I know a battery creates a potential difference, making an electric field that exerts a force on the electrons, who start moving. But why is there a potential drop after a resistor for example? How does it go in hand with electric potential being a scalar assigned to a point in space? How can a resistor change the potential of all the points in the conductor succeeding him? Or am I looking at it in a wrong way? I think I'm considering electric potential from an electrostatics point of view and it gets me nowhere.
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$\begingroup$ You can prove it mathematically by considering the line integral of electric field over the loop and applying some vector calculus identities. The intuition is that if it didn't drop then each time you looped you'd be gaining energy. $\endgroup$– Cantor Dust DrachenCommented May 11, 2020 at 23:55
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$\begingroup$ Very closely related: How do resistors form electric fields and thus potential drops? $\endgroup$– VishnuCommented May 12, 2020 at 4:22
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2 Answers
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The relations between currents, electric and magnetic fields in a circuit have to follow Maxwell equations. There is nothing there about resistances.
So it is possible to have a current in a closed circuit without any eletric field, (so no potential drop) since the relation between $\mathbf B$ and $\mathbf E$ are fulfilled. It is the case of superconducting.
So it is the other way around. If there is an electric potential drop (so an electric field) between two point of a circuit where a current is present, there is some resistance there. It is a definition of resistance. $$R = \frac {\Delta V}{I}$$
answered May 12, 2020 at 2:15
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Think of it in terms of impedance to a wave. Say there is a rope in air with some portion in the middle of it being in water. 
Now if we move one end of the rope continuously, the wave travels until it reaches the water-air interface. Here some of the wave reflects and other gets transmitted, constrained by energy conservation. And similarly at the second interface.
Under equilibrium condition, something called as the steady state is achieved. Here the flow rate is constant. This means that the reflections are coming in constantly to the constant wave generation done at the left end. The energy flow in the system is constant.
Here it’s easy to see that in the presence of the water region in between, the energy transferred to the other end has decreased. The flow of energy from left to right of the whole system has decreased due to the presence of impedance at some portion in the middle.
This is exactly what happens in the case of resistance present in the circuit. It provides an impedance to the flow (current) and in steady state, the reflections from the interface cause the current in the whole circuit to decrease.
answered May 12, 2020 at 15:13