# Maximum current carrying capacity (theoretically infinite?)

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1. Oct 4, 2016

### tim9000

I started simply looking at a circuit breaker connection diagram, then I fell down the rabbit hole.
So I wondered, if you had a piece of pure copper and getting rid of heat (structural integrity, gravity etc.) was not an issue, just how much current could you push through it before it hit it's physical limit.
So I went back to basics and looked up resistivity to re-acquaint myself with some units and give me some context:
https://en.wikipedia.org/wiki/Electrical_resistivity_and_conductivity
Annoyingly the maths gives it in terms of physical dimensions and resistance, which I suppose is fair enough, given that it states the cause of it is band-gaps etc. But doesn't really help my thought-experiment.

Then I got onto drift velocity:
u = I n A q u = 1 ( 8.5 × 10 28 ) ( 3.14 × 10 − 6 ) ( − 1.6 × 10 − 19 ) u = − 2.3 × 10 − 5 https://wikimedia.org/api/rest_v1/media/math/render/svg/b86b4325af264ae571d138befb7e1132988de612 https://wikimedia.org/api/rest_v1/media/math/render/svg/b86b4325af264ae571d138befb7e1132988de612
For detail and context see the Numerical Example in:
https://en.wikipedia.org/wiki/Drift_velocity

Then I thought, okay so to begin the thought-experiment I need to think of some physical parameters for my copper, using the resistivity calculate the resistance, then from that divide my supply voltage by that resistance to calculate the current? (And if I'm so inclined also the drift velocity)

However I then considered, what if the supply is pretty much infinitely large; what the electrons just get faster as they approach the speed of light? Resulting in this plasma-like copper conductor as electrons fly through it. That doesn't really sit well with me even in this extremely unrealistic thought experiment, but I concede it being a possibility. I always just expected there was some sort of electron charge density limit to a material to prevent any more electric field or something. But that was just my gut.

Any thoughts?

P.S. I have a second Irritant that I ran into on the drift velocity page at the bottom, it stated for the above example that the amplitude of an electron of the above drift velocity, in AC was:
[PLAIN]https://wikimedia.org/api/rest_v1/media/math/render/svg/161ff5356c65d6afc22d07317b295fce100c1cf3[PLAIN]https://wikimedia.org/api/rest_v1/media/math/render/svg/161ff5356c65d6afc22d07317b295fce100c1cf3 [Broken]
But I don't understand this numerically (I don't even see how they got that answer) or by derivation (I can't figure out how they got that equation).
I scribbled playing with integration, averages and RMS' of Sine waves but I'm having trouble linking them to speed. Anyone have any clue?

Thanks

Last edited by a moderator: May 8, 2017
2. Oct 4, 2016

### CWatters

Typically the amount of current you can pass through a wire is limited by the insulation. The power dissipated in the resistance of the wire makes the wire hot and that can damage the insulation either quickly or over time.

In high current applications uninsulated copper bus bars are sometimes used. There are quite complex issues that determines how much current they can carry...

3. Oct 4, 2016

### CWatters

The current in a conductor is given by....

I=nqVdA where

n is the electron number density
Q is the charge of an electron
Vd is the drift velocity
A is the cross sectional area

So I think you are asking if there is a limit to the drift velocity Vd.

Perhaps look up the Saturation Velocity. Above which the drift velocity is no longer proportional to the electric field. As I recall, above the Saturation Velocity the frequency with which electrons start colliding with atoms starts to reduce the conductivity significantly. I think this is significant for semiconductors but not sure about conductors like copper. I don't think the actual drift velocity in copper ever reaches its saturation velocity?

Edit: Found...

4. Oct 4, 2016

### tim9000

I have to go to bed now, so I haven't followed your links, but from what I read I think you interpreted my general exercise as well as I could articulate. A limit to drift velocity would certainly resolve my intuition, that there would be some physical limit to the amount of current a conductor could carry. Thanks
Did you per chance get to take a look at the "P.S." question I posed, regarding deriving the AC distance magnitude of an electron? (it was also here:https://en.wikipedia.org/wiki/Drift_velocity)

5. Oct 4, 2016

### CraigHB

Then there are super-conductors where things get really strange when you try to find theoretical limits.

6. Oct 4, 2016

### tim9000

I would imagine so, yeah, when I = V/R starts tending towards infinity I suppose.

7. Oct 4, 2016

### analogdesign

A DC current acts as an electron wind in any metal and can "blow" atoms out of the lattice and create opens. This is called Electromigration. It is the long term limit on any wire (the short term limit is vaporization).

8. Oct 4, 2016

### Simon Bridge

... in a nutshell: the theoretical limit depends on the theoretical model being used.
It is easy to propose a model that has no upper limit ... how well that model maps to Nature is a different question.
There are a bunch of accepted models that use different assumptions depending on the situation, which is why much of the replies so far are more about trying to find out what the question means. That should give you an idea of the scope of the field you have lifted the lid on. It's all good - enjoy :)

9. Oct 5, 2016

### tim9000

ugh, conduction is complicated, I'll have to look up effective mass and scattering time when I get a chance.

Yeah that is an interesting analogy and point. I've always been curious about electrolysis, so (do you know) is it like linearly proportional to amplitude and time. For instance if you had two pieces of conductors, some sort of junction, either with heaps of current flowing through it for a short time, or a minuscule current for a very...very long time, would they both equally 'blow' atoms through the junction?
I also wonder if this happens to silicon junctions over time too??
Wise words Simon. I suppose I am close(enough) to finding the upper limit, on paper there could be no theoretical limit until you look into a limit on the electron velocity, or a saturation of the electric field, or the atomic lattice being vapourised, whichever would actually be the case in reality....

So any ideas how to derive the amplitude of the AC electron(?):
https://en.wikipedia.org/wiki/Drift_velocity)
Cheers

10. Oct 5, 2016

### Tom.G

It's called "electromigration" and it does not need a junction, it occurs in conductors. In fact that's where it was noticed, in ICs, where high current density in on-chip interconnects would fail over time. Try a Google search for 'electromigration'.

11. Oct 9, 2016

### Baluncore

Getting rid of heat as you put it is the critical thing. Dismissing that makes it an unreal exercise. Electron current can flow both inside and outside a conductor.

The internal current conduction limit will be when the generation of heat is so great that the thermal gradient needed to transfer heat to the cooled outside of the copper melts the inside of the copper. The resistance of solid copper rises in proportion to absolute temperature.

If the temperature of the copper were allowed to rise significantly there would be a space charge of electrons above the surface of the conductor. The limit to surface current is then the number of electrons that can be accelerated by the voltage drop along the conductor. Electrons will be torn from the surface at the cathode end, accelerate along just above the surface, then dump their energy on impact at the anode end. The surface current limit will be determined by the electron emission at the cathode end, which is a function of temperature.

There are equations used to characterise copper fuses, but to be a fuse assumes heat dissipation is not infinite.
Google the Preece equation and the Onderdonk equation.