Aha, I believe I see the confusion now. You are confusing the potential energy of a circuit with the potential energy of an electron. Let me explain...
Potential energy is just the tendency of something to move. An object will want to move towards a spot of lower potential.
- Like a book on a shelf; it wants to fall down, since the potential energy is lower there.
- Same for an electron. If you put an electron close to the negative end of a battery, it will repel because the further away it gets, the smaller is the potential energy.
- But a positive charge would be attracted and would move closer to the negative battery end! The potential energy is in this case smaller the closer they are.
The point is that potential energies depend on the type of "object". In the case of electronics, it depends on the sign of the charge.
In general when we talk about potential energies or potentials or voltages of some circuit, we don't necessarily know what the moving charges are. In other words, we don't know what the charge carriers are.
- They can be negative such as electrons in usual metallic wires.
- They can be positive such as holes in semiconductors. A hole is a missing electron and corresponds to a positively charged carrier.
- They can be a mix such as ions in a conducting solution (an electrolyte). When there is a voltage over some electrodes stuck into this solution, negative ions move to the positive one and positive ions to the negative one.
Current flowing is the total motion of moving (or drifting) charge. As you see there are several types of current; which just means several types of charge carriers.
If we talked about potential of a specific type of charge, we would have to know which carrier is present. To avoid having to know this, people in the physics world have decided that whenever we talk about potential, we talk about it as if the charge carriers are positive. (The same goes for direction of current, direction of electric field etc.)
With this consensus, it is easy to just flip it around, if we know that we have negative charge carriers in a specific case. If I tell you that the current flows clockwise, but you know that it is metal wires so electrons are the carriers, then you know that the electrons are moving counterclockwise.
And the same applies for the potential: If I tell you that there is high potential at point A and lower potential at point B, then I mean that a positive charge sees it this way. A positive charge would want to move towards point B rather than A. But an electron would feel the exact opposite - whatever repels the positive charge from point A, attracts a negative charge. And vice versa. An electron would see a lower potential at point A and a higher at point B.
I hope I understood the point in your question and have addressed it in this answer. Otherwise please let me know.
When you move a specific amount of charge across a particular voltage difference, then a specific amount of work is done.
Since electrons have a constant charge, then moving an electron across an increasing voltage difference requires (or produces) more work.
Imagine there's two conducting wires exactly the same in every way.
The first has 1 volt times 1 amp of electrical power travelling
through it.
The second has 5,000,000 volts times 1 amp of electrical power
travelling through it.
The important part about the above is that the charge has to move across a specific voltage. "Wires" are normally conductors, and they usually have very little voltage difference from one end to the other. So a wire being at "1 volt" or any other amount isn't the big deal. It's the voltage difference between that wire and some other location. The energy is released or expended as the charges move to the new potential.
Think of it like trying to get energy from a falling weight. You can hook up a weight to a rope and get energy from it. But for the same rock and the same harness, you can get more energy if you can let it fall farther.
The charges in the wires are the same, but you can get more energy from them if you have a larger potential difference with some other location.
Best Answer
You write
This isn't correct. The statement should instead be (in more generality)
Fact. When there is a potential difference $\Delta V$ between two points, then if a charge $q$ moves from one of the points to the other, then its electrostatic potential energy will change by $\Delta U = q \Delta V$.
Therefore, if an electron moves from one point to the other, and if there are no other interactions around, then the change in its electrostatic potential energy in going from one point to the other will equal the (negative of) the change in its kinetic energy by energy conservation: $$ U_1 + K_1 = U_2 + K_2 \implies \Delta U = -\Delta K $$