electromagnetism - How does the speed of electrons change around a circuit?

I have been thinking about ways of teaching electronics and I'm wondering if the following is true...

For starters, when we talk about voltage as energy per unit charge, is this energy manifest simply as the kinetic energy of the electron?

So, all other things being equal, will the electrons coming out of a 4 volt battery will have twice the velocity of electrons coming out of a 1 volt battery because of ${1 \over 2}mv^2$?

This would seem to imply that the voltage drop across different parts of the circuit is essentially a comparison of the electron speeds at those points.

Furthermore doesn't this imply that the electrons are nearly halted by the time they reach the positive terminal of the battery? This makes intuitive sense to me, it seems pedagogically sound, and it provides an explanation for how electrons "know" to give up their energy across the circuit -- electrostatic repulsion communicates later resistances to earlier parts of the circuit.

How much of the above is correct? The one things that makes me suspicious is that, in this picture, electron density at the positive terminal of the battery is very high, and repulsive forces would prohibit this. On the other hand, if the Joules per Coulomb are not coming from kinetic energy, then what? The field? But wouldn't that just dissipate as light?

Something is wrong but I don't know what.

Answer

For starters, when we talk about voltage as energy per unit charge, is this energy manifest simply as the kinetic energy of the electron?

Voltage is potential energy per unit charge. An analogy: voltage is to charge as altitude (as on the surface of Earth) is to mass.

So if you lift a 1kg rock off the ground by a height of 1m, you've added some gravitational potential to that rock. Does it manifest as kinetic energy?

Well, if you drop it in a friction-less environment, maybe. But any number of other things could happen. The space within the 1m where the object falls could be filled of some viscous goo. In that case, most of the gravitational potential energy is converted to heat by way of friction. Or, we might design an apparatus to turn a generator as the mass falls, converting its gravitational potential energy into some kind of electrical energy. Maybe we charge a battery, and it becomes chemical energy.

And of course, at the bottom of the fall it probably hits the ground, transferring momentum to the Earth and sending off energy as sound, among other things. It doesn't stay as kinetic energy for long, it most real situations.

And so it is with electricity, too. You probably won't find anything analogous to a frictionless place to "drop" electric charge. If the current goes through a resistor, then it is converted to heat. If it goes through a motor, then it's converted to mechanical energy.

There are devices that launch electrons in essentially lossless conditions (vacuum tubes, CRTs), but most such devices have something at the end into which the electrons smack to do something else (in the case of the CRT, produce visible light). In these cases, the kinetic energy of the electrons immediately prior to impact would be proportional to the voltage through which they have fallen, but I don't suspect this is especially insightful for teaching electronics generally. If anything, it demonstrates that even if the current is constant in a series circuit, the speed of the electrons is not necessarily so.

To know what becomes of the electrical potential energy, you need to know what happened to the charge as it fell.