Would the real Ohm's Law please stand up?

There are so many to choose from. It's all a matter of context and application.

At the risk of boring all those who know all tis stuff:

The Ohms Law we all know and love, I = V / R, is all about a ideal resistor. It has the same resistance no matter what I and V are. It's resistance does not change with frequency or temperature or or time or anything else. It never blows up!

That perfect R is a non-existent component. A model we use to make it easy to reason about circuits. We also have perfect C's and perfect L's and so on. Not to mention perfect wires, switches, voltage sources, current sources etc. This all the basis of the "Lumped element model" we use to analyse circuits.

https://en.wikipedia.org/wiki/Lumped_element_model.

Things get tricky when we talk about alternating current circuits. Now a C or L has a "resistance" to the flow of current. But that "resistance" is now changing with the frequency we are using. It also messes with how the current and voltage are related at any instant, they need not be in phase any more. At this point we stop calling it "resistance", we call it "impedance" with the symbol Z. This has a lot of interesting maths using complex numbers which we won't even start to look into here.

But we can still use Ohms Law in AC circuits, just use that weird Z thing instead of R, I = V / Z. Sneaky ha?

http://hyperphysics.phy-astr.gsu.edu/hb ... cohml.html
What about that pesky negative resistance? It is certainly the case that you can build active circuits for which current goes down as voltage goes up. This also happens in gas discharge tubes and so on.

No problem. We had R = V / I. Well looking at the units of measurement here we can say that the unit of resistance is "volts per amp". So what if, instead of taking the actual voltages and currents as measured from zero we just change the voltage by a little bit and see how the current changes. Now we have a change in voltage dV = (V2 - V1) and a resulting change in current dI = (I2 - I1). We can use that little "d" to indicate difference.

Notice that the units of measurement of dI is amps and dV is volts. So if we write

X = dV / dI

We see that our mystery X has units of "volts per amp"

But hey, that's the same as the R in our old Ohms Law. X is a kind of R. And it can be negative! Oh my God.

How does current and resistance change in a plasma? In a plasma you have a soup of electrons and ions flying around not bound to each other as in normal materials. No worries there is an Ohms Law for that as well. The Generalized Ohms Law. See section 3.1 here:

http://people.duke.edu/~ad159/files/p142/12.pdf if you are brave

Memory in C++ is a leaky abstraction .