How can insulator paper take electrons from the ruler?

I came across a question in my textbook which was:

"When a charged ruler attracts small pieces of paper, sometimes a piece jumps quickly away after touching the ruler. Explain."

And the answer was:

"When a charged ruler attracts small pieces of paper, the charge on the ruler causes a separation of charge in the paper. For example, if the ruler is negatively charged, it will force the electrons in the paper to the edge of the paper farthest from the ruler, leaving the near edge positively charged. If the paper touches the ruler, electrons will be transferred from the ruler to the paper, neutralizing the positive charge. This action leaves the paper with a net negative charge, which will cause it to be repelled by the negatively charged ruler."

My question is how can the electrons will be transferred to the paper if it is an insulator? And it says that the electrons on the paper will be forced to the edge but again, the electrons are not free to move, so how can this occur?

3 Answers

Late answer, but interesting question, nonetheless. The reason why this happens is due to three factors:

1. the capacitance of the ruler relative to ground is extraordinarily small, which is why the separation of an extremely small charge requires/causes significant voltages
2. the charges that are exchanged are close to the surface of the paper/the ruler
3. the resistance of the paper relative to the ruler is determined by very small distances for the charges to be travelled, which causes only a moderate resistance, even though the specific resistivity of paper is very high

Suppose, the part of the ruler that is in contact with the paper can be (veeery) approximately considered a plate capacitor in air relative to ground, then its capacitance can be calculated by the formula $$C\approx\epsilon_0\frac{A}{D}$$ where $$A$$ is the area of the ruler in contact with the paper, and $$D$$ is the distance of the ruler relative to ground.

On the other hand, the resistance between the paper and the ruler can be calculated according to $$R=\rho \frac{d}{A}$$ where $$\rho$$ is the specific resistivity of paper (assuming that the ruler doesn't play a role here, but that is just for simplification), $$d$$ is the distance for the suface charges to be travelled and $$A$$ is the contact area.

Then the time constant of the RC-Circuit formed by the paper, the ruler and ground is approximately $$\tau=RC\approx\rho \frac{d}{A}\epsilon_0\frac{A}{D}=\rho\epsilon_0\frac{d}{D}$$ Assume that $$d=10^{-4}m$$ is given by the thickness of the paper, $$D=1m$$ is the distance of the ruler above ground (or its separation from the cat fur that charged it), and finally consider $$\epsilon_0=8.8\cdot 10^{-12}\frac{As}{Vm}\approx 10^{-11}\frac{As}{Vm}$$ and the specific resistance of paper $$\rho = 10^{10}\Omega m$$, then we obtain $$\tau\approx 10^{-5}s = 10\mu s$$ That is, within 10 microseconds, most of the charge from the paper has equalized with charges from the ruler, even though paper is a very bad conductor. This is because there is so little charge on the ruler and there is such a high voltage driving it over the small distance across the paper.

For reference, the capacitance and resistance can also be calculated, if for example we assume $$A=1cm^2=10^{-4}m^2$$: $$C\approx 10^{-15}F=1 fF$$ $$R=10^{10}\Omega=10 G\Omega$$ A capacitance of one femtofarad is extremely small, which compensates the high resistance of ten gigaohms.

Trying to make this or that part of the calculation more accurate does not change the result by orders of magnitude, so this is a decent estimation.

Insulators can be charged and can exchange charge (electrons) when they come into contact with another insulator. Keep in mind that a conductor is just better at conducting a flow of electricity whereas an insulator is a very bad conductor of electricity. In this case, we are however talking about static electricity, that is the buildup of excess charge on an insulator.

Basically what happens is that the ruler has picked up excess electrons from the material it was rubbed against, and now causes polarization of charge in the paper molecules. The positive charge is nearer to the negatively charged ruler than the negative charge in the paper is, due to the negative charge in the ruler "pushing away" the negative charge in the paper. The Coulomb force is therefore attractive and causes the small pieces of paper to be attracted by the ruler. Once they touch the ruler, some of them will stick and if they come next to each other they will repel since they have polarized charge. But some of the pieces will extract some negative charge from the ruler, again we are speaking of electrostatic discharge, meaning that if the voltage is high enough the charge can still travel between two insulators.

Think about two people, sometimes if one of them has gained some excess charge they will exchange this charge through an electrostatic discharge even though their skin is an insulator!

Back to the situation, we were in: the negatively charged pieces of paper are logically repelled by the negatively charged ruler as like charges repel.

I hope this has answered your question sufficiently!

The main thing you need to know is that insulators can conduct current along their surface. In other words, electrons (and other charged particles) can in principle move along surfaces, no matter what the nature of the material. In such conduction there is a still a lot of electrical resistance, just not infinite resistance. So you just get a tiny current even when the voltage is quite high.

One everyday way to notice this surface current is to watch (and listen) near high voltage lines when the air is damp or when it is raining. Obviously you should remain on the ground at a long distance from the lines! But you can hear a high-pitched crackling noise. This noise is made by a small current leaking from the lines along the surfaces of the ceramic insulting supports from which the high voltage cables are hanging. Sometimes you can see a slight bluish/violet effect as well, owing to little sparks going along the surface or out a short distance into the air. This is called a corona effect. But the surface current is there whether or not there is a corona effect.