What exactly happens in a circuit after it gains potential energy?

Question

When an electron moves across a resistor, it loses potential energy.

Now imagine a simple circuit (series connection) with the negative terminal having 10v as its potential and positive terminal having 20v as its potential. Now when the electron moves from the negative terminal of the battery to the positive terminal, the energy required to move per 1C from one to the other terminal is 10v.

Fine, but when its moving from the negative terminal of the battery and encounters a resistor in its path, that time its energy is 10v. Now when it passes the resistor (ideally it should lose potential but conventionally due to the positive terminal being at higher potential) it gains potential energy and it is this difference 20-10 which is the voltage.

Now my question is, when it's passing through the resistor after coming out of it, isn't the potential of each electron increasing from 10v to 20v? So shouldn't the number of electrons passing through a cross section in one second increase? I'm somewhat aware of the accumulation of charge thing, the transient period etc., and how it reaches equilibrium. But I would like a concise explanation of this because that doesn't really answer the fact that if there was no resistor there and only a load, then the potential will also increase after a certain point, so therefore the current should too.

Answer 1

Now when it passes the resistor (ideally it should lose potential but conventionally due to the positive terminal being at higher potential) it gains potential energy

The electron has a negative charge. The potential energy is the product of the charge and the potential. So when the electron moves to a location with a higher potential, it has lower potential energy.

Now my question is, when it's passing through the resistor after coming out of it, isn't the potential of each electron increasing from 10v to 20v?

Remember that an electron has a negative charge.

So an electron at a location with 20 V potential has less potential energy than an electron at a location with 10 V potential.

It is at a higher potential, but it has lost energy as it passed through the resistor, just as you expected.

So shouldn't the number of electrons passing through a cross section in one second increase?

The current through the resistor depends on the potential difference between its two terminals.

If the two terminals are at 20 V and 10 V, that's a 10 V difference. If they're at 10 V and 0 V, that's a 10 V difference. If they're at -160 V and -170 V, that's a 10 V difference. In any of those cases, the current through the resistor will be $10 {\rm V}/R$ for a resistor with resistance $R$.