Here's one example I use to show why one might graph data in various
ways.
With a variable-separation air capacitor and a digital capacitance
meter, acquire capacitance data as a function of plate separation. In
the spirit of the original discussion, plate separation (d) is the
independent variable, and capacitance (C) is the dependent variable.
The expected relation is C = eA/d where e is the permittivity and A is
the plate area. How should this be graphed? As shown, it might be
obvious to plot C versus 1/d to get a straight line with slope eA (and
ultimately this is the best). Another possibility would be to write
this as 1/C = d/eA and we could plot 1/C versus d to get a straight line
with slope 1/eA.
Does it make a difference which way we do it? Yes, very much. First,
again in the spirit of dependent/independent, I was taught that if
possible, it is best not to mess with the independent variable. In this
case that would mean the 1/C versus d plot would be preferred. This is
because when your audience views the graph, an "unperturbed independent
variable" is easier to understand. If the horizontal axis goes from 0
cm to 20 cm separation it is more comfortable for the viewer to get the
feel for the graph and the range of the data acquisition (as opposed to
having the horizontal axis plotting the reciprocal of the independent
variable and the audience has to deal with the fact that large
separations are then on the left and small separations are on the
right). In other words, if the independent variable is plotted as
itself, the audience is less likely to be confused by the graph, and of
course one purpose of a graph is to display the data in a nonconfusing
manner.
However, in this case, if you plot 1/C versus d you do not get a
straight line. Why not? Because there are artifacts in the data beyond
the expected data. Capacitance meters are difficult to zero,
particularly if the meter is connected to the capacitor with some
reasonable length of wires that add capacitance.
We can correct the equation by writing C = eA/d + C(wires) + C(meter)
where C(meter) is the capacitance due to a meter that is not perfectly
zeroed, and C(wires) is the wire capacitance. Let's combine these
additions as C(offsets) and write the relation as C(measured) = eA/d +
C(offsets).
With this relationship, which is actually what is happening, we don't
have the choice to plot 1/C versus d if we want a straight line. We
have to plot C versus 1/d. Oh...Oh... look what happens... the slope of
the line is still eA, but now we have a vertical-intercept of
C(offsets). Your linear graph is showing you the capacitance offsets of
your system.
Now, if you want, you can use your intercept as a correction to the
data. You can subtract the intercept from the data points, and your
corrected data points are very close to the actual capacitance of the
capacitor at the various separations.
On the other hand, perhaps your goal was to find the permittivity of the
gas between the plates from the slope of the plot because e = slope/A.
In that case, you don't even need to bother correcting the data because
we see that the slope of the C versus 1/d plot is not affected by
C(offsets).
I have the students plot C versus d, C versus 1/d, 1/C versus d and
discover that only the C versus 1/d graph yields a straight line with
raw data. Then they correct the data by subtracting the
intercept-determined C(offsets) from the raw data. Then they remake all
three plots with the corrected data. Now the 1/C versus d plot is
straight, and the C versus 1/d plot is still straight but also has a
zero intercept.
This actually comes as a surprise to many students, but most fairly
quickly can say, "Duh, what did I expect?" Those who have the graphing
epiphany have learned something worthwhile, and that is a big reason I
do this experiment, because the physics is pretty trivial.
Unfortunately, perhaps 10% of the class never see the light and the
whole graphing experience of plotting 1/C or 1/d or correcting the data
is just a big blur for them. But it seems to me that a few graphing
experiences like this are good experiences, and I do have a few more
experiments where we do very similar things with graphs.
I think one of the major lessons from this experience is learning that
the way you graph the data might divulge artifacts in your data. In
addition to showing you the artifact, a particular plot might render the
artifact inconsequential. That is, if experimental offsets only show up
in the intercept, or only in the slope, and you can get the information
you want from the unaffected parameter, then the proper graph choice can
render some experimental difficulties harmless.
Michael D. Edmiston, Ph.D.
Professor of Physics and Chemistry
Bluffton University
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu