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In general, plastics are composed of many chains of complex molecules. In a lot of cases, all the valence electrons of all the atoms of the material are in Fermi energy levels below the conduction band. That is, the energy required to move electrons in plastics is "high" because the energy levels that electrons would have to be in within the structure of the plastic are well above where the electrons are actually hanging out.

The conduction band is a term we apply to the energy band that electrons have to be in to support current flow. Remember that current flow is like musical chairs in that everyone has to "move over one" all along the current path for current to flow. It's isn't about one electron going "into" a circuit at one end and that same electron coming out the other end. The "willingness" of electrons to "move over" to support current flow is conductivity, and electrons that are in "too low" an energy level (because they are being "kept at home" by the chemical structure of the material - the plastic) won't help with conduction. Just as a quick contrast, in a metal, there are lots of electrons in energy levels high enough to support conduction. These are the so-called "free electrons" you hear about. Plastics don't have them.

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