Why is the electric potential difference independent of resistance?

According to wikipedia the definition of electric potential difference is as follows:

The difference in electric potential between two points (i.e., voltage) is defined as the work needed per unit of charge against a static electric field to move a test charge between the two points.

Imagine a simple circuit one variable resistor connected to a 12v battery. An electric field is created between the + and - pole of the battery. The difference in electric potential is apparently constant when measured across the resistor independent of it's resistance.

Intuitively I would say that the work needed for a unit of charge to pass a with resistance $$a$$ is larger than for a resistance $$b$$ where $$a>b$$. This implies that that the potential difference across the resistor is greater for $$a$$ than for $$b$$. Why am I wrong?

You are confusing work done with energy needed. The work done is equal to the change in potential and kinetic energy of the charges. This is O dependent of the resistance. The energy spent is the sum of the work and the heat losses whole performing the work. The latter do depend on the resistance.

• So we only look at the work done by the force in the electric field? Why is it then there is only work (a voltage) across this resistor and not across any other section of the wire? – Ruben23630 Nov 7 '18 at 11:14
• Because we consider that th wire is an ideal conductor, with 0 resistance and no electric field inside. – FGSUZ Nov 7 '18 at 11:55
• But if there is no electric field inside a wire, what causes the electrons to flow inside the circuit? – Ruben23630 Nov 7 '18 at 13:11
• @my2cts Can you explain why the change in potential/kinetic energy when a unit of charge passes a resistor is the same for every resistor? – Ruben23630 Nov 7 '18 at 13:13
• @Ruben23630 the work only depends on the difference between the initial and the final energy of the charge, so only on the voltage. – my2cts Nov 7 '18 at 13:45

There is no overall change in KE of the electron as it goes through the resistor. Both the average thermal energy as well as the drift velocity are the same at both end of the resistor (and everywhere in the resistor). The electrons do not travel through the resistor as in vacuum, in a straight line. I suppose this is one source of the confusion. It may be that the electrons entering the resistor never come out, during your experiment.

The potential and potential energy are defined only for the electric field. So the potential difference is related to the work done by the electric field only. This is the conservative field and can be associated with a potential. The work of other forces (dissipative) cannot be described by a potential energy.

There is an electric field inside the conductor when there is a current flowing through it. The zero field inside conductors applies only to electrostatic situations, those with no current, no net motion of charge. The field is produced by surface charges established on the conductors once the steady state was achieved.