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.