ideally) meaning your electrons aren't really accelerating at all (well, classically they do, but only for very short distance until they scatter, i.e. bump, into atoms and lose majority of their kinetic energy). So, the entire potential energy of the battery is slowly lost on these scattering processes, which is basically just heat.