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I know the fact that protons don't have the mobile ability like electrons to move freely because they are bounded to the nuclei due to the $VW$ forces apparent in the object but I've read that there exists positive charges that do move especially in a solid.

I'm confused as to how is it possible since if positive charges do move, wouldn't it alter the state or structure of a material.

One example that positive charges move is that of the electroscope experiment. When a negatively charged object is placed near the dome of the electroscope, the positive charges are attracted to the "negative charge" of the object and the electrons in the electroscope repel each other and we experience repulsion in the foils of an electroscope and when a positively charged object is placed near the dome, the same thing happens. In this situation, we see that positive charges do move.

How is this possible? Wouldn't we be able to observe an altered structure of the electroscope if this happens?

Also, are positive charges the protons itself or are they different?

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In the case of the electroscope experiment, you are mistaken. The charge carriers here are always electrons, they just move in different directions. In one case (negative charge close to the dome), they move down, accumulating a negative charge in the arms of the electroscope. In the other case (positive charge close to the dome) they move into the dome, leaving the positively charged protons (or rather ions; not all the electrons in a conductor are mobile) in the arms behind. For the charge flow this isn't really relevant: When electrons flow upwards, charge flows downwards. In both cases the arms are charged: Negatively if there are more electrons than protons present, positively if there are less.

That said, it is possible to move positively (or negatively) charged ions through solids. The important point here is that they cannot move freely, opposed to electrons in a conductor. But if you apply enough force (i.e. voltage), they can and will dig their way through a solid. Depending on the materials used, the solid's structure may be altered by this.

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I think this positive charge "traveling" observation could be just a virtual, due to the very fact, that negative charge traverses the material. As known, negative charge next to the dome not only attracts positive particles (indeed fixed in the material grid), but also repulses the free electrons away. Thus, that particular part of specimen material is positively charged due to insufficiency of electrons. Fixed electrons have been outnumbered by fixed protons, since free electrons have moved away due to the near negative charge.

Same effect is used in PN junctions in semiconductor technology.

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