I’m still unclear about the mechanisms behind static cling. I haven’t really considered it until recently. What confuses me is that I can take an object with a static charge (like a normal piece of plastic), and it will stick to my fingers, and to the fingers of others who touch it, as well as other surfaces. How does that work? If it was just my finger, I would guess that it was charged with the opposite charge of the plastic (kind of like rubbing a balloon, where you clearly separate positive and negative charges). But what I don’t get is why stuff will stick to surfaces which would not appear to have any charge imbalance above normal. Is it because of some polarization effect, or some other mechanism?
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Roughly speaking, when something (object A) is rubbed against something else (object B), some of that rubbing-energy (friction) makes electrons jump from one object to another. We call this triboelectrification. The wiki gives a basic overview.
Although this effect has been known for thousands of years, it is not fully understood from a fundamental aspect. Its exact working mechanism is debated with some models showing thermal excitation playing a role, some suggest an "abundance effect" property of materials, some say mechanical stress and agitation produce free radicals with high-enough energy electrons, etc. This is somewhat due to the lack of generalised predictability of models and the difficulty in accurately measuring the effect across materials.
Nonetheless, it is generally agreed that chemical bonds form between the objects and the "willingness" of the material to receive or donate electrons (electronegativity/positivity) is relevant. Take a more willing-donor-material and rub it against a more willing-receiver-material and you will have this effect. That's why glass (which is made up of mostly $SiO_2$) attracts electrons from silk (which is mostly proteins with lots of Hydrogen and Carbon) when rubbed with it.
As to why it sticks to things, you are right. It is due to a polarisation effect where the presence/influence of a charge induces a redistribution of charge elsewhere, thus it is called electrostatic induction.
Let's say object A has an excess of electrons/is negatively charged and is brought close to a neutral object, the neutral object will experience a push-away of the electrons at the point of contact. When you push away electrons, you get a small net positive charge at the place you pushed the electrons away from (you "minus" minus, you get plus). This induced positive charge then attracts the excess electrons in object A and they stick. After sticking, the electrons will try to "flow" from the more negative object to the more positive one (very slow flow in insulators and fast flow in conductors).
Here's a very good video on how induction makes things stick complete with diagrams and demonstrations. I think you will find minutes 7-10 and 12-16 particularly illuminating. This attraction isn't much and that's why all these experiments and demonstrations always use small and light objects.
