Electron accelerated in electric field

[GhostLoveScore]

We have this situation

an electron comes flying in electric field between two capacitor plates. Let's say that voltage is 12V and distance between plates is 1m. Electric field E=U/d=12V/m. Plates are connected to the battery just initially. The battery is later disconnected.

If electron comes flying in right in the center, between the plates, electric field accelerates electron and gives it some energy. Kinetic energy of the electron in the "y" direction is now E_k=E*q*d/2

Now I have one question.
If the electron doesn't hit positive plate, that means that energy from the field was transferred to electron and if we would continue accelerating electrons the field will obviously lose energy and it will have to weaken and disappear? So where have all the electrons from negative plate gone?What if the electron hits the positive plate? That means that it will return it's energy back into the positive plate. But if it hits positive plate it will reduce positive charge on that plate. That would mean that it will again weaken the electric field between two plates. I know there is a catch here, but I can't see it.

EDIT: correcting an equation

 

[Jonathan Scott]

If the electron doesn't hit the positive plate but flies past it and has enough kinetic energy not to get stopped, it will lose the gained kinetic energy as it flies away. It will be climbing out of the positive potential (as it is closer to the positive plate) back into a neutral potential.

If it hits the positive plate, then the energy of the collision becomes heat and/or chemical and mechanical energy, and is lost from the field.

 

[GhostLoveScore]

Ah yes, I forgot that there is electric field outside the plates. That makes sense. thanks

So, when the electron hits positive plate, why doesn't it neutralize some of the positive charges on it?

 

[Jonathan Scott]

So, when the electron hits positive plate, why doesn't it neutralize some of the positive charges on it?

It does cancel some positive charge. That's the mechanism by which the energy is lost from the field and dissipated in the collision.

 

[Dale]

Kinetic energy of the electron in the "y" direction is now Ek=E*q*d/2

There is no such thing as kinetic energy in the y direction, however you can just start with the electron at rest in the center so that the kinetic energy is given by your expression.

But if it hits positive plate it will reduce positive charge on that plate. That would mean that it will again weaken the electric field between two plates.

Yes, the field is weakened and some work is done on the plate, usually going quickly to thermal energy.

 

[GhostLoveScore]

There is no such thing as kinetic energy in the y direction.

I meant to say that I am only looking at the y component of the speed. Because y component of the speed is being affected by electric field.
But you are right, the initial question would be better if I said that electron was at rest in the middle of the plates and there is a small hole in the plate for the electron to pass through.

 

[sophiecentaur]

But if it hits positive plate it will reduce positive charge on that plate.

If the Anode(+) is maintained at a given Potential, the electrons will be removed from it by the power supply. By a possibly circuitous route, electrons will shuffle around - possibly via Earth and the electrons at the Cathode (-) will be replenished.
If the Anode is not maintained at its potential then electrons will build up and alter the Potential. This can cause an electron beam in a crt to spread out. If the accelerating voltage is not high, it is possible for so much negative charge to build up on an insulated plate that electrons will actually be repelled.