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Yes, the current consensus of the concept behind conduction of electricity is that the electron itself moves. It moves from one atom to the next, decreasing the charge behind it and increasing the charge in front of it. This is a daisy chain event, where an enormous number of electrons is constantly moving through the conductor, daisy chaining from atom to atom.

Current flow is measured in amperes, or coulombs per second. A coulomb is about 6.242×1018 elementary charge units, in this case electrons.

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