# Potential difference across a zero resistance wire

So I started off with electrostatics and everything seemed nice and mathematical and justified and then "DC circuits" happened!

I just cannot understand the model of electron flow in electrical circuits. Here are my specific doubts-:

1) If potential difference across a tiny cross section of conducting wire is zero, then why on earth does electron flow across that cross section at all? Never mind potential difference across the whole circuit.

2) Is there a constant electric field across a wire connected to a battery? If yes then how is potential difference across a zero resistance wire constant? Shouldnt it be increasing? Doesnt it violate ohms law? If no, then why do electrons flow at all?

Please take time to consider these doubts and relieve me of my frustration. I havs searched through the net for this but every answer seems like beating around the bush. All of the 4 books I have consulted do not address these facts to my satisfaction.

Frankly I think nobody understands this.

• This must have been asked before. – Qmechanic May 31 '18 at 11:04
• "Doesnt it violate ohms law?" - Ohm's law for a zero resistance is $V = I \cdot R = I\cdot 0 = 0$. Put another way, according to Ohm's law, the voltage across a zero resistance wire is zero for any current through. By what you've written, you seem to be expecting $V$ to be non-zero and increasing according to Ohm's law. Why? – Alfred Centauri May 31 '18 at 11:08
• I did not mean "according" to the ohms law. What I meant was that for constant electric field, V=Ed should increase. This in turn violates ohm law – IncludedExcluded May 31 '18 at 11:51

## 3 Answers

For a current to flow in a conventional wire (not a superconductor, vacuum, etc.), the potential difference across any segment of the wire and the electric field in it have to be greater than zero.

In most cases, the potential difference in the wires could be approximated as zero, because the resistance of the wires is much smaller than the resistance of other elements in a circuit, including the battery, and, therefore, most of the voltage drops on those other elements.

• Understood. But what about the electric field? is it constant in any configuration across a wire? How is it generated anyway? – IncludedExcluded May 31 '18 at 10:36
• It is not constant and it is generated automatically through redistribution of electrons, which will accumulate in front of sections of high resistance to support whatever current ends up flowing in a circuit. You can check out this post for more details: physics.stackexchange.com/questions/407558/… – V.F. May 31 '18 at 10:48
• If the charges redistribute and accumulate in regions of circuit then this violates Kirchoffs junction law. The fact that field is constant is used to derive J=§E which is another form of ohms law from drift velocity – IncludedExcluded May 31 '18 at 11:53
• The redistribution/accumulation is not continuous, i.e., the charges redistribute themselves as needed and stay that way until some conditions change. So there is no violation of Kirchoff''s law, which is applicable to the steady state. – V.F. May 31 '18 at 13:19
• @V.F., KCL also applies to dynamic situations, but to apply it to this situation you'd have to consider displacement currents (aka parasitic capacitance). – The Photon May 31 '18 at 15:45

Consider an electron that enters your piece of wire with some velocity $v_{in}$ and emerges at the other end with a velocity $v_{out}$: When we say the wire has a resistance what this means is that our electron will lose energy and slow down. That means we have to supply some energy to the electron to keep it moving at the same velocity. That energy is produced by generating a voltage difference $\Delta V$ between the ends of the wire. This gives the electron an energy $e\Delta V$ to replace the energy it loses to the wire.

Now suppose the wire has zero resistance. That means any electrons flowing through it don't slow down, so no extra energy needs to be added to the electrons, so the voltage difference between the ends of the wire can be zero.

This is why electrons will flow through a perfect conductor even though the potential difference is zero. The electrons have a non-zero velocity when they are pushed into the wire by whatever makes up the rest of the circuit, and they simply keep going through the wire at the same velocity.

In fact you could have a potential difference $\Delta V$ between the ends of your zero resistance wire, and what it would do is accelerate the electrons passing through it. You would in effect have built a linear accelerator. An electron would increase its energy by $e\Delta V$ on each pass through the wire. To maintain the voltage you'd need to keep pouring in power at a rate that matched the increase in kinetic energy of the electrons.

• But this works only in the case of straight loop. When the cross section considered is not parallel to the battery how is the potential difference genrated across its ends? – IncludedExcluded May 31 '18 at 11:56

In ordinary circuit analysis (no superconductors involved) when we say the resistance of a wire is zero, we really mean it's close enough to zero that the voltage drop across it doesn't significantly affect the circuit behavior.

If you want to understand exactly how this could work, you could model each wire as a low-valued resistor, and then take the limit as the resistance value goes to zero.

If you want to know the effect of the actual resistance of the wire, you can easily calculate the resistance from the resistivity of copper (~$1.72\times 10^{-8}{\rm\ \Omega\ m}$) and the geometry (cross-section area and length) of the wire. If you design high power circuits, you will certainly run into situations where wire resistance must be considered in the design.

• Thanks. This is the part I have now grasped. What i cannot understand is why "if potential difference between two points is zero then no current flow takes place b/w them". According to me zero field should be the basic condition not pd, for in some cases work by field across some distance cancels out work by another field between the points considered. So charge can actually move between those points. – IncludedExcluded Jun 1 '18 at 17:58
• @IncludedExcluded, if there are two fields in a conductive material exactly cancelling each other out, then the net current in that region will be 0. But when we talk about zero-resistance wires in a lumped circuit analysis, we're working at a much higher level of abstraction, and we shouldn't be thinking about the details of current density distribution on the wire, etc. – The Photon Jun 1 '18 at 18:06
• My doubt was not that fields cancel each other but there work done does. I get the part where you say Is shouldnt be too picky about the field part – IncludedExcluded Jun 1 '18 at 18:35
• How can the work cancel if the field contributions aren't equal and opposite? – The Photon Jun 1 '18 at 18:38
• If the fields act across different distances – IncludedExcluded Jun 1 '18 at 18:40