I was a little surprised by how many students had trouble with the following homework question, which was intended to be an easy point for them:

Estimate C2(touching) − C2(not touching), the capacitance of a finger touch on the packing-tape and foil sensor, by estimating the area of your finger that comes in contact with the tape, and assume that the tape is 2mil tape (0.002” thick) made of polypropylene (look up the dielectric constant of polypropylene on line). Warning: an inch is not a meter, and the area of your finger tip touching a plate is not a square meter—watch your units in your calculations!

Remember that capacitance can be computed with the formula

where is the dielectric constant, 8.854187817E-12 F/m is the permittivity of free space, A is the area, and d is the distance between the plates.

The problem is part of their preparation for making a capacitance touch sensor in lab—estimating about how much capacitance they are trying to sense.

There is a fairly wide range of different correct answers to this question, depending on how large an area is estimated for a finger touch. I considered any area from 0.5 (cm)^{2} to 4 (cm)^{2} reasonable, and might have accepted numbers outside that range with written justification from the students. Some students have no notion of area, apparently, trying to use something like the length of their finger times the thickness of the tape for A.

People did not have trouble looking up the relative dielectric constant of polypropylene (about 2.2)—it might have helped that I mentioned that plastics were generally around 2.2 when we discussed capacitors a week or so ago.

What people had trouble with was the arithmetic with units, a subject that is supposed to have been covered repeatedly since pre-algebra in 7th grade. Students wanted to give me area in meters or cm (not square meters), or thought that inches, cm, and m could all be mixed in the same formula without any conversions. Many students didn’t bother writing down the units in their formula, and just used raw numbers—this was a good way to forget to do the conversions into consistent units. This despite the warning in the question to watch out for units!

A lot of students thought that 1 (cm)^{2} was 0.01 m^{2}, rather than 1E-4 m^{2}. Others made conversion errors from inches to meters (getting the thickness of the tape wrong by factors of 10 to 1000).

A number of students either left units entirely off their answer (no credit) or had the units way off (some students reported capacitances in the farad range, rather than a few tens of picofarads).

A couple of students forgot what the floating-point notation 8.854187817E-12 meant, even though we had covered that earlier in the quarter, and they could easily have looked up the constant on the web to figure out the meaning if they forgot. I wish high-school teachers would cover this standard way of writing numbers, as most engineering and science faculty assume students already know how to read floating-point notation.

Many students left their answers in “scientific” notation (numbers like 3.3 10^{-11} F) instead of using more readable engineering notation (33pF). I didn’t take off anything for that, if the answer was correct, but I think that many students need a lot more practice with metric prefixes, so that they get in the habit of using them.

On the plus side, it seems that about a third of the class did get this question right, so there is some hope that students helping each other will spread the understanding to more students. (Unfortunately, the collaborations that are naturally forming seem to be good students together and clueless students together, which doesn’t help the bottom half of the class much.)