Increasing the Voltage, Ohms law, what am I missing?

Question

There is one thing in Ohms law I am missing I think.

Lets say you have a simple circuit with a single resistor, $R$. We apply voltage $V_1$ to it, by Ohms law the current is $I_1=V_1/R$.

Now we increase the voltage to $V_2$, I am wondering what actually happens then. Voltage is defined as energy per charge, so we increase the energy we are giving the charges over $R$, I assume this materializes as a stronger electric field? So then we accelerate the electrons up to current $I_2=V_2/R$? But I am wonder why the current/speed of electrons don't increase further? My rationale is this:

With the voltage $V_1$ we had an electric field that was strong enough to overcome the internal resistance and keep the current at $I_1$. I look at the internal resistance as a friction force $r$ on the electrons, and the work the internal resistance does is $rl$ where $l$ is the length of the resistor. When we increase the voltage to $V_2$ we also increase the force on the electrons, but the internal resistance is the same? So then the electrons should keep accelerating and the current should just increase and not find stop at $I_2=V_2/R$, because the other work on the electrons has increased, but the internal resistance and the friction work done by the resistor is the same? We keep feeding the electrons more and more energy?

Can you please explain the flaw in my argument and why the current doesn't just keep increasing?

Answer 1

What I don't understand is why we don't then get acceleration according to Newtons second law?, and the electrons just keep accelerating?

The electrons do accelerate and then collide with lattice ions, then accelerate between collisions.
On average one can assign a drift speed to the motion of the electrons under the influence of the external electric field.
If the electric field increases then the drift speed increases in proportion.

My answer to the post Drift velocity in Drude model explains how the averaging is done.

Answer 2

First, you should stop thinking about electrons at all in this context. They serve no value in basic circuit theory and they are clearly confusing you. Just think in terms of current and voltage. A resistor enforces that the current is proportional to the voltage. Simple.

The issue is that electrons are not simple entities. To really understand how they work requires QED, and the usual Drude and similar models are not particularly simple nor beneficial.

However, your mental model departs in a critical way from the Drude and other similar models. Specifically:

When we increase the voltage to V2 we also increase the force on the electrons, but the internal resistence is the same? So then the electrons should keep accelerating and the current should just increase and not find stop at I2=V2/R, because the outher work on the electrons has increased, but the internal resistance and the friction work done by the resistor is the same?

It is correct that the internal resistance is the same, but nothing else in here follows from that. You are envisioning the resistance as a fixed force acting on the electrons, which is not at all how electrical resistance behaves. As the current is increased the amount of "friction" work is increased also. This is not a fixed force, but rather a force which is depends on the current density. As you increase the voltage you increase the energy put in by the power source and you also increase the energy removed by the resistance.