# Is Ohm's Law always exactly true even at the microscopic level?

My probability textbook introduction just mentioned that Ohm's Law is not always precisely true at the microscopic level.

How is this possible? What is causing this to happen?

• Ohms law is, as any law, a generalization (and simplification) of (a class of) observations. Even at macro level ut does not hold for non-linear components, and it does not hold for noisy components. I would be surprised if it would hold in the small level in any other than a statistical (averaged) way. – Wouter van Ooijen Aug 27 '14 at 21:35
• Can you provide more context for the quote? Out of context like this, it's very difficult to guess what they might have in mind. I'll bet that there's a section within the book itself that addresses the issue that the introductory comment raises. – Dave Tweed Aug 27 '14 at 21:35
• @WoutervanOoijen with noisy components are you referring to noisy resistors? I'd expect the white (voltage) noise to produce a current in the resistor that obeys Ohm's law indeed. – Vladimir Cravero Aug 27 '14 at 21:39
• @Valdimir: only when averaged over time. – Wouter van Ooijen Aug 28 '14 at 8:26

Consider the rule that the volume of liquid in a vessel is equal to the height of the liquid times its horizontal cross sectional area. For vessels whose horizontal cross section is uniform from top to bottom, halving the height of the liquid will halve its volume. Doubling the amount of liquid, if there's room, will double the volume.

Suppose, however, one removes nearly all of the liquid from the vessel, leaving only two nanoliters. and then cuts the height of the liquid in half. Should one expect that exactly one nanoliter would remain?

The relationship between cross-sectional area, height, and volume remains true by definition at any scale, large or small, provided that those quantities are concretely defined numbers. At very small scales, however, concepts like "volume" become rather nebulous. If there are only a few hundred molecules bouncing around, it's unclear how much of the space outside each should count as part of the "volume" of liquid, and how much should be considered "empty space". In such cases, the relationship between height, volume, and cross sectional area doesn't "break down" so much as it becomes less and less meaningful, especially in cases where the cross-sectional area varies widely with height.

To put it another way, when quantities get very small, measurement uncertainties get very big. When quantities get so small that measurement uncertainties get to be as big as the measurements, Ohm's Law will still hold, but will cease to have much predictive value.

On a microscopic level everything changes. You can't measure precisely anything! Also the known form of Ohm's Law is used with good results only in circuit theory. This form is also an approximation. Ohm's Law is derived (like all electromagnetism equations) from Maxwell equations so its general form is:

$$J=\sigma \cdot E$$

Every theory that tries to describe a microscopic phenomenon uses probabilities and statistics (as you can see yourself by reading this in such a book) and as a result every quantity can only be an approximation. Exactly the same goes for Ohm's Law.

Also if someone tried to find precisely one of the quantities that appear in the equation he should take an infinite number of samples for the rest of the quantities so to have zero error. This is not only impossible given that no one can live forever but also impossible based on that in order to take infinite samples of J you have to be able to count all electrons which obviously is also impossible. All in all it is just an approximation.

I think Ohm's law works because the electrons do a lot of scattering.
For distances that are on the order of the scattering length, you might see non-ohmic behavior.

Ohm's Law is a simple equation that can be derived from the equations that describe the microscopic behaviour of electrons by averaging those equations over long time and spatial scales.

Although it is not often stated, the voltage, current and resistant that appear in Ohm's Law are average values.

On a microscopic level, the current flowing through some surface is not nice and smooth, rather it looks quite random as electrons move across that surface. If your surface is too small, you will have to wait a long time before you see any electrons (and hence any current). Similarly if you don't wait long enough, you may not see any electrons at all, or you may even see electrons moving in the opposite direction to what the average current actually is. Remember that electrons jiggle around in a conductor in a mostly random way - only on average do they move together in one direction.

So you have to wait a long time and look at big areas so you can build up the statistics to get a nice average theory. That theory is Ohm's Law.