How do we know that any two points of an ideal wire must be equipotential? [duplicate]

Question

How do we know that any two points of an ideal wire must be equipotential whether or not there is a current flowing through it?

I fully understand that how the electric field in a conductor in electrostatic condition will be zero. My question is primarily for a conductor which is not an electrostatic condition.

In any circuit in which circuit elements like batteries, resistors etc. are connected by "ideal wires" we assume that any two points on the ideal wires are at the same potential. What is the justification behind this assumption?

The popular explanation is as follows:

Since from Microscopic form of Ohm's Law, we know that "E = ρ J", and for an ideal wire, the ρ (resistivity) tends to zero, so E (electric field) must tend to zero and if electric field inside the ideal wire is zero, so potential drop across idea wire must be zero.

The question which then comes is, why and how exactly the electric field inside an ideal wire becomes zero? Mathematically I agree that it turns out to be zero, but how do we understand this idea "Intuitively"?

What makes electric field zero in a wire?

I read the above post on why electric field inside a wire becomes zero. It explains the following reason:

In the steady state condition, there is no electric field inside a perfect conductor because the electrons have moved to cancel out any electric field. That is because electrons will move whenever there is an electric field.

Does it and will it always happen that electrons rearrange themselves in such a way that electric field inside the ideal wire will always become zero?

Is there a fundamental law which governs this idea that electric field inside an ideal wire should be zero? If yes, then what that is?

Is it just an inductive/recurring fact that it always becomes zero (like Sun rises from the east) that we assume this fact to be true? Or there is some fundamental reality which is governing this fact?

Kindly help.

Answer 1

First, keep in mind that since these are ideal wires, what you are asking is fundamentally a mathematical question that comes down to the definition of "ideal." In any real, physical wire, there is a finite (non-zero) resistivity and there is a finite (non-zero) current. However, for typical real wires in typical real circuits, the resisitivity of the wire is much smaller than that of other components and so can be taken to be zero, and therefore the mathematics of an ideal wire/conductor is very useful for understanding real wires in real circuits.

Anyway...

What we mean by an ideal wire is that it is an ideal conductor, which is a material that does not resist moving charges. An ideal conductor must have a zero electric field in its bulk in a steady state situation. The "proof" of this is by contradiction. Imagine there were a non-zero net electric field inside an ideal conductor. Then, there would be a net motion of charge along the electric field lines, since the conductor does not resist the flow of charge. The charge would build up on the boundary of the conductor. This boundary charge would tend to cancel the effect of the non-zero electric field. Since the net electric field changes, the system can't be in a steady state. Therefore, the net electric field inside the conductor must be zero in steady state. The core property we needed for this argument was the definition of an ideal conductor -- the behavior is not really derived from a fundamental law, but rather a model for the electrical conductivity of certain kinds of materials.

In practice, real wires in real circuits are typically very good approximations to an ideal conductor/ideal wire, and reach steady state very quickly.

Bonus comments added due to comments

It might not be immediately obvious, but the static steady-state situation is still relevant for circuits with moving pieces.

• First, in very simple DC circuits, the circuit reaches a steady state where the current through each component is constant. This requires the electric field is not changing. In this sense, even though charges move through the circuit, the fields are static. Then the above considerations for ideal conductors apply directly.

• In more complex circuits where the flow of charge is dynamical (RC, LC, LCR circuits, or AC circuits), one may object to using the static argument. Then the point is that the timescale over which the field in the wires reach a steady state, is much faster than the timescale over which the voltages change. Roughly speaking, the condition for this to work is that the timescale for changes in the circuit (RC time constant, AC period, etc) are much longer than the light travel time across the wire you are using.

• If you really are driving a circuit very quickly compared to the light travel time, then indeed you do need to think more carefully about what is happening in the cable. Here is a website with a thought experiment along these lines.

• In addition to the light travel time, there are other situations where you do have to worry about the fact that conductors aren't perfect. For example, we often imagine that it's possible to connect the circuit to a common ground, or conductor with a fixed potential. In the real world, this isn't always possible, and a ground loop can form, leading to noise in the circuit. This phenomena isn't necessarily common in wires, but it is due to a breakdown in the assumption of having an ideal conductor.

Basically, the "vanilla" situation is that an ideal wire has zero internal electric field, for the same reasons a static conductor has a zero internal electric field. However, your mileage may vary as to whether this is a good description of the particular system you want to describe.

Answer 2

Assume the counterexample. You have a wire (a conductor, a material with mobile charges), and it has an electric field inside.

The electric field puts a force on the charges inside, and that force accelerates them. The charges that accumulate from the motion generate their own electric field. If this field does not cancel the original, then charges will continue to move. This state continues until the net electric field becomes zero everywhere within the conductor.

Any time there's a non-zero field, charges will be moving. The only way you can have a static solution with all the mobile charges is for the field inside to be zero.