I’ve been thinking a lot about introductory physics labs over break. In particular, I’ve been thinking about the strategy of having student construct plots (often linear ones), in order to get a best-fit line. We often do this in such a way as to have the slope and/or y-intercept provide a quantity of physical significance. I’ve been thinking a lot about when its best to use this “slope method” vs. just using the “average all your data”. Here is an interesting case to consider.

Electrical resistance, R, is defined as the following ratio:

R ≡ V / I , where V is voltage and I is current. This pretty much means you first measure the voltage across some circuit element, and then you measure the current through that element. Finding the resistance is to construct the ratio from your measurements.

Despite its algebraic similarity, the definition is quite different from the statement of Ohm’s Law, V = I R. Ohm’s law expresses a particular relationship between current and voltage that holds given that R is constant. Not all materials are Ohmic, however. For example, lightbulbs are typically not very Ohmic. If you double the voltage, you don’t typically get double the current. Or, for every amount you increase the voltage, you don’t always get same amount of increase in current. Of course, this doesn’t mean that a lightbulb doesn’t have resistance. It’s resistance changes depending on how much current is flowing through it. Thus, it doesn’t have a characteristic resistance.

There is an interesting consequence of this definition of resistance. The interesting consequence arises because resistance is a ratio of the value of two quantities (V/I) not a ratio of changes to value of those quantities (ΔV/ΔI) or to their infinitesimals (dV/dI).

Because of this, and despite the fact that it is common practice, it is sort of a mistake to find the resistance of a circuit element by plotting V vs I and finding the slope. Obviously, if the circuit element is non-ohmic it doesn’t make any sense to report a single R value. However, even if the circuit element can be nicely approximated by Ohm’s Law, it actually makes more sense to report the value of R as an average of all V/I ratios than it does to calculate the resistance by determining the slope of the best fit line. This is especially the case if your best fit-line has a non-zero y-intercept.

This really gets at the heart of what resistance is supposed to be about epistemologically–what is resistance supposed to tell us. It’s supposed to tell us how much current we’re going to get for a set amount of voltage. It’s not necessarily supposed to tell us how rapidly that current will change as we change the voltage, although it will if the element is perfectly Ohmic. Now, we certainly could be interested in knowing that rate of change, but it would be a subtly different quantity than what we typically want resistance to tell us.

Anyway, I’m interested in what others think.

Brian,

Super interesting, and I never thought of this before. Of course, this is different from the procedure in just about every graphing exercise in modeling. You get the best measure of the velocity of the buggy not by averaging x/t for a bunch of points, but instead from finding the slope of the best fit line. Do the modeling materials or modeling workshops explicitly point out this subtle conceptual point in the circuits units?

I’m not very familiar with the modeling materials for circuits… but yes I agree this is mostly a non-issue for mechanics.

It did make me think when would the quantity x/t be a valuable quantity. Imagine you had a race with handicaps, where different racers started at different positions and times, but all had to end at the same position. I have to think about it more, but the quantity x/t could tell you something important about the race/racers.

Since R is only defined for ohmic materials, as you point out, I think that we can choose how to generalize it to non-ohmic materials. I would choose to do the instantaneous slope rather than the average you discuss.

For me, Ohm’s law if J=E which tries to talk about how charges move given an electrical force. It’s similar to terminal velocity, which changes depending on the conditions (and so slope is better than average).

I’m not sure if R is only defined for Ohmic materials… It’s just easily defined. I think it’s that R for non-ohmic is a function of voltage you put across it. With ohmic materials, we can be sloppy about whether R means “this much current for this much voltage” or “adding this much more voltage will add this much more current”. Thus, with non-ohmic materials, I think we need to define two different kinds of resistances… one that captures the first definition (V/I) or and another the second (dV/dI). Both quantities are useful to know about… the first one to perhaps to compare two different resistors (even at different voltages), the other to tell you how hard it would be to change the current locally for a given resistor. It that sense, I agree its a choice, but it’s not an arbitrary choice… it’s a question of what need/want that quantity to tell you.

I guess I feel that R is only ever defined (in textbooks, anyways) for ohmic materials. Certainly the concepts are closely tied. I’m trying to decide if we need the V/I definition. It seems like it’s a map: tell me your voltage, I’ll tell you the current. But if that’s the case, then the whole plot (or map) should be reported. I suppose you could call that the R-plot, but you wouldn’t have to.

There is an old electricity lab experiment used where I work that refers to DC resistance and AC resistance, with the first being V/I and the second being the inverse slope of the I-V characteristic curve. As I had never seen this terminology before being assigned to teach a section of the lab, I looked through all my textbooks and then some but did not find references to r_AC and R_DC. The lab document is quite old so perhaps this nomenclature was once common or perhaps it was common in engineering (I did check with a couple of EE folks and they claimed not to use it now but it was perhaps vaguely familiar to them). I would be curious to know if anyone else has ever encountered this.