Why is there no electric field in the ideal wire?

Why is there an electric field only inside the resistor in the circuit (assuming that the wire has no resistance)?

If there is no electric field inside the wire, why does the electron move to the positive pole?

There is no electric field in a zero resistance wire, but with the exception of superconductors, all real wires have some resistance. In a superconductor electrons can freely move with no energy loss and therefore require no electric field.

The resistance of wires in a circuit is typically so low compared to resistors in the circuit that they are generally approximated as zero resistance conductors for the purpose of circuit analysis.

Hope this helps.

• Thank you for the answer, but I still have a question. Even a superconductor, there must have a force to drive the electrons to flow in a uniform direction when the circuit is turned on, then after the electrons start the motion, the force disappear. Commented Jun 15, 2021 at 23:02
• Sure you need an energy source to start the process and thereafter it can be removed and the electrons will continue to move. A mechanical analogy is pushing a box on a friction less surface (analogous to a zero resistance wire). You give it a push (do work to accelerate it and give it a velocity) and the box theoretically moves on at constant velocity forever without any additional work. Commented Jun 15, 2021 at 23:15

The wire has the same potential everywhere if its resistance is zero. There is no force needed to keep the electrons flowing in it because of the very fact that its resistance zero. The positive pole has the same voltage as at the resistor. So inside the wire, there is a constant voltage and no electric field. But how can the electrons flow? What gives them their motion? They are kicked into the wire and battery pole by the inside potential and beacause the wire has no resitance they continue to have this motion. They are not slowed down by the resistance of the wire, as this is zero.

• Thank you, but I don't really understand the reason why the electrons flow, where does that force come from? Commented Jun 15, 2021 at 23:16
• "So inside the wire, there is a constant voltage and no electric field", how can this be if voltage is a measure of electric field strength? Voltage is necessarily coupled with the electric field. The electric field is defined as $E = v/L$ where L is the length of the wire and v is the voltage. If there is voltage, then there is an electric field. Commented Jun 15, 2021 at 23:17
• @Mr.Bohner The electrons get a first kick from the inside of the battery after which they keep this momentum, as there is no resistance in the wire.
– user304539
Commented Jun 15, 2021 at 23:20
• @Tachyon The voltage can be constant over the wire. Only compared to Earth there is a difference in voltage. Maybe you mean something different with voltage. I see it as the field with constant potential energy. So the voltage in the wire is the same as the at the battery pole.
– user304539
Commented Jun 15, 2021 at 23:22
• I think the force is always directed to lower potentials and as the potential is constant there is no force.
– user304539
Commented Jun 15, 2021 at 23:25

Why there is an electric field only inside the resistor in the circuit (assuming that the wire has no resistance)

There is an electric field inside a resistor, or a capacitor for that matter is because there exists a potential difference across it. The potential difference exists because the battery deems so.

The wire doesn't have any potential difference across it -its an equipotential, an ideal conductor with zero resistance and hence can't hold an electric field inside.

If there is no electric field inside the wire, why does the electron move to the positive pole?

Oh no. The electron moves precisely because its experiencing an electric field - that created from the battery. Where does the electron go after leaving the negative terminal of a battery in a circuit? In the connected ideal wire? No. It goes straight to the resistor, then to the next component with the field.

But then why would one draw wires in circuit diagrams at all? That's because in real world circuits charge carriers do travel through wires to other components - they can' t jump through the intervening air gap$$^1$$ and need a conductive path. We carry forth this intuition in the form of an idealization while drawing circuit diagrams, drawing finite length ideal wires between circuit elements and showing currents even though, for this idealized diagram, the wires don't develop a field inside and play no role beyond that of showing connections.

$$^1$$ at lower than breakdown voltages

• "Where does the electron go after leaving the negative terminal of a battery in a circuit? In the connected ideal wire? No. It goes straight to the resistor, then to the next component with the field." If the current doesn't go via the superconducting wires to reach the resistors with the field, then how does it get there in the first place? Commented Jun 15, 2021 at 23:31
• @Tachyon current transport in superconductors is different than the ideal $0$ LCR wires one draws in circuit diagrams. My point is that the wires drawn are symbolic - even without them with components connected end to end, the circuit would be the same. Whether one can actually do this or not is a real-life question where we use wires anyways. To reiterate, in ideal circuit diagrams, charges move only in regions of field. Commented Jun 15, 2021 at 23:40
• @Tachyon and they are a quantum phenomenon. I am not sure if a classical idealization which is just a simplification tool, can even be extended to the much deeper phenomenon of resistance less flow, electrical or otherwise=e, merely because they share a characteristic. Commented Jun 15, 2021 at 23:49
• @Tachyon they are both $0$ res. But there are no ideal wires in the world - things we deem wires have low but finite resistances, ideal wires can't and don't carry current, and superconductors are much richer and $\ne$ "a real-world ideal wire". Commented Jun 15, 2021 at 23:51
• @Tachyon see physics.stackexchange.com/a/179386/226560 Commented Jun 15, 2021 at 23:52

It doesn't take force to keep up motion. It takes force to generate motion. This is Newton's 2nd law.

Only if there are frictions, resistances and the like along the path will you have to exert a forwards force to maintain the motion. In an ideal wire, there are no sich resistances or frictions. So in an ideal wire, no electric force is needed. Charges will continue moving with constant speed effortlessly.

No wire is ideal, though. So no wire contains zero electric field. But it can be very small. The closest to zero is within superconductors.