# Effect of electron discrete nature on current density

Coulomb’s law states that the current that flows through a wire is proportional to the applied voltage. For sure there are many physical effects (linear and non-linear) that participates in the voltage-current relation making this law valid only in ideal systems. I’m familiar with many of them, for example the temperature coefficient of a material. But I never heard of any effect that suggests the current density is somehow affected by the fact that electrons are discrete entities. I mean, my common sense says the higher the current density the higher the probability of electrons interacting to each other which could lead to a higher resistance and eventual saturation. Am I making any sense here? Does such behavior happen?

## 2 Answers

Current through a cross section is

$$I = \int_A \left(\sigma \vec{E} + \varepsilon \frac{\partial \vec{E}}{\partial t}\right) \cdot \mathrm{d}\vec{A}$$

The first term in the sum is for conductors, their conductance sigma is that high the second term does not matter. You can't "saturate" conductors electrically. Before that happens, you saturate them magnetically. (But that's another story.)

The second term in the sum is for isolators. Their sigma is so tiny it does matter. It's the dielectric displacement current. You can see the dielectric constant epsilon and the change of the electric field over time rather than the electric field matters.

If you have boundaries, the electron work function on the conductor also matters. This is e.g. important for tubes, but also for semiconductors.

• Hello Janka, tks for all these great information, but I’m afraid that doesn’t quite answer my question. My question is about current density being or not (and why not) affected by electron interference with each other – PDuarte Sep 2 '18 at 1:57
• In that case, you'll have to consult the magnetic equation. It isn't about single electrons then, but about the magnetic field the current creates as a whole. That one hinders the rate current may rise or fall. Inverse to the dielectric. It doesn't put a limit on the current itself. Static magnetic fields do nothing to static current in a non-moving conductor. (Again, they do something in isolators and semiconductors, that's where it gets interesting.) – Janka Sep 2 '18 at 12:52
• So your question is put wrongly. It's the lack of moveable electrons that makes things interesting. Not the surplus. – Janka Sep 2 '18 at 12:53

I'm a logic designer and not a circuit designer, but when I think of the discrete nature of electrons I think of the opposite... Low current regime, not high current.

For an extreme example, consider a circuit with a current of 1 attoamp. That's about 10 electrons per second. How many electrons in a microsecond? Probably 0. Maybe 1. Very unlikely to get 2. That's an unrealistically small current, but this effect happens in picoamp circuits too. This is the famous "shot noise".

It may have an effect on high power circuits too, I don't know. But definitely low power.