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An electric field in solar cells is created through photons knocking electrons off of atoms. What happens to the positively charged atoms once the electrons have been knocked off? Furthermore, are the atoms the photons collide with in order to knock the electron off of the atom any specific kind of atom, and are there any chemical equations modelling this?

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What happens to the positively charged atoms once the electrons have been knocked off?

The atom grabs the next closest electron, usually from its neighbor; which grabs an electron from its neighbor, etc. Eventually the electron comes from the terminal that completes the circuit (usually) or from the leakage current (when no current is flowing).

Furthermore, are the atoms the photons collide with in order to knock the electron off of the atom any specific kind of atom, and are there any chemical equations modelling this?

Yes, it depends on the material making up the photo cell. Look up photoelectric effect. But instead of the electrons being kicked free from the metal, the silicon layer captures the electron and transfers it to the anode to make the electrical circuit.

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