In addition to making a toy model to show the tipping-point behavior of charged pieces of sticky tape, I spent some time on Tuesday trying to do something quantitative with this. Of course, Tuesday is the one day of the week that I don't teach, and I didn't want to go to campus to do the experiment, so I put it together from the incredibly sophisticated materials I had available at home: Lego bricks and a tape measure belonging to SteelyKid and The Pip.

Having built this high-tech rig, I set up my new video camera on the tripod, and shot some videos of the key phenomena. First, there's the attraction between two tapes with opposite charges:

In this, you can see the tipping point thing I mentioned-- as I push them closer together, there's an extremely narrow range where the electrostatic attraction pulls the tapes together by a perceptible amount without them flying up and sticking to each other. Once I managed to find that range, I used it to demonstrate the effect that set this whole thing off, namely that when you stick another object in between the tapes, the net electrostatic force on each increases.

I wanted video of this because I used it as a discussion question when talking about the polarization of matter in response to electric fields. The seemingly intuitive answer is to say that the force should decrease because it's partially "blocked" in some sense, but that's not how electromagnetism works. The electric field from a charge is not directly impeded by any intervening matter-- the net field can change because of new sources of fields, but the original charge still contributes exactly the same field and thus force.

So why the change? Because you can think of neutral matter as being made up of atoms with an electron cloud outside a positive nucleus. In an electric field, these polarize, and become little dipoles aligned with the local field. If the tap on the left is positively charged, the electrons in a piece of paper stuck between the tapes will shift a little to the left. When they do that, the paper is no longer perfectly inert from the perspective of the tapes, but produces its own field.

The effect of that field is to attract both of the tapes. The electrons have shifted left by a tiny amount, exerting an attractive force on the positive tape on the left, while the positive nuclei don't move at all, and end up a bit more to the right, where they exert an attractive force on the negative tape on the right.

My original hope with this was to see if there is a measurable difference between an insulator like paper and a conductor like aluminum foil. Unfortunately, as you can see, both of them just increase the attractive force to the point where the tapes cross the tipping point, and get sucked onto the paper or foil. There isn't much difference between them.

The same effect, though, happens between charged and neutral tapes, so I repeated this with one charged tape and one neutral:

The tape on the left has a charge on it, which makes it attracted to my hand, while the tape on the right is uncharged, and not attracted to my hand. When I bring the two tapes close together, though, you get the same tipping point effect-- they twitch a little, then get sucked together. The distance involved is much smaller, though-- cranking these into Tracker Video, I estimated about a factor of 4 difference (roughly 3cm between the support points for only one charged, and about 12cm for both charged). So, what can we get from that?

Well, the equations giving the forces are pretty straightforward. In the case where both tapes are charged, we just have a Coulomb's Law sort of thing:

$latex F_{both} = \frac{1}{4 \pi \epsilon_0} \frac{q^2}{r^2} $

(where I've assumed that the two tapes have the same magnitude of charge, q but opposite signs-- this is a pretty good assumption, as the charging process involves quickly separating a neutral pair of tapes, so whatever charge one picks up had to come from the other). If only one tape is charged, the force comes from the polarization of the neutral tape, which is generally expressed in terms of a "polarizability," which gets the symbol $latex \alpha $, because physicists are lazy and don't want to go any farther into the Greek alphabet than they have to. The force between a charge and a polarizable object is something we derive in class, and is given by the formula:

$latex F_{one} = (\frac{1}{4 \pi \epsilon_0})^2 2 \alpha \frac{q^2}{r^5} $

This depends on the fifth power of r, so it's a much shorter range force than the case where both tapes are charged-- if you double the distance, the force between charged tapes drops by a factor of 4, but the force between one charged and one neutral tape drops by a factor of 32. So the qualitative behavior in the videos above is exactly right.

Can we get something quantitative out of this, though? Well, if we make some simplifying assumptions, sure. And this is physics-- we're all about simplifying assumptions...

The main assumptions to make are 1) that the charges involved have the same magnitude in both cases, and 2) that the force at the "tipping point" is the same in both cases. I think these are both fairly reasonable-- the charging process is the same in both cases, so the q should be pretty similar, and the range of the effect is small enough that I think it's not completely ridiculous to say that the force needed to start the tape moving by enough to get tipping point behavior is the same in both cases.

If we do that, then we can just set the two forces above equal to each other, with two different values of r:

$latex \frac{1}{4 \pi \epsilon_0} \frac{q^2}{r_{both}^2} = (\frac{1}{4 \pi \epsilon_0})^2 2 \alpha \frac{q^2}{r_{one}^5} $

The q is the same on both sides, so we don't need to worry about those. which means the only thing in this that we haven't measured is $latex \alpha $, the polarizability of the tape. So, we can solve for that, and get:

$latex \alpha = \frac{1}{2} 4 \pi \epsilon_0 \frac{r_{one}^5}{r_{both}^2} $

Using the fact that the tipping point for the case where both tapes were charged was about 12cm and the tipping point for the case with only one charged was 3cm, we get a value of:

$latex \alpha = 9.4 \times 10^{-17} $ C-m/(N/C)

Which, um, yeah. That's a number all right. Is it a reasonable number? Well...

We're saved, though, by the fact that the textbook makes several references to the polarizability of a single carbon atom, which is about $latex \alpha = 2 \times 10^{-40} $ C-m/(N/C). That might actually seem disastrously wrong-- we're 20-odd orders of magnitude off-- but that's the value for a single atom. A piece of tape is made up of quite a few atoms, and that would scale the effective polarizability of the tape up by roughly that number.

So, how many atoms in a piece of tape? I didn't measure these specifically, lacking a milligram scale in Chateau Steelypips, but as part of the lab we did last week, the students measured the tapes they were using, and a fairly typical mass is something like 300 milligrams. If I assume the entire thing is carbon atoms, that would be around $latex 1.5 \times 10^{22} $ atoms, each with a polarizability of $latex \alpha = 6.4 \times 10^{-39} $ C-m/(N/C).

"You're still wrong by a factor of 32," you say. And that's true. But, dude, look at how many crude assumptions went into this measurement-- you only need five factor-of-two errors to account for a factor of 32, and I've got at least three assumptions (the identical charge in the two different experiments, the identical force at the tipping point, and the mass-to-number-of-atoms process) that aren't any better than that. I'd say this does remarkably well.

So, it turns out you can measure fundamental atomic properties using Duplo blocks and sticky tape. I think that's pretty awesome. If you don't, why are you reading this blog, anyway?

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For homework, use the "featured image" at the top of this post to estimate the total amount of charge on a single tape. Send your answers to Rhett for grading, and make sure to show all the steps of your calculation.