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A very simple circuit consist of a resistor, a battery, and some conducting wires connecting the resistor to the battery(assuming resistance of the conducting wire is $0$). It is always assumed that both the ends of the conducting wire, i.e. one connecting to battery's terminal and other connecting to the terminal of resistor, are at equal potential and hence the potential difference is zero.

If the potential difference is zero inside conducting wire, then it implies that electric field inside the conductor must be zero. If this is the case then why does an electron inside the conducting wire move, and why there is current flowing through the wire?

My Idea

If the electric field inside conducting wire is zero then it only means that electrons inside conducting wire are not accelerating but can move with a constant speed and since current is the rate of flow of charge per unit time, so it is possible for electric current to exist in the conductor.

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    $\begingroup$ Superconductors exist. But in your case, it is just an idealized case to make learning the fundamentals easier. $\endgroup$ – Jon Custer Sep 24 '19 at 15:05
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Their are two points to make here:

(1) In an idealised circuit model the resistance is modelled as zero. This means that it is not actually zero, it is just very low and not worth the bother of including in calculations.

(2) In a superconducting wire you don't need a voltage to have a current. A loop of superconducting wire with no other things on the circuit can carry a current (where the electrons just go round and round the loop). Clearly in this loop their is no potential difference between any two points.

This arises (exactly as you guessed) from the fact that electric fields are only needed to accelerate electrons. When we talk about normal (non superconducting) circuits we are always talking about the steady-state, where the interia (mass) of the electrons is ignored and we match the resistance against the force of the push (voltage).


As a mechanical analogue: ordinary electric circuit equations can be compared to a lightweight bead being pushed through treacle. The force pushing the bead forwards (Voltage) is immediately matched by the drag from the treacle (drag coefficient times the bead's speed $\sim$ resistance $\times$ current). The time taken to accelerate the bead up to this equilibrium speed is neglected because the treacle is thick and the bead light. (IE electrons are not very heavy).

In a superconductor it is now like a bead in vacuum, where in the presence of a constant force it will undergo constant acceleration.

Useful link:

See the equations immediately under the header "London Theory" on this Wikipedia page. You will see that the rate of change of current (speed-up of electrons) is related to the electric field.

https://en.wikipedia.org/wiki/Superconductivity

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My Idea

If the electric field inside conducting wire is zero then it only means that electrons inside conducting wire are not accelerating but can move with a constant speed and since current is the rate of flow of charge per unit time, so it is possible for electric current to exist in the conductor.

Essentially, that is correct. I like the mechanical analogy of pushing a box on a surface with friction vs one without friction. When pushing the box on a surface with friction I have to apply a force equal to the opposing kinetic friction force which makes the box move at constant velocity. I do work which is dissipated as friction heating. This is analogous to the force of an electric field needed to do work to overcome electrical resistance, resulting in resistance heating.

But if I then encounter an ideal frictionless surface (the ideal zero resistance conductor) in the same path as the surface with friction, I can release the box and it will continue at the same constant velocity it had on the surface with friction, but now without the need of a force and work to keep the box going.

The two situations taken together are analogous to the same current flowing in a series circuit comprised of a resistor and an ideal zero resistance conductor.

Hope this helps.

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